Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant
EEFFECTIVE APPROXIMATION OF HEAT FLOW EVOLUTION OF THERIEMANN ξ FUNCTION, AND A NEW UPPER BOUND FOR THE DEBRUIJN-NEWMAN CONSTANT
D.H.J. POLYMATHA bstract . For each t ∈ R , define the entire function H t ( z ) (cid:66) (cid:90) ∞ e tu Φ ( u ) cos( zu ) du where Φ is the super-exponentially decaying function Φ ( u ) (cid:66) ∞ (cid:88) n = (2 π n e u − π n e u ) exp( − π n e u ) . This is essentially the heat flow evolution of the Riemann ξ function. From the work of de Bruijnand Newman, there exists a finite constant Λ (the de Bruijn-Newman constant ) such that thezeroes of H t are all real precisely when t ≥ Λ . The Riemann hypothesis is equivalent to theassertion Λ ≤
0; recently, Rodgers and Tao established the matching lower bound Λ ≥
0. Ki,Kim and Lee established the upper bound Λ < .In this paper we establish several e ff ective estimates on H t ( x + iy ) for t ≥
0, including somethat are accurate for small or medium values of x . By combining these estimates with numericalcomputations, we are able to obtain a new upper bound Λ ≤ .
22 unconditionally, as well asimprovements conditional on further numerical verification of the Riemann hypothesis. We alsoobtain some new estimates controlling the asymptotic behavior of zeroes of H t ( x + iy ) as x → ∞ .
1. I ntroduction
Let H : C → C denote the function(1) H ( z ) (cid:66) ξ (cid:32) + iz (cid:33) , where ξ : C → C denotes the Riemann ξ function(2) ξ ( s ) (cid:66) s ( s − π − s / Γ (cid:18) s (cid:19) ζ ( s )(which is an entire function after removing all singularities) and ζ is the Riemann ζ function.Then H is an entire even function with functional equation H ( z ) = H ( z ), and the Riemannhypothesis (RH) is equivalent to the assertion that all the zeroes of H are real.It is a classical fact (see [27, p. 255]) that H has the Fourier representation H ( z ) = (cid:90) ∞ Φ ( u ) cos( zu ) du where Φ is the super-exponentially decaying function(3) Φ ( u ) (cid:66) ∞ (cid:88) n = (2 π n e u − π n e u ) exp( − π n e u ) . a r X i v : . [ m a t h . N T ] A ug D.H.J. POLYMATH
The sum defining Φ ( u ) converges absolutely for negative u also. From Poisson summation onecan verify that Φ satisfies the functional equation Φ ( u ) = Φ ( − u ) (i.e., Φ is even); this fact is ofcourse closely related to the functional equation for ζ .De Bruijn [5] introduced (with somewhat di ff erent notation) the more general family of func-tions H t : C → C for t ∈ R , defined by the formula(4) H t ( z ) (cid:66) (cid:90) ∞ e tu Φ ( u ) cos( zu ) du . As noted in [9, p.114], one can view H t as the evolution of H under the backwards heat equation ∂ t H t ( z ) = − ∂ zz H t ( z ). As with H , each of the H t are entire even functions with functionalequation H t ( z ) = H t ( z ); from the super-exponential decay of e tu Φ ( u ) we see that the H t are infact entire of order 1. It follows from the work of P´olya [19] that if H t has purely real zeroesfor some t , then H t (cid:48) has purely real zeroes for all t (cid:48) > t ; de Bruijn showed that the zeroes of H t are purely real for t ≥ /
2. Newman [14] strengthened this result by showing that there isan absolute constant −∞ < Λ ≤ /
2, now known as the
De Bruijn-Newman constant , with theproperty that H t has purely real zeroes if and only if t ≥ Λ . The Riemann hypothesis is thenclearly equivalent to the upper bound Λ ≤
0. Recently in [22] the complementary bound Λ ≥ Λ [6, 15, 8, 7, 16, 23]. Furthermore, Ki, Kim, and Lee [10] sharpened theupper bound Λ ≤ / Λ < / Theorem 1.1 (New upper bound) . We have Λ ≤ . . The proof of Theorem 1.1 combines numerical verification with some new asymptotics andobservations about the H t which may be of independent interest. Firstly, by analyzing the dy-namics of the zeroes of H t , we establish in Section 3 the following criterion for obtaining upperbounds on Λ : Theorem 1.2 (Upper bound criterion) . Suppose that t , X > and < y ≤ obey the followinghypotheses: (i) (Numerical verification of RH at initial time ) There are no zeroes ζ ( σ + iT ) = with + y ≤ σ ≤ and ≤ T ≤ X . (ii) (Asymptotic zero-free region at final time t ) There are no zeroes H t ( x + iy ) = withx ≥ X + (cid:113) − y and y ≤ y ≤ √ − t . (iii) (Barrier at intermediate times) There are no zeroes H t ( x + iy ) = with X ≤ x ≤ X + (cid:113) − y , (cid:113) y + t − t ) ≤ y ≤ √ − t, and ≤ t ≤ t .Then Λ ≤ t + y . Informally, hypothesis (i) implies that at time t =
0, there are no zeroes H t ( x + iy ) = y to the left of the barrier region in (iii). The absence of zeroes in that barrier,together with a continuity argument and an analysis of the time derivative of each zero, can thenbe used to show that for later times 0 < t ≤ t , there continue to be no zeroes H t ( x + iy ) = y to the left of the barrier; see Figure 1. Hypothesis (ii) then gives thecomplementary assertion to the right of the barrier, and one can use an existing theorem of deBruijn (Theorem 3.2) to conclude. PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 3
In practice, we have found it convenient numerically to replace the barrier region in Theorem1.2 with the larger and simpler region X ≤ x ≤ X + y ≤ y ≤
1; 0 ≤ t ≤ t . F igure
1. A visualization of Theorem 1.2. If at time t one can show that thereare no zeroes H t ( x + iy ) = ≤ x < ∞ , y ≤ y ≤
1, then atheorem of de Bruijn allows one to conclude the desired bound Λ ≤ t + y .The hypothesis (i) prevents zeroes hitting this canopy from an initial positionto the left of the barrier; the hypothesis (ii) prevents zeroes from lying in thecanopy to the right of the barrier; and the hypothesis (iii) prevents zeroes fromstarting to the right of the barrier and reaching the canopy to the left of thebarrier.We will obtain Theorem 1.1 by applying Theorem 1.2 with the specific numerical choices t = . X = × + − .
5, and y = .
2. The reason we choose X close to 6 × is that this is near the limit of known numerical verifications of the Riemann hypothesis suchas [18], which we need for the hypothesis (i) of the above theorem; the shift 83952 − . (cid:81) p ≤ (cid:32) − p − iX (cid:33) − large, which helps in keepingthe functions H t ( x + iy ) , H t ( x + iy ) large in magnitude, which in turn is helpful for numericalverifications of (ii) and (iii); see also Figure 11. The choices t = . , y = . X . (The hypothesis(iii) is also verified numerically, but can be done quite quickly compared to (ii), and so does notpresent the main bottleneck to further improvements to Theorem 1.1.) Further upper bounds to Λ can be obtained if one assumes the Riemann hypothesis to hold up to larger heights than thatin [18]: see Section 10.To verify (ii) and (iii), we need e ffi cient approximations (of Riemann-Siegel type) for H t ( x + iy ) in the regime where t , y are bounded and x is large. For sake of numerically explicit constants,we will focus attention on the region(5) 0 < t ≤
12 ; 0 ≤ y ≤ x ≥ , D.H.J. POLYMATH though the results here would also hold (with di ff erent explicit constants) if the numerical quan-tities , ,
200 were replaced by other quantities.A key di ffi culty here is that H t ( x + iy ) decays exponentially fast in x (basically because of theGamma factor in (2)); see Figure 3. This means that any direct attempt to numerically establisha zero-free region for H t ( x + iy ) for large x would require enormous amounts of numericalprecision. To get around this, we will first renormalise the function H t ( x + iy ) by dividing itby a nowhere vanishing explicit function B t ( x + iy ) (basically a variant of the aforementionedGamma factor) that removes this decay. To describe this function, we first introduce the function M : C \ ( −∞ , → C \{ } defined by the formula(6) M ( s ) (cid:66) s ( s − π − s / √ π exp (cid:32)(cid:32) s − (cid:33) Log s − s (cid:33) , where Log denotes the standard branch of the complex logarithm, with branch cut at the negativeaxis and imaginary part in ( − π, π ]. One may interpret M ( s ) as the Stirling approximation to thefactor s ( s − π − s / Γ (cid:16) s (cid:17) appearing in (1), (2); it decays exponentially as one moves to infinity s → ± i ∞ along the critical strip. We may form a holomorphic branch log M : C \ ( −∞ , → C of the logarithm of M by the formula(7) log M ( s ) (cid:66) Log s + Log( s − − s π + log √ π + (cid:32) s − (cid:33) Log s − s ff erentiating this, we see that the logarithmic derivative α : C \ ( −∞ , → C of this function,defined by(8) α (cid:66) (log M ) (cid:48) = M (cid:48) M is given explicitly by the formula α ( s ) = s + s − −
12 log π +
12 Log s − s = s + s − +
12 Log s π . (9)For any time t ∈ R , we then define the deformation M t : C \ ( −∞ ,
1] of M by the formula(10) M t ( s ) (cid:66) exp (cid:18) t α ( s ) (cid:19) M ( s )for any t ≥
0. In the region (5), we introduce the quantity(11) B t ( x + iy ) (cid:66) M t (cid:32) + y − ix (cid:33) . For fixed t ≥ y > B t ( x + iy ) is non-vanishing, and it is easy to verify the asymptotic | B t ( x + iy ) | = e − ( π + o (1)) x . As it turns out, B t ( x + iy ) is an asymptotic approximation to H t ( x + iy )in the region (5), in the sense that(12) lim x →∞ H t ( x + iy ) B t ( x + iy ) = t > y >
0; see Figure 2. (However, the convergence of (12) is not uniform as t approaches zero.) PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 5 F igure
2. The quantity | H t ( x + iy ) / B t ( x + iy ) − | when y = t = . x ≤
12 (left) and when for 1000 ≤ x ≤ y , t ) (right).F igure
3. The quantities log | H t ( x + iy ) | and log | B t ( x + iy ) | when y = t = . x ≤
7. Both quantities decay like − π x ≈ − . x .In fact we have the following significantly more accurate approximation (of Riemann-Siegeltype) with e ff ective error estimates. For any real number X , let O ≤ ( X ) denote a quantity that isbounded in magnitude by X . We also use x + = max( x ,
0) to denote the positive part of a realnumber x . D.H.J. POLYMATH
Theorem 1.3 (E ff ective Riemann-Siegel approximation to H t ( x + iy )) . Let t , x , y lie in the region (5) . Then we have (13) H t ( x + iy ) B t ( x + iy ) = f t ( x + iy ) + O ≤ (cid:0) e A + e B + e C , (cid:1) where f t ( x + iy ) (cid:66) N (cid:88) n = b tn n s ∗ + γ N (cid:88) n = n y b tn n s ∗ + κ (14) b tn (cid:66) exp( t n )(15) γ = γ ( x + iy ) (cid:66) M t (cid:16) − y + ix (cid:17) M t (cid:16) + y − ix (cid:17) (16) s ∗ = s ∗ ( x + iy ) (cid:66) + y − ix + t α (cid:32) + y − ix (cid:33) (17) κ = κ ( x + iy ) (cid:66) t (cid:32) α (cid:32) − y + ix (cid:33) − α (cid:32) + y + ix (cid:33)(cid:33) (18) N (cid:66) (cid:114) x π + t (19) and e A , e B , e C , are certain explicitly computable positive quantities depending on t and x + iy.Furthermore, we have the following bounds: | γ | ≤ e . y (cid:18) x π (cid:19) − y / (20) Re s ∗ ≥ + y + t x π − t x (cid:32) − y + y (1 + y ) x (cid:33) + (21) | κ | ≤ ty x − e A + e B ≤ N (cid:88) n = (1 + | γ | N | κ | n y ) b tn n Re s ∗ exp t log x π n + . x − . − (23) e C , ≤ (cid:18) x π (cid:19) − + y exp (cid:32) − t
16 log x π + . × (3 y + − y ) N − . + | log x π + i π | + . x − (cid:33) (24)This theorem will be proven in Section 6; see Figures 4, 5 for a numerical illustration of theapproximation. The strategy is to express H t as a convolution of H with a gaussian heat kernel,then apply an e ff ective Riemann-Siegel expansion to H to rewrite H t as the sum of variouscontour integrals; see Section 4 for details. One then uses the saddle point method to shift eachsuch contour to a location that is suitable for e ff ective estimation. We remark that f t ( x + iy )is a holomorphic function of x + iy in the region (5) as long as N is constant, but has jumpdiscontinuities when N is incremented. See (71)-(74) for the precise definition of these quantities.
PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 7 F igure
4. Comparison of | f t ( x + iy ) | and | H t ( x + iy ) | / | B t ( x + iy ) | for y = . t = .
1, 1000 ≤ x ≤ ≤ x ≤ +
30 (right). Theapproximation improves as x gets larger.F igure
5. Comparison of | f t ( x + iy ) | and | H t ( x + iy ) | / | B t ( x + iy ) | for y = t = . ≤ x ≤ ≤ x ≤ +
30 (right). Again notice theimproving approximation with x .From (13) and the triangle inequality, we have a numerically verifiable criterion to establishnon-vanishing of H t at a given point: Corollary 1.4 (Criterion for non-vanishing) . Let t , x , y lie in the region (5) , and let f t , e A , e B , e C , be as in Theorem 1.3. If one has the inequality (25) | f t ( x + iy ) | > e A + e B + e C , then H t ( x + iy ) (cid:44) . Actually, for some regions of x , y , t we will use a more complicated criterion than (25), inorder to exploit the argument principle. To numerically estimate f t ( x + iy ) in a feasible amountof time, we will use Taylor expansion to be able to e ffi ciently compute many values of f t ( x + iy ) D.H.J. POLYMATH F igure
6. The error upper bound | e A + e B + e C , | versus | f t − H t / B t | when y = t = . ≤ log x ≤ igure
7. The individual error upper bounds | e A | , | e B | and | e C , | with y = t = . ≤ log x ≤
5. The e C , -term clearly dominates.simultaneously (see Section 7), and for some ranges of the parameters t , x , y we will also use anEuler product mollifier to reduce the amount of oscillation in the sum f t ( x + iy ) (see Section 8.5). PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 9
For y , t fixed and x su ffi ciently large, we have the asymptotics e A , e B , e C , = O ( x − ct ) f t ( x + iy ) = + O ( x − ct )for some absolute constant c >
0; see Proposition 9.1(i) and its proof. This gives the crudeasymptotic (12) in the region (5) at least. In practice, the e C , term numerically dominates the e A + e B term, although both errors will be quite small in the ranges of x under consideration;in particular, for the ranges needed to verify conditions (ii) and (iii) of Theorem 1.2, we canmake e A + e B and e C , both significantly smaller than | f t ( x + iy ) | . In the spirit of expanding theRiemann-Siegel approximation to higher order, we also obtain an even more accurate explicitapproximation in which a correction term − C t B t is added to f t , and the error term e C , is replacedby a smaller quantity e C ; see (69) and Figure 8.F igure
8. Comparison of | f t ( x + iy ) | , | f t ( x + iy ) − C t ( x + iy ) B t ( x + iy ) | and | H t ( x + iy ) | / | B t ( x + iy ) | for y = t = .
3, 1000 ≤ x ≤ y isallowed to be as large as 10) to obtain asymptotic control on the zeroes of H t , refining previouswork of Ki, Kim, and Lee [10]. Indeed, in Section 9 we will establish Theorem 1.5 (Distribution of zeroes of H t ) . Let < t ≤ / , let C > be a su ffi ciently largeabsolute constant, and let c > be a su ffi ciently small absolute constant. For x ≥ π , defineg ( x , t ) (cid:66) x π log x π − x π + + t
16 log x π and for all n ≥ C, let x n be the unique real number greater than π such that (26) g ( x n , t ) = n . (This is well-defined since the g ( x , t ) is increasing in x for x ≥ π .) (i) If x ≥ exp( Ct ) and H t ( x + iy ) = , then y = , andx = x n + O ( x − ct ) for some n. (ii) Conversely, for each n ≥ exp( Ct ) there is exactly one zero H t in the disk { x + iy : | x + iy − x n | ≤ c log x n } (and by part (i), this zero will be real and lie within O ( x − ct ) of x n ). (iii) If X ≥ exp( Ct ) , the number N t ( X ) of zeroes with real part between and X (countingmultiplicity) is N t ( X ) = g ( X , t ) + O (1) . (iv) For any X ≥ , one hasN t ( X + − N t ( X ) ≤ O (log(2 + X )) and N t ( X ) = g ( X , t ) + O (log(2 + X )) . Here and in the sequel we use X = O ( Y ) to denote the estimate | X | ≤ AY for some constant Athat is absolute (in particular, A is independent of t and C).
Roughly speaking, these estimates tell us that the zeroes of H t behave (on macroscopic scales)like those of H in the region x = O (exp( O (1 / t ))), and are very evenly spaced (and on the realaxis) outside of this range. The factor t log x n π in (26) indicates that as time t advances, thezeroes (or at least those with large values of x ) will tend to move towards the origin at a speedof approximately π . Although we will not prove this here, the conclusions (i) and (iii) suggestthat one in fact has an asymptotic of the form N t ( X ) = (cid:106) g ( X , t ) + O ( X − ct ) (cid:107) when X ≥ exp( C / t ); in particular (since the sawtooth function x − (cid:98) x (cid:99) has average value ) onewould have the heuristic approximation N t ( X ) ≈ X π log X π − X π + + t
16 log X π after performing some averaging in X , thus recovering the familiar term in the usual averagedasymptotics for N ( X ).The results in Theorem 1.5 refine previous results of Ki, Kim, and Lee [10, Theorems 1.3,1.4], which gave similar results but with constants that depended on t in a non-uniform (andine ff ective) fashion, and error terms that were of shape o (1) rather than O ( x − ct ) in the limit x → ∞ (holding t fixed). The results may also be compared with those in [3], who (in ournotation) show that assuming RH, the zeroes of H are precisely the solutions x n to the equation12 π arg − e i ϑ ( x n / ζ (cid:48) ( − ix n ) ζ (cid:48) ( + ix n ) = n for integer n , where − ϑ ( t ) is the phase of ζ ( + it ) and one chooses a branch of the argument sothat the left-hand side is − when x n = Remark 1.6.
One can draw an analogy between the various potential behaviours of zeroes of H t and the three classical states of matter. A “gaseous” state corresponds to the situation in whichsome fraction of the zeroes of H t are strictly complex. A “liquid” state corresponds to a situation PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 11 F igure
9. The real zeroes x i of H t will converge to integer values of g ( x i , t )when t (left) and / or x (right) increases. in which the zeroes are real, but disordered (with highly unequal spacings between zeroes). A“solid” state corresponds to a situation in which the zeroes are real and arranged roughly inan arithmetic progression. Thus for instance the Riemann hypothesis and the GUE hypothesisassert (roughly speaking) that the zeroes of H should exhibit liquid behaviour everywhere, whileTheorem 1.5 asserts that the zeroes of H t , t > “solidify” in the region x ≥ exp( C / t ) . Below thisregion we expect liquid behaviour. In general, as the parameter t increases, the zeroes appear to “cool” down, transitioning from gaseous to liquid to solid type states; see [22] for someformalisations of this intuition. About this project.
This paper is part of the
Polymath project , which was launched byTimothy Gowers in February 2009 as an experiment to see if research mathematics could beconducted by a massive online collaboration. The current project (which was administered byTerence Tao) is the fifteenth project in this series. Further information on the Polymath projectcan be found on the web site michaelnielsen.org/polymath1 . Information about this spe-cific project may be found at michaelnielsen.org/polymath1/index.php?title=De Bruijn-Newman constant and a full list of participants and their grant acknowledgments may be found at michaelnielsen.org/polymath1/index.php?title=Polymath15 grant acknowledgments
We thank the anonymous referees for their careful reading of the paper and for several usefulcorrections and suggestions. 2. N otation
We use the standard branch Log of the logarithm to define the standard complex powers z w (cid:66) exp( w Log z ), and in particular define the standard square root √ z (cid:66) z / = exp( Log z ). This is the picture for positive t at least. As t becomes very negative, it appears that the “gaseous” zeroesbecome more ordered again, for instance organizing themselves into curves in the complex plane. See [21] forfurther discussion of this phenomenon. We record the familiar gaussian identity(27) (cid:90) R exp (cid:16) − ( au + bu + c ) (cid:17) du = (cid:114) π a exp (cid:32) b a − c (cid:33) for any complex numbers a , b , c with Re a > O ≤ ( X ), any expression of the form A = B using this notation should be interpreted as the assertion that any quantity of the form A is alsoof the form B , thus for instance O ≤ (1) + O ≤ (1) = O ≤ (3). (In particular, the equality relation isno longer symmetric with this notation.)If F is a meromorphic function, we use F (cid:48) to denote its derivative. We also use F ∗ to denotethe reflection F ∗ ( s ) : = F ( s ) of F . Observe from analytic continuation that if F : Ω → C isholomorphic on a connected open domain Ω ⊂ C containing an interval in R , and is real-valuedon Ω ∩ R , then it is equal to its own reflection: F = F ∗ (since the holomorphic function F − F ∗ has an uncountable number of zeroes).3. D ynamics of zeroes In this section we control the dynamics of the zeroes of H t in order to establish Theorem 1.2.As H t is even with functional equation H t = H ∗ t , the zeroes are symmetric around the origin andthe real axis; from (4) and the positivity of Φ , we also see that H t ( iy ) > y ∈ R , so thereare no zeroes on the imaginary axis. From the super-exponential decay of Φ and (4) we see thatthe entire function H t is of order 1; by Jensen’s formula, this implies that the number of zeroesin a large disk D (0 , R ) is at most O ( R + o (1) ) as R → ∞ .We begin with the analysis of the dynamics of a single zero of H t : Proposition 3.1 (Dynamics of a single zero) . Let t ∈ R , and let ( z k ( t )) k ∈ Z \{ } be an enumerationof the zeroes of H t in C (counting multiplicity), with the symmetry condition z − k ( t ) = − z k ( t ) . (i) If j ∈ Z \{ } is such that z j ( t ) is a simple zero of H t , then there exists a neighbourhoodU of z j ( t ) , a neighbourhood I of t in R , and a smooth map z j : I → U such that forevery t ∈ I, z j ( t ) is the unique zero of H t in U. Furthermore one has the equation (28) dz j dt ( t ) = (cid:48) (cid:88) k (cid:44) j z j ( t ) − z k ( t ) where the sum is over those k ∈ Z \{ } with k (cid:44) j, and the prime means that the k and − kterms are summed together (except for the k = − j term, which is summed separately) inorder to make the sum convergent. (ii) If j ∈ Z \{ } is such that z j ( t ) is a repeated zero of H t of order m ≥ , then there is aneighbourhood U of z j ( t ) such that for t su ffi ciently close to t , there are precisely mzeroes of H t in U, and they take the formz j ( t ) + √ t − t ) / λ j + O ( | t − t | ) PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 13 for j = , . . . , m as t → t , where λ < · · · < λ m are the roots of the m th Hermitepolynomial He m ( z ) (cid:66) ( − m exp (cid:32) z (cid:33) d m dz m exp (cid:32) − z (cid:33) (29) = (cid:88) ≤ l ≤ m / m ! l !( m − l )! ( − l z m − l l (30) and the implied constant in the O () notation can depend on t , j, and m. The di ff erential equation (28) was previously derived in [9, Lemma 2.4] in the case t > Λ (inwhich all zeroes are real and simple); however, in our applications we also need to consider theregime t ≤ Λ in which the zeroes are permitted to be complex or repeated. The roots λ , . . . , λ m appearing in Proposition 3.1(ii) can be given explicitly for small values of m as λ = − λ = + m = λ = − √ x = λ = + √ m =
3, and x λ = − (cid:113) + √ λ = − (cid:113) − √ λ = (cid:113) − √ λ = (cid:113) + √ m =
4. From (29) and iterating Rolle’s theorem we see that all the roots λ , . . . , λ m ofHe m are real; from the Hermite equation (cid:16) d dz − z ddz + m (cid:17) He m ( z ) = Proof.
First suppose we are in the situation of (i). As z j ( t ) is simple, ∂∂ z H t is non-zero at z j ( t );since H t ( z ) is a smooth function of both t and z , we conclude from the implicit function theoremthat there is a unique solution z j ( t ) ∈ U to the equation H t ( z j ( t )) = z j ( t ) in a su ffi ciently small neighbourhood U of z j ( t ), if t is in a su ffi ciently small neigh-bourhood I of t ; furthermore, z j ( t ) depends smoothly on t , and agrees with z j ( t ) when t = t .Di ff erentiating the above equation at t , we obtain ∂ H t ∂ t | t = t ( z j ( t )) + dz j dt ( t ) H (cid:48) t ( z j ( t )) = , where the primes denote di ff erentiation in the z variable. On the other hand, from (4) and di ff er-entiation under the integral sign (which can be justified using the rapid decrease of Φ ) we havethe backwards heat equation(31) ∂ H t ∂ t = − H (cid:48)(cid:48) t for all t ≥
0. Inserting this into the previous equation, we conclude that(32) dz j dt ( t ) = H (cid:48)(cid:48) t H (cid:48) t ( z j ( t )) , noting that the denominator H (cid:48) t ( z j ( t )) is non-vanishing by the hypothesis that the zero at z j ( t )is simple. Henceforth we omit the dependence on t for brevity. From Taylor expansion of H t , H (cid:48) t , and H (cid:48)(cid:48) t around the simple zero z j we see that(33) H (cid:48)(cid:48) t H (cid:48) t ( z j ) = z → z j (cid:32) H (cid:48) t H t ( z ) − z − z j (cid:33) . On the other hand, as H t is even, non-zero at the origin (as follows from (4) and the positivity of Φ ), and entire of order 1, we see from the Hadamard factorization theorem that H t ( z ) = H t (0) (cid:48) (cid:89) k (cid:32) − zz k (cid:33) , where the prime indicates that the k and − k factors are multiplied together. The product is locallyuniformly convergent, so we may take logarithmic derivatives and conclude that H (cid:48) t H t ( z ) = (cid:48) (cid:88) k z − z k . Inserting this into (32), (33) and using the continuity of z (cid:55)→ (cid:80) (cid:48) k : k (cid:44) j z − z k at z j (which followsfrom the growth in the number of zeroes, either from the dominated convergence theorem or theWeierstrass M -test), we obtain the claim (i).Now we prove (ii). We abbreviate z j ( t ) as z j . By Taylor expansion we have ∂ k H t ∂ z k ( z ) = m ( m − . . . ( m − k + a m ( z − z j ) m − k + O ( | z − z j | max( m − k + , )as z → z j for any fixed integer k ≥ a m = a m ( z j , t ) (withthe implied constant in the O () notation allowed to depend on k , z j , t ); applying the backwardsheat equation (31) we thus have ∂ k H t ∂ t k | t = t ( z ) = ( − k m ( m − . . . ( m − k + a m ( z − z j ) m − k + O ( | z − z j | max( m − k + , ) . Performing Taylor expansion in time and using (30), we conclude that in the regime z − z j = O ( | t − t | / ), one has the bound H t ( z ) = m a m (( t − t ) / ) m (cid:32) He m (cid:32) z − z j √ t − t ) / (cid:33) + O (cid:16) | t − t | / (cid:17)(cid:33) as t → t , using (say) the standard branch of the square root. By the inverse function theorem(and the simple nature of the zeroes of He m ), we conclude that for t su ffi ciently close but notequal to t , we have m zeroes of H t of the form z j + √ t − t ) / λ j + O ( | t − t | ) . By Rouche’s theorem, if U is a su ffi ciently small neighborhood of z j then these are the onlyzeroes of H t in U for t su ffi ciently close to t . The claim follows. (cid:3) Next, we recall the following bound of de Bruijn:
Theorem 3.2.
Suppose that t ∈ R and y > is such that there are no zeroes H t ( x + iy ) = with x ∈ R and y > y . Then for any t > t , there are no zeroes H t ( x + iy ) = with x ∈ R andy > max( y − t − t ) , / . In particular one has Λ ≤ t + y . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 15
Proof.
See [5, Theorem 13]. (cid:3)
We are now ready to prove Theorem 1.2. The main step is to establish
Proposition 3.3 (Zero-free region criterion) . Suppose that t , X > and < y ≤ obey thefollowing hypotheses: (i) There are no zeroes H ( x + iy ) = with ≤ x ≤ X and (cid:113) y + t ≤ y ≤ . (ii) There are no zeroes H t ( x + iy ) = with x ≥ X + (cid:113) − y and y ≤ y ≤ √ − t . (iii) There are no zeroes H t ( x + iy ) = with X ≤ x ≤ X + (cid:113) − y , (cid:113) y + t − t ) ≤ y ≤√ − t, and ≤ t ≤ t .Then there are no zeroes H t ( x + iy ) = with x ∈ R and y ≥ y .Proof. It is well known that the Riemann ξ function has no zeroes outside of the strip { ≤ Re s ≤ } , hence there are no zeroes H ( x + iy ) = y >
1. By Theorem 3.2, we may thusremove the upper bound constraints y ≤ y ≤ √ − t , and y ≤ √ − t from (i), (ii), and (iii)respectively.By hypotheses (ii), (iii) and the symmetry properties of H t , it su ffi ces to show that for every0 ≤ t ≤ t , there are no zeroes H t ( x + iy ) = ≤ x ≤ X and y ≥ Y ( t ), where Y ( t ) (cid:66) (cid:113) y + t − t ). By hypothesis (i), this is true at time t =
0. Suppose the claim failed for sometime 0 < t ≤ t . Let t ∈ (0 , t ] be the minimal time in which this occurred (such a time existsbecause H t varies continuously in t , and there are no zeroes H t ( x + iy ) = y > H t ( x + iy ) = x + iy on the boundary of the region { x + iy : 0 ≤ x ≤ X , y ≥ Y ( t ) } . The right side x = X ofthis boundary is ruled out by hypothesis (iii), and (as mentioned at the start of the section) theleft side x = Φ . Thus by the symmetry properties of H t we must have H t ( x + iY ( t )) = < x < X .Suppose first that H t has a repeated zero at x + iy . Using Proposition 3.1(ii) and observing(from the symmetry of He m ) that at least one of the roots x , . . . , x m is positive, we then seethat for t < t su ffi ciently close to t , H t has a zero in the region { x + iy : 0 ≤ x ≤ X , y ≥ Y ( t ) } ,contradicting the minimality of t . Thus the zero x + iY ( t ) of H t must be simple. In particular, byProposition 3.1(i) we can write x + iY ( t ) = z j ( t ) for some smooth function z j in a neighbourhoodof t obeying (28), such that z j ( t ) is a zero of H t for all t close to t . We will prove that(34) Im ddt z j ( t ) < ddt Y ( t ) , which implies that there is a zero of H t in the region { x + iy : 0 ≤ x ≤ X , y ≥ Y ( t ) } for t < t su ffi ciently close to t , giving the required contradiction.The right-hand side of (34) is ddt Y ( t ) = − Y ( t ) . By Proposition 3.1(i), the left-hand side of (34) is − (cid:48) (cid:88) k (cid:44) j Y ( t ) − y k ( x − x k ) + ( Y ( t ) − y k ) where we write z k = x k + iy k . Clearly any zero x k + iy k with imaginary part y k in [ − Y ( t ) , Y ( t )]gives a non-positive contribution to this sum, the contribution of the zero x − iY ( t ) is − Y ( t ) ,the contribution of the zero − x + iY ( t ) vanishes, and the contribution of − x − iY ( t ) is negative.Grouping the remaining zeroes with their complex conjugates, it then su ffi ces to show that Y ( t ) − y k ( x − x k ) + ( Y ( t ) − y k ) − Y ( t ) + y k ( x − x k ) + ( Y ( t ) + y k ) ≤ y k > Y ( t ). Cross-multiplying and canceling like terms, this inequality eventuallysimplifies to y k ≤ ( x − x k ) + Y ( t ) . But from the hypothesis (iii) and the assumption y k > Y ( t ), we have | x k | ≥ X + (cid:112) − Y ( t ) , so( x − x k ) ≥ − Y ( t ) . On the other hand from Theorem 3.2 one has y k <
1, giving the requiredcontradiction. (cid:3)
By combining Proposition 3.3 with Theorem 3.2, we obtain Theorem 1.2, noting from (1),(2) that condition (i) of Proposition 3.3 is implied by condition (i) of Theorem 1.2.4. A pplying the fundamental solution for the heat equation
As discussed in the introduction, we will establish Theorem 1.3 by writing H t in terms of H using the fundamental solution to the heat equation. Namely, for any t >
0, we have from (27)that e tu = (cid:90) R e ± √ tvu √ π e − v dv for any complex u and either choice of sign ± . Multiplying by e ± izu and averaging, we concludethat e tu cos( zu ) = (cid:90) R cos (cid:16)(cid:16) z − i √ tv (cid:17) u (cid:17) √ π e − v dv for any complex z , u . Multiplying by Φ ( u ) and using Fubini’s theorem, we conclude the heatkernel representation H t ( z ) = (cid:90) R H ( z − i √ tv ) 1 √ π e − v dv for any complex z . Using (1), we thus have(35) H t ( z ) = (cid:90) R ξ (cid:32) + iz + √ tv (cid:33) √ π e − v dv . Remark 4.1.
We have found numerically that the formula (35) gives a fast and accurate meansto compute H t ( z ) when z is of moderate size, e.g., if z = x + iy with | x | ≤ and | y | ≤ . However,we will not need to directly compute the right-hand side of (35) for our application to bounding Λ , as we will only need to control H t ( x + iy ) for large values of x, and we will shortly developtractable approximations of Riemann-Siegel type that are more suitable for this regime. PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 17
We now combine this formula with expansions of the Riemann ξ -function. From [27, (2.10.6)]we have the Riemann-Siegel formula(36) 18 ξ ( s ) = R , ( s ) + R ∗ , (1 − s )for any complex s that is not an integer (in order to avoid the poles of the Gamma function),where R , ( s ) is the contour integral R , ( s ) : = s ( s − π − s / Γ (cid:18) s (cid:19) (cid:90) (cid:46) w − s e i π w e π iw − e − π iw dw with 0 (cid:46) e π i / that crosses the interval [0 , e i π w along the e π i / and e π i / directions) wemay expand R , ( s ) = N (cid:88) n = r , n ( s ) + R , N ( s )for any non-negative integer N , where r , n , R , N are the meromorphic functions r , n ( s ) : = s ( s − π − s / Γ (cid:18) s (cid:19) n − s , (37) R , N ( s ) : = s ( s − π − s / Γ (cid:18) s (cid:19) (cid:90) N (cid:46) N + w − s e i π w e π iw − e − π iw dw (38)and N (cid:46) N + e π i / that crosses the interval[ N , N + z that is not purely imaginary, we see from Stirling’s approximation that thefunctions r , n ( + iz + √ tv ) and R , N ( + iz + √ tv ) grow slower than gaussian as v → ±∞ (indeedthey grow like exp( O ( | v | log | v | )), where the implied constants depend on t , z ). From this and(35), (36) we conclude that(39) H t ( z ) = N (cid:88) n = r t , n (cid:32) + iz (cid:33) + N (cid:88) n = r ∗ t , n (cid:32) − iz (cid:33) + R t , N (cid:32) + iz (cid:33) + R ∗ t , N (cid:32) − iz (cid:33) for any t >
0, any z that is not purely imaginary, and any non-negative integer N , where r t , n ( s ) , R t , N ( s ) are defined for non-real s by the formulae r t , n ( s ) : = (cid:90) R r , n (cid:16) s + √ tv (cid:17) √ π e − v dvR t , N ( s ) : = (cid:90) R R , N (cid:16) s + √ tv (cid:17) √ π e − v dv ;these can be thought of as the evolutions of r , n , R , N respectively under the forward heat equa-tion.The functions r , n ( s ) , R , N ( s ) grow slower than gaussian as long as the imaginary part of s isbounded and bounded away from zero. As a consequence, we may shift contours (replacing v by v + √ t α n ) and write(40) r t , n ( s ) = exp (cid:18) − t α n (cid:19) (cid:90) R exp (cid:16) − √ tv α n (cid:17) r , n (cid:18) s + √ tv + t α n (cid:19) √ π e − v dv for any complex number α n with Im ( s ) , Im ( s + t α n ) having the same sign. Similarly we maywrite(41) R t , N ( s ) = exp (cid:18) − t β N (cid:19) (cid:90) R exp (cid:16) − √ tv β N (cid:17) R , N (cid:18) s + √ tv + t β N (cid:19) √ π e − v dv for any complex number β N with Im s , Im ( s + t β N ) having the same sign. In the spirit of thesaddle point method, we will select the parameters α n , β N later in the paper in order to make theintegrands in (40), (41) close to stationary in phase at v =
0, in order to obtain good estimatesand approximations for these terms.5. E lementary estimates
In order to explicitly estimate various error terms arising in the proof of Theorem 1.3, we willneed the following elementary estimates:
Lemma 5.1 (Elementary estimates) . Let x > . (i) If a > and b ≥ are such that x > b / a, thenO ≤ (cid:18) ax (cid:19) + O ≤ (cid:32) bx (cid:33) = O ≤ (cid:32) ax − b / a (cid:33) . More generally, if a > and b , c ≥ are such that x > b / a , √ c / a, thenO ≤ (cid:18) ax (cid:19) + O ≤ (cid:32) bx (cid:33) + O ≤ (cid:18) cx (cid:19) = O ≤ (cid:32) ax − max( b / a , √ c / a ) (cid:33) . (ii) If x > , then log (cid:32) + O ≤ (cid:32) x (cid:33)(cid:33) = O ≤ (cid:32) x − (cid:33) . or equivalently + O ≤ (cid:32) x (cid:33) = exp (cid:32) O ≤ (cid:32) x − (cid:33)(cid:33) . (iii) If x > / , then exp (cid:32) O ≤ (cid:32) x (cid:33)(cid:33) = + O ≤ (cid:32) x − . (cid:33) . (iv) We have exp ( O ≤ ( x )) = + O ≤ ( e x − . (v) If z is a complex number with | Im z | ≥ or Re z ≥ , then Γ ( z ) = √ π exp (cid:32)(cid:32) z − (cid:33) log z − z + O ≤ (cid:32) | z | − . (cid:33)(cid:33) . (vi) If a , b > , y ≥ and x ≥ x ≥ exp( a / b ) and x > c ≥ , then log a | x + iy | ( x − c ) b ≤ log a | x + iy | ( x − c ) b . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 19
Proof.
Claim (i) follows from the geometric series formula ax − t = ax + atx + at x + . . . whenever 0 ≤ t < x .For Claim (ii), we use the Taylor expansion of the logarithm to note thatlog (cid:32) + O ≤ (cid:32) x (cid:33)(cid:33) = O ≤ (cid:32) x + x + x + . . . (cid:33) which on comparison with the geometric series formula1 x − = x + x + x + . . . gives the claim. Similarly for Claim (iii), we may compare the Taylor expansionexp (cid:32) O ≤ (cid:32) x (cid:33)(cid:33) = + O ≤ (cid:32) x + x + x + . . . (cid:33) with the geometric series formula 1 x − . = x + x + x + . . . and note that k ! ≥ k for all k ≥ e x = + ( e x −
1) and the elementary inequality e − x ≥ − ( e x − Γ = Γ ∗ to assume thatIm z ≥
0. From the work of Boyd [4, (1.13), (3.1), (3.14), (3.15)] we have the e ff ective Stirlingapproximation Γ ( z ) = √ π exp (cid:32)(cid:32) z − (cid:33) log z − z (cid:33) (cid:32) + z + R ( z ) (cid:33) where the remainder R ( z ) obeys the bound | R ( z ) | ≤ (2 √ + C Γ (2)(2 π ) | z | for Re z ≥ | R ( z ) | ≤ (2 √ + C Γ (2)(2 π ) | z | | − e π iz | for Re z ≤
0, where C is the constant C : =
12 (1 + ζ (2)) = (cid:32) + π (cid:33) . In the latter case, we have Im z ≥ | − e π iz | ≥ − e − π . We concludethat in all ranges of z of interest, we have | R ( z ) | ≤ (2 √ + C Γ (2)(2 π ) | z | (1 − e − π ) ≤ . | z | and hence by Claim (i) Γ ( z ) = √ π exp (cid:32)(cid:32) z − (cid:33) log z − z (cid:33) (cid:32) + O ≤ (cid:32) | z | − . (cid:33)(cid:33) and the claim then follows by Claim (ii). For Claim (vi), it su ffi ces to show that the function x (cid:55)→ log a | x + iy | ( x − c ) b is non-increasing for x ≥ exp( a / b ). Since log | x + iy | = (log x )(1 + log(1 + y x )2 log x ) and the second factor is monotone decreasingin x , it su ffi ces to show that x (cid:55)→ log a x ( x − c ) b is non-increasing in this region. Taking logarithmsand di ff erentiating, we wish to show that ax log x − bx − c ≤
0. But this is clear since bx − c ≥ bx andlog x ≥ a / b . (cid:3)
6. P roof of T heorem ff ective approximation in the region (5), since we will be able to ensure thatquantities such as s + √ tv + t α n or s + √ tv + t β N , with s = + i ( x + iy )2 , stay away from the realaxis where the poles of Γ are located (and also where the error terms in the Riemann-Siegelapproximation deteriorate).Accordingly, we will need e ff ective estimates on the functions r t , n , R t , N appearing in Section4. We will treat these two functions separately.6.1. Estimation of r t , n . We recall the function α ( s ) defined in (8). From di ff erentiating (9) wesee that(42) α (cid:48) ( s ) = − s − s − + s whenever s ∈ C \ ( −∞ , s >
3, we conclude in particular the useful bound α (cid:48) ( s ) = O ≤ (cid:32) s ) (cid:33) + O ≤ (cid:32) s ) (cid:33) + O ≤ (cid:32) s ) (cid:33) = O ≤ (cid:32) s ) − (cid:33) (43)thanks to Lemma 5.1(i).We also recall the function M t and the coe ffi cients b tn from (10), (15) respectively. It turns outwe have a good approximation r t , n ( σ + iT ) ≈ M t ( σ + iT ) b tn n σ + iT + t α ( σ + iT ) . More precisely, we have
Proposition 6.1 (Estimate for r t , n ) . Let σ be real, let T > , let n be a positive integer, and let < t ≤ / . Thenr t , n ( σ + iT ) = M t ( σ + iT ) b tn n σ + iT + t α ( σ + iT ) (cid:0) + O ≤ ( ε t , n ( σ + iT )) (cid:1) where (44) ε t , n ( σ + iT ) (cid:66) exp t | α ( σ + iT ) − log n | + t + T − . − . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 21
Proof.
From (37), (6) and Lemma 5.1(v) one has r , n ( s ) = M ( s ) n − s exp (cid:32) O ≤ (cid:32) | s | − . (cid:33)(cid:33) whenever Im s >
2. Let α n denote the quantity(45) α n (cid:66) α ( σ + iT ) − log n ;this is the logarithmic derivative of M ( s ) n − s at s = σ + iT . By (9) and the hypothesis T ≥ α n may be lower bounded by(46) Im α n ≥ − T − T ≥ − . σ + iT and σ + iT + t α n have imaginary parts of the same sign. We can now apply(40) to obtain r t , n ( σ + iT ) = exp (cid:18) − t α n (cid:19) (cid:90) R exp (cid:16) − √ tv α n (cid:17) M (cid:18) σ + iT + √ tv + t α n (cid:19) ×× exp − (cid:18) σ + iT + √ tv + t α n (cid:19) log n + O ≤ | σ + iT + √ tv + t α n | − . √ π e − v dv . From (46) we see that σ + iT + √ tv + t α n has imaginary part at least T − .
08. Thus O ≤ | σ + iT + √ tv + t α n | − . = O ≤ (cid:32) T − . (cid:33) = O ≤ (cid:32) T − . (cid:33) . From (43) we have α (cid:48) ( s ) = O ≤ (cid:32) T − . (cid:33) for all s on the line segment between σ + iT and σ + iT + √ tv + t α n . Applying Taylor’s theoremwith remainder to the branch of the logarithm log M defined in (7), we conclude that M ( σ + iT + √ tv + t α n ) = M ( σ + iT ) exp α ( σ + iT )( √ tv + t α n ) + O ≤ | √ tv + t α n | T − . . Combining these estimates, writing α ( σ + iT ) = α n + log n , estimating | √ tv + t α n | by 2 tv + t | α n | , and simplifying, we conclude that r t , n ( s ) = M ( σ + iT ) exp (cid:18) t α n − ( σ + iT ) log n (cid:19) × (cid:90) R exp O ≤ t v + t | α n | + T − . √ π e − v dv . Using (45), (10), (15) we see that M ( σ + iT ) exp (cid:18) t α n − ( σ + iT ) log n (cid:19) = M t ( σ + iT ) b tn n σ + iT + t α ( σ + iT ) and so it su ffi ces to show that (cid:90) R exp O ≤ t v + t | α n | + T − . √ π e − v dv = + O exp t | α n | + t + T − . − . Since √ π e − v dv integrates to one, it su ffi ces by Lemma 5.1(iv) to show that (cid:90) R exp t v + t | α n | + T − . √ π e − v dv ≤ exp t | α n | + t + T − . . Since T − . < T − . , we can remove the t | α n | + terms from both sides and reduce to showingthat(47) (cid:90) R exp (cid:32) tv T − . (cid:33) √ π e − v dv ≤ exp (cid:32) t T − . (cid:33) . Using (27), the left-hand side may be calculated exactly as (cid:32) − t T − . (cid:33) − / . Applying Lemma 5.1(ii) and using the hypotheses t ≤ / T ≥
10, one has1 − t T − . = exp (cid:32) O ≤ (cid:32) t T − . (cid:33)(cid:33) and the claim follows. (cid:3) Estimation of R t , N . We begin with the following estimates of Arias de Reyna [2] on theterm (cid:82) N (cid:46) N + w − s e i π w e π iw − e − π iw appearing in (38): Proposition 6.2.
Let σ be real and T (cid:48) > , and define the quantitiess (cid:66) σ + iT (cid:48) (48) a (cid:66) (cid:114) T (cid:48) π (49) N (cid:66) (cid:98) a (cid:99) (50) p (cid:66) − a − N )(51) U (cid:66) exp (cid:32) − i (cid:32) T (cid:48) T (cid:48) π − T (cid:48) − π (cid:33)(cid:33) . (52) Let K be a positive integer. Then we have the expansion (cid:90) N (cid:46) N + w − s e i π w e π iw − e − π iw dw = ( − N − Ua − σ K (cid:88) k = C k ( p , σ ) a k + RS K ( s ) where C ( p , σ ) = C ( p ) is independent of σ and is given explicitly by the formula (53) C ( p ) (cid:66) e π i ( p + ) − i √ π p π p ) (removing the singularities at p = ± / ), while for k ≥ the C k ( p , σ ) are complex numbersobeying the bounds (54) | C k ( p , σ ) | ≤ √ π σ Γ ( k / k PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 23 for σ > and (55) | C k ( p , σ ) | ≤ − σ π Γ ( k / π ((3 − π ) k / for σ ≤ , while the error term RS K ( s ) is a complex number obeying the bounds (56) | RS K ( s ) | ≤
17 2 σ/ Γ (( K + / a / . K + for σ ≥ , and (57) | RS K ( s ) | ≤ (cid:32) (cid:33) (cid:100)− σ (cid:101) Γ (( K + / a / . K + if σ < and K + σ ≥ .Proof. This follows from [2, Theorems 3.1, 4.1, 4.2] combined with [2, (3.2), (5.2)]. Thedependence of C k ( p , σ ) , k ≥ σ and the dependence of RS K ( s ) on s is suppressed in[2], but can be discerned from the definitions of these quantities (and the related quantities g ( τ, z ) , P k ( z ) = P k ( z , σ ) , Rg K ( τ, z )) in [2, (3.9), (3.10), (3.7), (3.6)]. (cid:3) Note that p ranges in the interval [ − , | C ( p ) | ≤ p ∈ [ − , n = igure
10. Plot of | C ( p ) | for − ≤ p ≤ Informally, the above proposition (and (38), (6)) yield the approximation R , N ( s ) ≈ s ( s − π − s / Γ (cid:18) s (cid:19) ( − N − Ua − σ C ( p ) ≈ ( − N − U M ( s ) a − σ C ( p ) . If one writes s = σ + iT , then by using the approximation α ( s ) ≈ log iT π for the log-derivativeof M , one can then obtain the approximate formula R , N ( s ) ≈ ( − N − Ue π i σ/ M ( iT ) C ( p ) . In fact we have the more general approximation R t , N ( s ) ≈ ( − N − Ue π i σ/ exp (cid:32) t π (cid:33) M ( iT (cid:48) ) C ( p )where T (cid:48) (cid:66) T + π t . More precisely, we have Proposition 6.3 (Estimate for R t , N ) . Let ≤ σ ≤ , let T ≥ , and let < t ≤ / . SetT (cid:48) (cid:66) T + π t and then define a , N , p , U , C ( p ) using (49) , (50) , (52) , (53) . ThenR t , N ( σ + iT ) = ( − N − Ue π i σ/ exp (cid:32) t π (cid:33) M ( iT (cid:48) ) ( C ( p ) + O ≤ ( ˜ ε ( σ + iT ))) where (59) ˜ ε ( σ + iT ) (cid:66) (cid:32) . × σ a − . + T − (cid:33) exp (cid:32) . T − (cid:33) . Proof.
We apply (41) with β N : = π i / R t , N ( σ + iT ) = exp (cid:32) t π (cid:33) (cid:90) R exp (cid:32) − √ tv π i (cid:33) R , N ( σ + iT (cid:48) + √ tv ) 1 √ π e − v dv . From (38) we have R , N ( σ + iT (cid:48) + √ tv ) = s v ( s v − π − s v / Γ (cid:18) s v (cid:19) ( − N − Ua − σ − √ tv K v (cid:88) k = C k ( p , σ + √ tv ) a k + RS K v ( s v ) for any positive integer K v that we permit to depend (in a measurable fashion) on v , where s v (cid:66) σ + iT (cid:48) + √ tv . From (6) and Lemma 5.1(v) we thus have R , N ( σ + iT (cid:48) + √ tv ) = M ( s v ) exp O ≤ T (cid:48) − . ( − N − Ua − σ − √ tv K v (cid:88) k = C k ( p , σ + √ tv ) a k + RS K v ( s v ) . From (43) and Taylor expansion of the logarithm log M defined in (7), we have M ( s v ) = M ( iT (cid:48) ) exp (cid:32) α ( iT (cid:48) )( σ + √ tv ) + O ≤ (cid:32) ( σ + √ tv ) T − (cid:33)(cid:33) . From (9), (49) one has α ( iT (cid:48) ) = O ≤ (cid:32) T (cid:48) (cid:33) + O ≤ (cid:32) T (cid:48) (cid:33) +
12 Log iT (cid:48) π = log a + i π + O ≤ (cid:32) T (cid:48) (cid:33) PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 25 and hence (bounding T (cid:48) by T − ) α ( iT (cid:48) )( σ + √ tv ) = ( σ + √ tv ) log a + π i σ + √ tv π i + O ≤ (cid:32) | σ + √ tv | T − (cid:33) . We conclude (bounding T (cid:48) − . ≤ / T − ) thatexp (cid:32) − √ tv π i (cid:33) R , N ( σ + iT (cid:48) + √ tv ) = M ( iT (cid:48) ) exp O ≤ ( σ + √ tv ) + | σ + √ tv | + T − ×× ( − N − Ue π i σ/ K v (cid:88) k = C k ( p , σ + √ tv ) a k + RS K v ( s v ) . Bounding 6 | σ + √ tv | ≤ σ + √ tv ) +
3, we have( σ + √ tv ) + | σ + √ tv | + T (cid:48) − . ≤ ( σ + √ tv ) + T − . Putting all this together, we obtain R t , N ( σ + iT ) = ( − N − Ue π i σ/ exp (cid:32) t π (cid:33) M ( iT (cid:48) ) ×× (cid:90) R exp O ≤ ( σ + √ tv ) + T − K v (cid:88) k = C k ( p , σ + √ tv ) a k + RS K v ( s v ) √ π e − v dv . We separate the k = √ π e − v integratesto one, we can write the above expression as(60) R t , N ( σ + iT ) = ( − N − Ue π i σ/ exp (cid:32) t π (cid:33) M ( iT (cid:48) ) ( C ( p )(1 + O ≤ ( (cid:15) )) + O ≤ ( δ ))where (cid:15) : = (cid:90) R exp ( σ + √ tv ) + T − − √ π e − v dv and δ : = (cid:90) R exp ( σ + √ tv ) + T − K v (cid:88) k = | C k ( p , σ + √ tv ) | a k + | RS K v ( s v ) | √ π e − v dv . Bounding ( σ + √ tv ) ≤ σ + tv and using (27) we obtain (cid:15) ≤ exp σ + T − (cid:32) − tT (cid:48) − . (cid:33) − / − . Applying Lemma 5.1(ii) and using the hypotheses t ≤ / T ≥ − tT − = exp (cid:32) O ≤ (cid:32) tT − (cid:33)(cid:33) and hence (cid:15) ≤ exp σ + t + T − − . With t ≤ / ≤ σ ≤
1, one has 2 σ + t + ≤ . By the mean value theorem we then have(61) (cid:15) ≤ T −
6) exp (cid:32) T − (cid:33) . Now we work on δ . Making the change of variables u (cid:66) σ + √ tv , we have δ = (cid:90) R exp u + T − ˜ K u (cid:88) k = | C k ( p , u ) | a k + | RS ˜ K u ( u + iT (cid:48) ) | √ π t e − ( u − σ ) / t du , where ˜ K u is a positive integer parameter that can depend arbitrarily on u (as long as it is measur-able, of course).We choose ˜ K u to equal 1 when u ≥ (cid:98)− u (cid:99) + , (cid:98) T (cid:48) π (cid:99) ) when u <
0, so that Proposition6.2 applies. The expression ˜ K u (cid:88) k = | C k ( p , u ) | a k + | RS ˜ K u ( u + iT (cid:48) ) | is then bounded by(62) √ π u Γ (1 / a +
17 2 u / Γ (1)( a / . ≤ . × u a + . × u / a for u ≥ (cid:88) ≤ k ≤ ˜ K u − u π Γ ( k / π ((3 − π ) k / a k +
12 (9 / (cid:100)− u (cid:101) Γ (( ˜ K u + / a / . ˜ K u + for u <
0. One can calculate that 2 π π ≤ . ≤ − π ) / ≤ . ≤ . . − u (cid:88) ≤ k ≤ T (cid:48) π (0 . k Γ ( k / a k +
12 2 − u (cid:88) T (cid:48) π ≤ k ≤− u + Γ ( k / a / . k . For u ≥
0, we can estimate (62) by0 . × u (cid:32) a + . a (cid:33) ≤ . × u a − . u <
0, we observe that if k ≤ a = T (cid:48) π then Γ ( k + ) a k + = k a Γ ( k / a k ≤ Γ ( k / a k and hence by the geometric series formula (cid:88) ≤ k ≤ T (cid:48) π , k even (0 . k Γ ( k / a k ≤ (0 . − (0 . Γ (2 / a ≤ . a and similarly (cid:88) ≤ k ≤ T (cid:48) π , k odd (0 . k Γ ( k / a k ≤ (0 . − (0 . Γ (3 / a ≤ . a and hence we can bound (63) by(0 . − u (cid:32) . √ π a + . a + . a (cid:33) +
12 2 − u (cid:88) T (cid:48) π ≤ k ≤− u + Γ ( k / a / . k . By Lemma 5.1(i) we have0 . (cid:32) . √ π a + . a + . a (cid:33) ≤ . a − . . × − u a − . +
12 2 − u (cid:88) T (cid:48) π ≤ k ≤− u + (1 . k Γ ( k / a k . Putting this together, we conclude that ˜ K u (cid:88) k = | C k ( p , u ) | a k + | RS ˜ K u ( u + iT (cid:48) ) | ≤ . × u a − . + . × − u a − . + − u (cid:88) T (cid:48) π ≤ k ≤− u + (1 . k Γ ( k / a k for all u (positive or negative). We conclude that δ ≤ δ + δ + δ , where δ (cid:66) (cid:90) R exp u + T − . × u a − .
865 1 √ π t e − ( u − σ ) / t du δ (cid:66) (cid:90) R exp u + T − . × − u a − .
353 1 √ π t e − ( u − σ ) / t du δ (cid:66) (cid:90) R exp u + T − − u (cid:88) T (cid:48) π ≤ k ≤− u + (1 . k Γ ( k / a k √ π t e − ( u − σ ) / t du . (64)For δ , we translate u by σ to obtain δ = . × σ a − . (cid:90) R exp u + σ u + σ + T (cid:48) − . + u log 3 √ π t e − u / t du and hence by (27)(65) δ = . × σ a − .
865 exp σ + T (cid:48) − . + t (log 3 + σ T (cid:48) − . ) − tT − (cid:18) − tT − (cid:19) − / . One can write(66) 11 − tT − = + tT − − t ≤ + tT − . − tT − = exp (cid:18) O ≤ (cid:18) tT − − t (cid:19)(cid:19) = exp (cid:18) O ≤ (cid:18) tT − . (cid:19)(cid:19) . We conclude that δ ≤ . × σ a − .
865 exp (cid:32) + t + σ T − . + t (cid:18) log 3 + σ T − (cid:19) (cid:18) + tT − . (cid:19)(cid:33) . From Lemma 5.1(i) and the hypothesis 0 ≤ σ ≤
1, we have (cid:18) log 3 + σ T − (cid:19) ≤ (log + σ/ log 3 T − − σ ≤ (log (cid:32) + σ/ log 3 T − . (cid:33) and therefore by a further application of Lemma 5.1(i) (cid:18) log 3 + σ T − (cid:19) (cid:18) + tT − . (cid:19) ≤ log + σ log 3 + tT − . − σ t / log 32 σ/ log 3 + t ≤ log + σ log 3 + tT − . − t ≤ log + σ log 3 + tT − and thus δ ≤ . × σ exp( t log a − .
865 exp (cid:32) + t + σ + t σ log 3 + t log T − (cid:33) . By repeating the proof of (65), we have δ = . × − σ a − .
353 exp σ + T − + t (cid:16) − log √ + σ T − (cid:17) − tT − (cid:18) − tT − (cid:19) − / . We can bound ( − log √ + σ T − ) by log √
2. Using (66), (67) we thus have δ ≤ . × − σ exp( t log √ a − .
353 exp + t + σ + t log √ T − . With t ≤ / ≤ σ ≤ . t log ≤ . .
029 exp( t log √ ≤ . + t + σ + t log √ ≤ + t + σ + t σ log 3 + t log ≤ . δ ≤ . × σ a − .
865 exp (cid:32) . T − (cid:33) and δ ≤ . × − σ a − .
353 exp (cid:32) . T − (cid:33) . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 29
Now we turn to δ , which will end up being extremely small compared to δ or δ . By (64)and the Fubini-Tonelli theorem, we have δ = √ π t (cid:88) k ≥ T (cid:48) . π (1 . k Γ ( k / a k (cid:90) − k −∞ exp u + T (cid:48) − . − ( u − σ ) t − u log 2 du . Since u ≤ − k , k ≥ T (cid:48) . π , and T (cid:48) ≥ T ≥ k ≥
14 and u ≤ −
10; since σ ≥
0, we maythus lower bound ( u − σ ) / t by u / t . Since t ≤ /
2, we can upper bound u + T (cid:48) − . − u t by (say) − u t , thus δ ≤ √ π t (cid:88) k ≥ T (cid:48) . π (1 . k Γ ( k / a k (cid:90) − k −∞ e − u t − u log 2 du . We can bound e − u t ≤ e ( k − u t , in the range of integration and thus (cid:90) − k −∞ e − u t − u log 2 du ≤ k − t − log 2 e − ( k − t + ( k −
4) log 2 ≤ k − t − log 2 e − ( k − + ( k −
4) log 2 ;bounding k − t − log 2 = k − − t log 22 t ≥ k − t we conclude that δ ≤ √ t √ π (cid:88) k ≥ T (cid:48) . π (1 . k Γ ( k / k − a k e − ( k − + ( k −
4) log 2 . For k ≥
14 one can easily verify that (1 . k Γ ( k / e − ( k − + ( k −
4) log 2 ≤ − ; discarding the √ t √ π and k − factors we thus have δ ≤ (cid:88) k ≥ − a k ≤ × − a (say). Since 0 . × − σ a − . + × − a ≤ . × − σ a − . δ ≤ δ + δ + δ ≤ . × σ + . × − σ a − .
865 exp (cid:32) . T − (cid:33) . Inserting this and (61), (58) into (60), and crudely bounding 2 − σ by 9 σ , we obtain the claim. (cid:3) Combining the estimates.
Combining Propositions 6.1, 6.3 with (39) and the triangleinequality (and noting that M = M ∗ , M t = M ∗ t and α = α ∗ , and that U has magnitude 1), weconclude the following “ A + B − C approximation to H t ”: Corollary 6.4 ( A + B − C approximation) . Let t , x , y obey (5) . Set (68) T (cid:48) (cid:66) x + π t and then define a , N , p , U , C ( p ) using (49) , (50) , (52) , (53) . Define the quantitiess + = s + ( x + iy ) (cid:66) + y − ix s − = s − ( x + iy ) (cid:66) − y + ix A t , N ( x + iy ) (cid:66) M t ( s − ) N (cid:88) n = b tn n s − + t α ( s − ) B t , N ( x + iy ) (cid:66) M t ( s + ) N (cid:88) n = b tn n s + + t α ( s + ) C t ( x + iy ) (cid:66) e − π iy / ( − N exp (cid:32) t π (cid:33) Re ( M ( iT (cid:48) ) C ( p ) Ue π i / ) where M , b tn were defined in (10) , (15) . ThenH t ( x + iy ) = A t , N ( x + iy ) + B t , N ( x + iy ) − C t ( x + iy ) + O ≤ ( E A ( x + iy ) + E B ( x + iy ) + E C ( x + iy )) where E A ( x + iy ) (cid:66) | M t ( s − ) | N (cid:88) n = b tn n − y + t Re α ( s − ) ε t , n ( s − ) E B ( x + iy ) (cid:66) | M t ( s + ) | N (cid:88) n = b tn n + y + t Re α ( s + ) ε t , n ( s + ) E C ( x + iy ) (cid:66) exp (cid:32) t π (cid:33) | M ( iT (cid:48) ) | ( ˜ ε ( s − ) + ˜ ε ( s + )) and ε t , n , ˜ ε were defined in (44) , (59) . In our applications, we will just use the cruder “ A + B ” approximation that is immediate fromthe above corollary and (58): Corollary 6.5 ( A + B approximation) . With the notation and hypotheses as in Corollary 6.4, wehave H t ( x + iy ) = A t , N ( x + iy ) + B t , N ( x + iy ) + O ≤ ( E A ( x + iy ) + E B ( x + iy ) + E C , ( x + iy )) where E C , ( x + iy ) (cid:66) exp (cid:32) t π (cid:33) | M ( iT (cid:48) ) | (1 + ˜ ε ( s − ) + ˜ ε ( s + )) . We can now prove Theorem 1.3. Dividing by the expression B t from (11), and using (14), weconclude that(69) H t ( x + iy ) B t ( x + iy ) = f t ( x + iy ) − C t ( x + iy ) B t ( x + iy ) + O ≤ ( e A + e B + e C )and(70) H t ( x + iy ) B t ( x + iy ) = f t ( x + iy ) + O ≤ (cid:0) e A + e B + e C , (cid:1) PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 31 where e A (cid:66) e A ( x + iy ) (cid:66) | γ | N (cid:88) n = n y b tn n Re s + Re κ ε t , n ( s − )(71) e B (cid:66) e B ( x + iy ) (cid:66) N (cid:88) n = b tn n Re s ε t , n ( s + )(72) e C (cid:66) e C ( x + iy ) (cid:66) exp (cid:16) t π (cid:17) | M ( iT (cid:48) ) || M t ( s + ) | ( ˜ ε ( s − ) + ˜ ε ( s + )) . (73) e C , (cid:66) e C , ( x + iy ) (cid:66) exp (cid:16) t π (cid:17) | M ( iT (cid:48) ) || M t ( s + ) | (1 + ˜ ε ( s − ) + ˜ ε ( s + )) , (74)and where γ, s ∗ , κ were defined in (16), (17), (18). Note also from (68), (49), (50) that N is givenby (19).To conclude the proof of Theorem 1.3 it thus su ffi ces to obtain the following estimates. Proposition 6.6 (Estimates) . Let the notation and hypotheses be as above. (i)
One has | γ | ≤ e . y (cid:18) x π (cid:19) − y / (ii) One has Re s ∗ ≥ + y + t x π − (1 − y + y (1 − y ) x ) + t x . (iii) One has κ = O ≤ (cid:32) ty x − (cid:33) . (iv) One has e A ≤ | γ | N | κ | N (cid:88) n = n y b tn n Re s ∗ exp t log x π n + . x − . − . (v) One has e B ≤ N (cid:88) n = b tn n Re s ∗ exp t log x π n + . x − . − . (vi) One hase C ≤ (cid:18) x π (cid:19) − + y exp (cid:32) − t
16 log x π + | log x π + i π | + . x − . (cid:33) (cid:32) . × (3 y + − y ) N − . + . x − (cid:33) . ande C , ≤ (cid:18) x π (cid:19) − + y exp (cid:32) − t
16 log x π + | log x π + i π | + . x − . (cid:33) (cid:32) + . × (3 y + − y ) N − . + . x − (cid:33) . Note that to obtain the bound (24) from Proposition 6.6(vi) we may simply use the inequality1 + u ≤ exp( u ) for any u ∈ R , and then bound x − . ≤ x − . Proof.
From the mean value theorem (and noting that M t = M ∗ t , so that (cid:12)(cid:12)(cid:12)(cid:12) M t (cid:16) + y − ix (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) M t (cid:16) + y + ix (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ),we have log | γ | = − y dd σ log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M t (cid:18) σ + ix (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for some − y ≤ σ ≤ + y . From (8), (10) we have dd σ log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M t (cid:18) σ + ix (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Re (cid:18) t α (cid:18) σ + ix (cid:19) α (cid:48) (cid:18) σ + ix (cid:19) + α (cid:18) σ + ix (cid:19)(cid:19) . From (43) one has(75) α (cid:48) (cid:18) σ + ix (cid:19) = O ≤ (cid:32) x − (cid:33) and from Taylor expansion we also have α ( σ + ix = α (cid:18) ix (cid:19) + O ≤ (cid:18) σ x − (cid:19) ;from (9) one has α (cid:18) ix (cid:19) = O ≤ (cid:32) x (cid:33) + O ≤ (cid:32) x (cid:33) +
12 Log ix π =
12 log x π + i π + O ≤ (cid:32) x (cid:33) and hence(76) α ( σ + ix =
12 log x π + i π + O ≤ (cid:32) + σ x − (cid:33) . Inserting these bounds, we conclude thatlog | γ | = − y Re (cid:32)(cid:32)
12 log x π + i π + O ≤ (cid:32) + σ x − (cid:33)(cid:33) (cid:32) + O ≤ (cid:32) t x − (cid:33)(cid:33)(cid:33) . Expanding this out, we havelog | γ | = − y
12 log x π + O ≤ + σ + t log x π + t π + t (2 + σ )2( x − x − . In the region (5), which implies that 0 ≤ σ ≤
1, we have2 + σ + t π + t (2 + σ )2( x − ≤ . | γ | ≤ − y x π + y t log x π + . x − . The function x (cid:55)→ log x π x − is decreasing for x ≥
200 thanks to Lemma 5.1(vi), hence y t log x π + . x − ≤ y t log π + . − ≤ . y . Claim (i) follows. We remark that one can improve the e . y factor here by Taylor expanding α to second order rather than first order, but we will not need to do so here.To prove claim (ii), it su ffi ces by (21) to show thatRe α ( s + ) ≥
12 log x π − (1 − y ) + x − y (1 + y ) x . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 33
By (9) one has Re α ( s + ) = + y (1 + y ) + x − − y )(1 − y ) + x +
12 log (cid:112) (1 + y ) + x π . We bound (cid:112) (1 + y ) + x ≥ x and calculate1 + y (1 + y ) + x − − y )(1 − y ) + x = − − y (1 + y ) + x − y (1 − y )((1 + y ) + x )((1 − y ) + x ) ≥ − − y + y (1 − y ) x (1 + y ) + x . Lower bounding the numerator by its nonnegative part and then lower bounding (1 + y ) + x by x , we obtain the claim.Claim (iii) is immediate from (75) and the fundamental theorem of calculus. Now we turn to(iv), (v). From (76) one has α (cid:32) ± y + ix (cid:33) − log n =
12 log x π n + i π + O ≤ (cid:32) x − (cid:33) for either choice of sign ± . In particular, we have(77) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (cid:32) ± y + ix (cid:33) − log n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =
14 log x π n + π + O ≤ (cid:32) | log x π n + i π | x − + x − (cid:33) . For any 1 ≤ n ≤ N , we have 1 ≤ n ≤ N ≤ a = x + π t π ;in the region (5), the right-hand side is certainly bounded by ( x π ) , so that4 π x ≤ x π n ≤ x π and hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log x π n + i π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log x π + i π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . In the region (5) we have x ≥ | log x π + i π | x − isdecreasing in x . Thus π + | log x π n + i π | x − + x − ≤ π + | log π + i π | − + − ≤ . . Similarly, in (5) we also have t × . + t + ≤ . . We conclude from (44) that ε t , n (cid:32) ± y + ix (cid:33) ≤ exp t log x π n + . T − . − . Inserting this bound into (71), (72), we obtain claims (iv), (v).
Now we establish (vi). From (10) we haveexp (cid:16) t π (cid:17) | M ( iT (cid:48) ) || M t ( s + ) | = exp (cid:32) t π − t α ( s + ) ) (cid:33) | M ( iT (cid:48) ) || M ( s + ) | . Note that + y + ix = iT (cid:48) + + y − π it . From (43) we see that | α (cid:48) ( s ) | ≤ x − for any s on the linesegment between iT (cid:48) and + y + ix . From Taylor’s theorem with remainder applied to a branch oflog M , and noting that | M ( s + ) | = | M ( + y + ix ) | , we conclude that | M ( iT (cid:48) ) || M ( s + ) | = exp Re (cid:32)(cid:32) − + y + π it (cid:33) α ( iT (cid:48) ) (cid:33) + O ≤ | − + y + π it | x − . For 0 ≤ y ≤ < t ≤ we have | − + y + π it | ≤ . α ( iT (cid:48) ) = O ≤ (cid:32) T (cid:48) (cid:33) + O ≤ (cid:32) T (cid:48) (cid:33) +
12 Log iT (cid:48) π =
12 log T (cid:48) π + i π + O ≤ ( 32 T (cid:48) )and hence | M ( iT (cid:48) ) || M ( s + ) | = exp − + y T (cid:48) π − t π + O ≤ | − + y + π it | T (cid:48) + . x − . Bounding T (cid:48) ≤ x − and | − + y + π it | ≤ .
02, this becomes | M ( iT (cid:48) ) || M ( s + ) | = (cid:32) T (cid:48) π (cid:33) − + y exp (cid:32) − t π + O ≤ (cid:32) . x − (cid:33)(cid:33) and henceexp (cid:16) t π (cid:17) | M ( iT (cid:48) ) || M t ( + y + ix ) | = (cid:32) T (cid:48) π (cid:33) − + y exp − t π − t α (cid:32) + y + ix (cid:33) ) + O ≤ (cid:32) . x − (cid:33) . By repeating the proof of (77) we haveRe ( α ( 1 ± y + ix ) =
14 log x π − π + O ≤ (cid:32) | log x π + i π | x − + x − (cid:33) . As before, in the region (5) we have3 | log x π n + i π | x − + x − ≤ | log x π + i π | x − + x − and thusexp (cid:16) t π (cid:17) | M ( iT (cid:48) ) || M t ( s + ) | = (cid:32) T (cid:48) π (cid:33) − + y exp (cid:32) − t
16 log x π + O ≤ (cid:32) | log x π + i π | + . x − + x − (cid:33)(cid:33) = (cid:32) T (cid:48) π (cid:33) − + y exp (cid:32) − t
16 log x π + O ≤ (cid:32) | log x π + i π | + . x − . (cid:33)(cid:33) PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 35 thanks to Lemma 5.1(i). Finally, since T (cid:48) ≥ x ≥
100 in (5), one hasexp (cid:32) . T − (cid:33) ≤ . ε (cid:32) ± y + ix (cid:33) ≤ . × ± y a − . + . T − . Hence ˜ ε ( s + ) + ˜ ε ( s − ) ≤ . × (3 y + − y ) a − . + . T − T (cid:48) = x / a ≥ N ). (cid:3)
7. F ast evaluation of multiple sums
Fix t ≥
0. For the verification of the barrier criterion (Theorem 1.2(iii)) using Corollary 1.4,we will need to evaluate the quantity f t ( s ) to reasonable accuracy for a large number of valuesof s in the vicinity of a fixed complex number X + iy . From (14) we have(78) f t ( s ) = N (cid:88) n = n b b tn n + y − iX + γ ( s ) N (cid:88) n = n a b tn n − y + iX , where b tn is given by (15), γ ( s ) is given by (16), N is given by (19) and b = b ( s ) (cid:66) + y − iX − s ∗ and a = a ( s ) (cid:66) − y − iX − s ∗ − κ with s ∗ , κ defined by (17), (18). In practice the exponents a , b will be rather small, and N will befixed (in our main verification we will in fact have N = f t ( s ) for M values of s would take time O ( N M ), which turns out tobe somewhat impractical for for the ranges of N , M we will need; indeed, for our main theorem,the total number of pairs ( t , s ) at which we need to perform the evaluation is 785052 (spread outover 152 values of t ), and direct computation of all this data required 78 . X but would not scale to significantlyhigher magnitudes. However, one can significantly speed up the computation (to about 0 . s and thus can be computed in advance.We turn to the details. To make the Taylor series converge faster, we recenter the sum in n ,writing N (cid:88) n = F ( n ) = (cid:98) ( N + / (cid:99) (cid:88) h = −(cid:98) N / (cid:99) + F ( n + h )for any function F , where n (cid:66) (cid:98) N / (cid:99) . We thus have f t ( s ) = B ( b ) + γ ( s ) A ( a ) One can obtain even faster speedups here by splitting the summation range (cid:80) Nn = into shorter intervals and usinga Taylor expansion for each interval, although ultimately we did not need to exploit this. where B ( b ) (cid:66) (cid:98) ( N + / (cid:99) (cid:88) h = −(cid:98) N / (cid:99) + ( n + h ) b b tn + h ( n + h ) + y − iX and A ( a ) (cid:66) (cid:98) ( N + / (cid:99) (cid:88) h = −(cid:98) N / (cid:99) + ( n + h ) a b tn + h ( n + h ) − y + iX . We discuss the fast computation of B ( b ) for multiple values of b ; the discussion for A ( a ) isanalogous. We can write the numerator ( n + h ) b b tn + h asexp( b log( n + h ) + t ( n + h ));writing log( n + h ) = log n + log(1 + hn ), this becomes n b + t log n exp( t (1 + hn )) exp(( b + t n ) log(1 + hn )) . By Taylor expanding the exponentials, we can write this as n b + t log n ∞ (cid:88) i = ∞ (cid:88) j = ( t log (1 + hn )) i i ! log j (1 + hn ) ( b + t log n ) j j !and thus the expression B ( b ) can be written as B ( b ) = n b + t log n ∞ (cid:88) i = ∞ (cid:88) j = B i , j ( b + t log n ) j j !where B i , j (cid:66) (cid:98) ( N + / (cid:99) (cid:88) h = −(cid:98) N / (cid:99) + ( t log (1 + hn )) i i ! log j (1 + hn )( n + h ) + y − iX . If we truncate the i , j summations at some cuto ff E , we obtain the approximation B ( b ) ≈ n b + t log n E − (cid:88) i = E − (cid:88) j = B i , j ( n ) ( b + t log n ) i i ! . The quantities B i , j , i , j = , . . . , E − O ( NE ), and then the sums B ( b )for M values of b may be evaluated in time O ( ME ), leading to a total computation time of O (( N + M ) E ) which can be significantly faster than O ( N M ) even for relatively large values of E . We took E =
50, which is more than adequate to obtain extremely high accuracy ; for f t ( s );see Figure 13. The code for implementing this may be found in the file dbn upper bound/pari/barrier multieval t agnostic.txt in the github repository [20]. It is also possible to proceed by just performing Taylor expansion on the second exponential and leaving the firstexponential untouched; this turns out to lead to a comparable numerical run time. One can obtain more than adequate analytic bounds for the error (which are several orders of magnitude morethan necessary) for the parameter ranges of interest by very crude bounds, e.g., bounding b and log(1 + hn ) by (say) O ≤ (2), and relying primarily on the i ! and j ! terms in the denominator to make the tail terms small. We omit thedetails as they are somewhat tedious. PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 37
8. A new upper bound for the de B ruijn -N ewman constant In this section we prove Theorem 1.1.8.1.
Selection of parameters.
As stated in the introduction, it su ffi ces to verify the conditions(i), (ii), (iii) of Theorem 1.2 t (cid:66) . X (cid:66) X − .
5, and y (cid:66) .
2, where X (cid:66) × + t = y = . X . Recall the familiarEuler product factorization ζ ( s ) = (cid:89) p (cid:32) − p s (cid:33) − for the Riemann zeta function. This leads to the heuristic H ( x + iy ) ∝ (cid:89) p ≤ P (cid:32) − p s (cid:33) − for some small prime cuto ff P , where s = + y + ix and we are extremely vague as to what theproportionality symbol ∝ means. This heuristic extends to non-zero times t as H t ( x + iy ) ∝ (cid:89) p ≤ P − b tp p s − and we also have(79) f t ( x + iy ) ∝ (cid:89) p ≤ P − b tp p s − . One can non-rigorously justify the latter assertion by by inspecting the first series of f t ( x + iy ) in(14) and ignoring the fact that the sequence n (cid:55)→ b tn is not multiplicative when t (cid:44) | f t ( x + iy ) | is aslarge as possible. It would therefore seem to be advantageous to try to work as much as possiblein regions where Euler product (cid:89) p ≤ P (cid:32) − b tn p s (cid:33) , is small, which heuristically corresponds to x π log p being close to an integer for p ≤ P (so that p s has argument close to zero). If one chooses x to lie in the vicinity of X (cid:66) × + − . then indeed the fractional parts { X π log p } for p ≤
11 are somewhat close to zero: { X π log 2 } = . . . . { X π log 3 } = . . . . { X π log 5 } = . . . . { X π log 7 } = . . . . { X π log 11 } = . . . . We found this shift by the following somewhat ad hoc procedure. We first introduced the quan-tity eulerprod( x , p n ) (cid:66) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:89) p ≤ p n − p − ix / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which is the exponent corresponding to y = | f t ( x + iy ) | in thebarrier region is expected to occur). We numerically located candidate integers 1 ≤ q ≤ forwhich the quantity min x − × − q ∈{− . , , . } | eulerprod( x , | exceeded a threshold (we chose 4), to obtain seven candidates for q : 1046, 22402, 24198, 52806,77752, 83952, and 99108. Among these candidates, we selected the value of q which maximisedthe quantity min x − × − q ∈{− . , , . } | f ( x + i ) | , namely q = ≈ .
32 for this value of q ).8.2. Verifications of claims.
Claim (i) of Theorem 1.2 is immediate from the result of Platt[18] that all the non-trivial zeroes of ζ with imaginary part between 0 and 3 . × lie on thecritical line { Re s = / } . For the remaining claims (ii), (iii) of Theorem 1.2, it will su ffi ce toverify that H t ( x + iy ) (cid:44) x , y , t ):(ii) x ≥ X − . + √ .
96, 0 . ≤ y ≤ √ .
6, and t = . X − . ≤ x ≤ X + .
5, 0 . ≤ y ≤ √ .
6, and 0 ≤ t ≤ . H t ( x + iy ) by H t ( x + iy ) / B t ( x + iy ).Set N (cid:66) (cid:114) x π + t so in particular(80) x N ≤ x < x N + where x N (cid:66) π N − π t . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 39 F igure
11. Improving approximation of ep n ( x ) w.r.t. | f ( x + iy ) | as n is increased,where ep n ( x ) = eulerprod( x , p n ), shown near X = × + .
5, whichwas chosen as a barrier location.Write N (cid:66) N (cid:66) . × . In region (ii) we then have N ≥ N , while in region (iii)we have N = N . It will now su ffi ce to verify H t ( x + iy ) B t ( x + iy ) (cid:44) X − . ≤ x ≤ X + . N = N , 0 ≤ t ≤ .
2, and 0 . ≤ y ≤ x ≥ X − . N ≤ N ≤ N , t = .
2, and 0 . ≤ y ≤ N ≥ N , t = .
2, and 0 . ≤ y ≤ Proposition 8.1.
Let ( x , y , t ) lie in one of the regions (a), (b), (c). Definef t ( x + iy ) (cid:66) N (cid:88) n = b tn n s ∗ + γ N (cid:88) n = n y b tn n s ∗ + κ b tn (cid:66) exp( t n ) , where κ, s ∗ , γ are as in Theorem 1.3. Then (81) H t ( x + iy ) B t ( x + iy ) = f t ( x + iy ) + O ≤ (1 . × − ) . Proof.
By Theorem 1.3 it su ffi ces to show that e A + e B + e C , ≤ . × − . From Theorem 1.3 again, we have(82) e A + e B ≤ ( e δ − F N , t (Re s ∗ ) + | γ | N | κ | F N , t (Re s ∗ − y )) where(83) F N , t ( σ ) : = N (cid:88) n = b tn n σ . and(84) δ (cid:66) t log x π + . x − . . From Lemma 5.1(vi), the quantity δ is monotone decreasing in x in the region (5). Thus wehave(85) δ ≤ (0 . log X − . π + . X − . − . x ≥ X − . ≥
200 and 0 ≤ t ≤ .
2. Computing the right-hand side, we conclude that δ ≤ . × − and hence by Taylor expansion e δ − ≤ . δ (say). Also, from Theorem 1.3 and (83) we can bound | γ | F N , t (Re s ∗ − y − | κ | ) ≤ | γ | N y N | κ | F N , t (Re s ∗ ) ≤ exp (cid:32) . y + y (cid:32) log N −
12 log x π (cid:33) + ty x −
6) log N (cid:33) F N , t (Re s ∗ ) ≤ exp (cid:32) . y + (cid:32) y + ty x − (cid:33)
12 log(1 + π t x ) + ty x −
6) log x π (cid:33) F N , t (Re s ∗ ) . For 0 . ≤ y ≤
1, 0 ≤ t ≤ .
2, and x ≥ X − . ty x −
6) log x π ≤ . X − . −
6) log X − . π ≤ . × − and (cid:32) y + ty x − (cid:33)
12 log(1 + π t x ) ≤ (cid:32) + . X − . − (cid:33)
12 log (cid:32) + . π X − . (cid:33) ≤ . × − and thus(86) | γ | F N , t (Re s ∗ − y − | κ | ) ≤ . F N , t (Re s ∗ ) . Thus e A + e B ≤ . δ F N , t (Re s ∗ ) . To estimate Re s ∗ , we use Proposition 6.6(ii) or (22), together with the inequality t x (cid:32) − y + y (1 − y ) x (cid:33) + ≤ . X − . (cid:32) − × . + X − . (cid:33) + ≤ . × − to obtain Re s ∗ ≥ . + t x π (say). Since F N , t ( σ ) is non-increasing in σ , we conclude e A + e B ≤ . δ F N , t (cid:18) . + t x π (cid:19) . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 41
Since N = (cid:114) x π + t , ≤ t ≤ .
2, and x ≥ X − .
5, it is easy to see that N ≤ x π and hence by (15) b tn n t log x π ≤ ≤ n ≤ N . Therefore F N , t (cid:18) . + t x π (cid:19) ≤ N (cid:88) n = n . and hence by the integral test F N , t (cid:18) . + t x π (cid:19) ≤ (cid:90) N + dss . = . N + . so that (by (84)) e A + e B ≤ . (0 . log x π + . x − .
66 10 . N + . . We have N + ≤ (1 . (cid:32) x − . π (cid:33) / (say), and hence e A + e B ≤ . . x π + . x − . . From Lemma 5.1(vi), the right-hand side is monotone decreasing in the region x ≥ X − . e A + e B ≤ . . X − . π + . X − . − . . ≤ . × − . Meanwhile, from Proposition 6.6(vi) one has e C , ≤ (cid:18) x π (cid:19) − + y exp (cid:32) − t
16 log x π + | log x π + i π | + . x − . (cid:33) (cid:32) + . × (3 y + − y ) N − . + . x − . (cid:33) . From Lemma 5.1(vi), the quantity log x π + π ( x − . is monotone decreasing in x in (5), hence | log x π + i π | x − . is also monotone decreasing. Also the expression is monotone decreasing in y . We conclude that e C , ≤ (cid:32) X − . π (cid:33) − + . exp | log X − . π + i π | + . X − . − . + . × (3 √ . + − √ . ) N − . + . X − . − . ≤ . × − (87)where we have discarded the negative term − t log X − . π . Combining the estimates, we obtainthe claim. (cid:3) Now we attend to the three claims.8.3.
Proof of claim (c).
We begin with claim (c), which is the easiest. By Proposition 8.1, itsu ffi ces to establish the bound | f t ( x + iy ) | > . × − . In fact we will establish the stronger estimate(88) f t ( x + iy ) = + O ≤ (0 . . In the region (c) we have from (80) that x ≥ x N ≥ . × . Our main tool here is the triangle inequality. From (14) one has f t ( x + iy ) = + N (cid:88) n = b tn n s ∗ + γ N (cid:88) n = n y b tn n s ∗ + κ and hence f t ( x + y ) = + O ≤ N (cid:88) n = b tn n σ + | γ | N (cid:88) n = n y b tn n σ −| κ | where σ (cid:66) Re s ∗ . Restoring the n = t = .
2, it thussu ffi ces to show that(89) N (cid:88) n = b . n n σ + | γ | N (cid:88) n = n y b . n n σ −| κ | < . . Our main tool here will be
Lemma 8.2.
Let N ≥ N ≥ be natural numbers, and let σ, t > be such that σ > t N . Then N (cid:88) n = b tn n σ ≤ N (cid:88) n = b tn n σ + max( N − σ b tN , N − σ b tN ) log NN . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 43
Proof.
From the identity b tn n σ = exp (cid:16) t (log N − log n ) − t (log N ) (cid:17) n σ − t log N we see that the summands b tn n σ are decreasing for 1 ≤ n ≤ N , hence by the integral test one has(90) N (cid:88) n = b tn n σ ≤ N (cid:88) n = b tn n σ + (cid:90) NN b ta a σ da . Making the change of variables a = e u , the right-hand side becomes N (cid:88) n = b tn n σ (cid:90) NN exp((1 − σ ) u + t u ) du . The expression (1 − σ ) u + t u is convex in u , and is thus bounded by the maximum of its valuesat the endpoints u = log N , log N ; thusexp((1 − σ ) u + t u ) ≤ N − σ b tN , N − σ b tN . The claim follows. (cid:3)
Remark 8.3.
The right-hand side of (90) can be evaluated exactly as N (cid:88) n = b tn n σ + √ π √ t exp( − ( σ − t ) (cid:32) erfi (cid:32) t log N − σ + √ t (cid:33) − erfi (cid:32) t log N − σ + √ t (cid:33)(cid:33) where erfi( z ) = − i erf( iz ) is the imaginary error function, with erf( z ) (cid:66) √ π (cid:82) z e − t dt.In practice, this upper bound for (cid:80) Nn = b tn n σ is slightly more accurate than the one in Lemma 8.2,and is a good approximation even for relatively small values of N (e.g., N = ). However,the cruder bound above su ffi ces for the numerical values of parameters needed to establish thebound Λ ≤ . . Observe from (22) that | κ | ≤ . × . × − ≤ . × − while from (20) one has | γ | ≤ . (cid:18) x N π (cid:19) − y / so in particular since 0 . ≤ y ≤ n ≤ . x N π ) / ≤ . N | γ | n y ≤ . N − . n . . Also from (21) one has σ ≥ . + .
24 log x N π − . x N . + . x N + ≥ . + . N + .
05 log (cid:32) − t N ) (cid:33) − . x N . + . x . × + ≥ . + . N − . × − . We can then apply Lemma 8.2 twice to bound the left-hand side of (89) by A + B , where A (cid:66) N (cid:88) n = b . n n σ (cid:48) + . (cid:18) x N π (cid:19) − . N (cid:88) n = b . n n σ (cid:48)(cid:48) , B (cid:66) (max( N − σ (cid:48) b . N , N − σ (cid:48) b . N ) + . N − . max( N − σ (cid:48)(cid:48) b . N , N − σ (cid:48)(cid:48) b . N )) log NN σ (cid:48) : = . + . N − . × − σ (cid:48)(cid:48) : = . + . N − . × − . The quantity A is decreasing in N , so we may bound it by its value at N = N . Performing thesum numerically, we obtain A ≤ . . Finally, the quantity B can also be seen to be decreasing in N in the range N ≥ N , and obeysthe bound B ≤ . . The claim (88) follows.8.4.
Proof of claim (a). As N = N is constant in this region, the function f t ( x + iy ) is holo-morphic, so by Rouche’s theorem, it su ffi ces to show that for each time 0 ≤ t ≤ .
2, as x + iy traverses the boundary ∂ R of the rectangle R (cid:66) { x + iy : X − . ≤ x ≤ X + .
5; 0 . ≤ y ≤ } , the function f t ( x + iy ) stays outside of the ball B (cid:66) { z : | z | ≤ . × − } , and furthermore hasa winding number of zero around the origin.To verify this claim numerically for a given value of t , we subdivide each edge of ∂ R intosome number n of equally spaced mesh points, thus approximating ∂ R by a discrete mesh x j + iy j , j = , . . . , n , with any two adjacent points on this mesh separated by a distance at most 1 / n .Using the techniques in Section 7, we evaluate f t ( x j + iy j ) numerically for each such value of j .The polygonal path connecting these points then winds around the origin with winding number12 π n (cid:88) j = arg( f t ( x j + + iy j + ) / f t ( x j + iy j )) This is visually apparent from Figure 12, but to prove it analytically, it su ffi ces to show that the quantities N − σ (cid:48) log NN , N − σ (cid:48) b . N log NN , N − . N − σ (cid:48)(cid:48) log NN , N − . N − σ (cid:48)(cid:48) b . N log NN are decreasing in N for N ≥ N . This can inturn be established by computing the log-derivative of all these quantities, multiplied by N ; there will be a negativeterm − . N or − . N which dominates all the other terms when N ≥ N . We leave the details to theinterested reader. PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 45 F igure
12. Plot of B vs N which can be easily computed (and verified to be zero). To pass from this polygonal path to thetrue trajectory f t ( ∂ R ) of f t ( x + iy ) on ∂ R , we again use Rouche’s theorem. If one has a derivativebound | ∂∂ z f t ( z ) | ≤ D z on the boundary of the rectangle, the polygonal path and the true trajectory f t ( ∂ R ) di ff er by a distance of at most D z n , and the latter will have the same winding numberaround the origin (and stay outside of the ball B ) as long as | f t ( x j + iy j ) | > . × − + D z n . Furthermore, the same is true for nearby times t ≤ t (cid:48) ≤ . t , as long as one has the strongerbound(91) | f t ( x j + iy j ) | > . × − + D z n + D t | t (cid:48) − t | n and a bound of the form | ∂∂ t f ˜ t ( z ) | ≤ D t for t ≤ ˜ t ≤ . z ∈ ∂ R .This gives the following algorithm to verify (a) for the entire range 0 ≤ t ≤ .
2. We start with t = D t , D z for the derivatives of f ˜ t ( z ) in the indicated ranges. Because of theway the barrier location X was selected, we expect | f t ( x + iy ) | to stay well above 1 in magnitude.We thus choose n so that D z n ≤
1, and evaluate f t ( x j + iy j ) at all the mesh points (in particularconfirming that | f t ( x j + iy j ) | does stay well above 1). Using the minimum value of | f t ( x j + iy j ) | ,we can then use the condition (91) to establish the claim for times in the interval [ t , t (cid:48) ) where t (cid:48) > t is chosen so that (91) holds (or t (cid:48) = .
2, if that is also possible); the most aggressive choiceof t (cid:48) would be one in which (91) held with equality, but in practice we can a ff ord to take moreconservative values of t (cid:48) and still obtain good runtime performance. If t (cid:48) < .
2, we then repeatthe process, replacing t by t (cid:48) , until the entire range 0 ≤ t ≤ . D t and D z . This is achieved by the followinglemma, which gives bounds which are somewhat complicated but which can be easily upperbounded numerically on ∂ R : Lemma 8.4.
In the region (5) , and away from the jump discontinuities of N, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ f t ∂ z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (cid:88) n = b tn n Re s ∗ (cid:32) log n + t log n x − (cid:33) + | γ | N | κ | N (cid:88) n = b tn n y n Re s ∗ (cid:32) t log n x − + (cid:32) log | + y + ix | π + π + x (cid:33) (cid:32) + t x − (cid:33)(cid:33) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ f t ∂ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (cid:88) n = b tn n Re s ∗ (cid:32)
14 log n log x π n + π n + nx − (cid:33) + | γ | N | κ | N (cid:88) n = b tn n y n Re s ∗ (cid:32)
14 log n log x π n + π n + nx − + (cid:32) π + x − (cid:33) (cid:32) log x π + x − (cid:33)(cid:33) . Proof.
We begin with the first estimate. Write s ∗∗ (cid:66) s ∗ − y + κ = − y + ix + t α (cid:32) − y + ix (cid:33) then(92) f t = N (cid:88) n = b tn n s ∗ + γ N (cid:88) n = b tn n s ∗∗ . One can check that s ∗ , s ∗∗ , γ are holomorphic functions of x + iy , hence by the Cauchy-Riemannequations (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ f t ∂ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ f t ∂ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By the product and chain rules, we may calculate ∂ f t ∂ x = − N (cid:88) n = b tn n s ∗ ∂ s ∗ ∂ x log n + γ N (cid:88) n = b tn n s ∗∗ (cid:32) ∂∂ x log γ − ∂ s ∗∗ ∂ x log n (cid:33) . From (17), (43) we have ∂ s ∗ ∂ x = − i − it α (cid:48) (cid:32) + y − ix (cid:33) = − i + O ≤ (cid:32) t x − (cid:33) . Similarly we have ∂ s ∗∗ ∂ x = i + O ≤ (cid:32) t x − (cid:33) . Writing s = − y + ix , we have from (16), (10) thatlog γ = t α ( s ) − α (1 − s ) ) + log M ( s ) − log M (1 − s )and hence by (8) ∂∂ x log γ = it α ( s ) α (cid:48) ( s ) + α (1 − s ) α (cid:48) (1 − s )) + i α ( s ) + i α (1 − s ) . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 47
From the triangle inequality and (43), we thus have | ∂ f t ∂ x | ≤ N (cid:88) n = b tn n Re s ∗ (cid:32) log n + t log n x − (cid:33) + | γ | N (cid:88) n = b tn n Re s ∗∗ (cid:32) t log n x − + | α ( s ) | + | α (1 − s ) | − log n + t ( | α ( s ) | + | α (1 − s ) | )4( x − (cid:33) . We have from (9) that | α ( s ) | , | α ( s ) −
12 log n | ≤
12 log | − y + ix | π + π + x since n ≤ N ≤ x π ≤ | − y + ix | π . Similarly | α (1 − s ) | , | α (1 − s ) −
12 log n | ≤
12 log | + y + ix | π + π + x and thus | α ( s ) + α (1 − s ) | , | α ( s ) + α (1 − s ) − log n | ≤ log | + y + ix | π + π + x . Writing Re s ∗∗ = Re s ∗ − y + Re κ , we then have the first estimate.Now we estimate the time derivative. Since ∂∂ t log b tn =
14 log n ∂∂ t s ∗ = α (1 − s ) ∂∂ t s ∗∗ = α ( s ) ∂∂ t log γ = (cid:16) α ( s ) − α (1 − s ) (cid:17) we see from di ff erentiating (92) that, we obtain ∂ f t ∂ t = N (cid:88) n = b tn n s ∗ (cid:32) log n − α (1 − s )2 log n (cid:33) + γ N (cid:88) n = b tn n s ∗∗ (cid:32) log n − α ( s )2 log n +
14 ( α ( s ) − α (1 − s )) (cid:33) . From (43), (9) we have α (cid:32) ± y + ix (cid:33) = α (cid:18) ix (cid:19) + O ≤ (cid:32) x − (cid:33) =
12 log x π + π i + O ≤ (cid:32) x − (cid:33) and hence (since α = α ∗ ) α (cid:32) ± y − ix (cid:33) =
12 log x π − π i + O ≤ (cid:32) x − (cid:33) so in particular (recalling that 1 − s = + y − ix and s = − y + ix ) α ( s ) − α (1 − s ) = π i + O ≤ (cid:32) x − (cid:33) and α ( s ) + α (1 − s ) = log x π + O ≤ (cid:32) x − (cid:33) so that (cid:12)(cid:12)(cid:12) α ( s ) − α (1 − s ) (cid:12)(cid:12)(cid:12) ≤ (cid:32) π + x − (cid:33) (cid:32) log x π + x − (cid:33) . We conclude from the triangle inequality that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ f t ∂ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (cid:88) n = b tn n Re s ∗ (cid:32)
14 log n log x π n + π n + nx − (cid:33) + | γ | N (cid:88) n = b tn n Re s ∗∗ (cid:32)
14 log n log x π n + π n + nx − + (cid:32) π + x − (cid:33) (cid:32) log x π + x − (cid:33)(cid:33) giving the second claim. (cid:3) The next few graphs summarize the numerical output of the algorithm for the following barrierparameters: x = × + ± . , y = . . . . , t = . . . . f t ( x + iy ) during the execution of the algorithmas per Section 7. The number of Taylor terms required was determined through an iterative pro-cess targeted to achieve a 20 decimal accuracy. Figure 13 illustrates that the achieved accuracyfor all rectangular mesh points at t = igure
13. Achieved error term in the Taylor expansion at t =
0. Target was setat 20 decimal places accuracy.
PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 49
The derivative bounds determine the number of mesh points required on each xy -rectangle andin the t -direction. Figure 14 illustrates that these bounds have been chosen quite conservatively:F igure
14. Derivative bounds versus their actual values at all required steps of t .The number of rectangle mesh points varies with t ranging from 11076 at t = t = . igure
15. The number of mesh points required per rectangle for each step of t .The overall winding number for the barrier at this specific location came out at 0. Figures 16,17 show the winding process at t = , . x variable, little oscillation isexpected to occur in each rectangle.The code for implementing these computations may be found in the directory F igure
16. Winding the rectangle at t = igure
17. Winding the rectangle at t = . dbn upper bound/arb in the github repository [20].8.5. Proof of claim (b).
Fix t = .
2, and let R denote the rectangle R (cid:66) { x + iy : 0 . ≤ y ≤ x ≥ X − . N ≤ N } . We wish to show that the holomorphic function H t ( x + iy ) / B t ( x + iy ) does not vanish in thisrectangle. We would like to establish this by the argument principle, however this turns out tobe di ffi cult to accomplish due to the oscillation in H t ( x + iy ) / B t ( x + iy ) ≈ f t ( x + iy ) indicated by PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 51 the heuristic (79). To damp out this oscillation, we introduce an “Euler mollifier” E t , ( x + iy ) (cid:66) (cid:89) p ≤ − b tp p s ∗ , where s ∗ is given by (17) the choice to use the first three primes 2 , , t , x , y . We have the upperbound | E t , ( x + iy ) | ≤ (cid:89) p ≤ + b tp p Re s ∗ . When N ≥ N , 0 . ≤ y ≤ t = .
2, one easily verifies from (21) thatRe s ∗ ≥ . | E t , ( x + iy ) | ≤ . E t , ( x + iy ) H t ( x + iy ) B t ( x + iy ) = E t , ( x + iy ) f t ( x + iy ) + O ≤ (2 . × − ) . The left-hand side remains holomorphic in x + iy (even though f t ( x + iy ) has jump discontinuities).Thus, by the argument principle, it will su ffi ce to show that the left-hand side avoids the negativereal axis ( −∞ ,
0] as x + iy traverses the boundary ∂ R of the rectangle R . In other words, it willsu ffi ce to show that(93) dist( E t , ( x + iy ) f t ( x + iy ) , ( −∞ , > . × − for x + iy ∈ ∂ R . For future reference we also observe that for any x + iy ∈ R , the magnitude of E t , ( x + iy ) can be bounded below by(94) | E t , ( x + iy ) | ≥ (cid:89) p ≤ − b tp p Re s ∗ ≥ . | arg( E t , ( x + iy )) | ≤ (cid:88) p ≤ sin − b tp p Re s ∗ ≤ . . Of the four sides of the rectangle ∂ R , the required estimate (93) will hold with significantroom to spare, and we can proceed using rather crude estimates. We first attend to the right edgeof ∂ R , in which N = N . From (88) we have f t ( x + iy ) = + O ≤ (0 . f t ( x + iy ) has argument of magnitude at most sin − . ≤ . E t , ( x + iy ) f t ( x + iy ) has magnitude at least 0 . .
823 in magnitude, giving the claim (93) on this side from elementarytrigonometry. In the literature one also sees other choices of mollifier than this Euler product used, for instance to controlthe extreme values of Dirichlet polynomials; however our numerical experimentations with alternative mollifiers to f t ( x + iy ) turned out to give inferior results for our application. Now we attend to the left edge of ∂ R , in which x = X − . . ≤ y ≤
1. From thecalculations for part (a) (see in particular Figure 17) one can verify that (for instance) | f t ( x + iy ) | ≥ | arg f t ( x + iy ) | ≤ π in this region. Combining this with (94), (95), we conclude that E t , ( x + iy ) f t ( x + iy ) has magnitude at least 0 .
534 and argument at most 2 .
13 in magnitude, againgiving the claim (93) on this side from elementary trigonometry.Now we attend to the upper edge of ∂ R , in which N ≤ N ≤ N and y =
1. From (21) onenow has Re s ∗ ≥ . . In particular N (cid:88) n = b tn n s ∗ = + O ≤ N (cid:88) n = b tn n . = + O ≤ (0 . . Meanwhile, from (20) one has | γ | ≤ e . (cid:18) x π (cid:19) − / ≤ . N − and from (22) one has | κ | ≤ ty x − ≤ × − and hence | γ | n y n s ∗ + κ = O ≤ (cid:32) . Nn . (cid:33) and γ N (cid:88) n = n y b tn n s ∗ + κ = O ≤ N (cid:88) n = . b tn N n . + N (cid:88) n = N + . b tn n . = O ≤ (0 . f t ( x + iy ) = + O ≤ (0 . . In particular f t ( x + iy ) has magnitude at least 0 . .
928 in magnitude, hence E t ( x + iy ) f t ( x + iy ) has magnitude at least 0 . . ∂ R , in which N ≤ N ≤ N and y = .
2. This isby far the most delicate side of the rectangle for the purposes of verifying (93). We will splitthe range [ N , N ] into a number of subintervals [ N − , N + ] and obtain a uniform lower bound fordist( E t , ( x + iy ) f t ( x + iy ) , ( −∞ , N is in one of these subintervals [ N − , N + ].Fix [ N − , N + ] ⊂ [ N , N ], suppose that N ∈ [ N − , N + ], and write s ∗ = σ + iT . We first deal withthe κ term in the definition of f t ( x + iy ) by writing n − κ = + O ≤ ( n | κ | − f t ( x + iy ) = N (cid:88) n = b tn n σ + iT + γ N (cid:88) n = n y b tn n σ − iT + O ≤ | γ | N (cid:88) n = n y b tn n σ ( n | κ | − . The sum (cid:80) N n = b tn n . can be numerically computed directly, but one could also use Lemma 8.2 (using for instance N = PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 53
In particular, using (20) to bound | γ | n y ≤ e . y ( x N / π n ) − y / ≤ e . (cid:32) ( N −
180 ) / n (cid:33) − y / ≤ . N / n ) − . ≤ . n / N − ) − . . we have E t , ( x + iy ) f t ( x + iy ) = E t , ( x + iy ) N (cid:88) n = b tn n σ + iT + O ≤ . | E t , ( x + iy ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = ( n / N − ) . b tn n σ − iT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ≤ ( Z )where Z (cid:66) . N (cid:88) n = ( n / N ) . b tn n σ ( n | κ | − . We can write E t , ( x + iy ) = (cid:88) d | D λ d d σ + iT where D (cid:66) × × λ d (cid:66) (cid:89) p | d ( − b tp ) . As a consequence, we have E t , ( x + iy ) N (cid:88) n = b tn n σ + iT = DN (cid:88) n = β n n σ + iT where β n (cid:66) (cid:88) d | n , D λ d b tn / d . Thus for instance β = β p = p = , ,
5, so that the Dirichlet series (cid:80)
DNn = β n n σ + iT isexpected to experience less oscillation than the series (cid:80) Nn = b tn n σ + iT .The product of E t , ( x + iy ) and (cid:80) Nn = ( n / N − ) . b tn n σ − iT is not favorable due to the negative sign inthe σ − iT exponent. But since | z || w | = | zw | , we have1 . | E t , ( x + iy ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = ( n / N − ) . b tn n σ − iT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E t , ( x + iy ) N (cid:88) n = ( n / N − ) . b tn n σ + iT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) DN (cid:88) n = N − . − α n n σ + iT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where α n (cid:66) . (cid:88) d | n , D λ d ( n / d ) . b tn / d . Note that the coe ffi cients α n , β n are both real. We now have(97) E t , ( x + iy ) f t ( x + iy ) = DN (cid:88) n = β n n σ + iT + O ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) DN (cid:88) n = N − . − α n n σ + iT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ≤ ( Z ) . A naive application of the triangle inequality (using β =
1) would give the lower bound(98) dist( E t , ( x + iy ) f t ( x + iy ) , ( −∞ , ≥ − N − . − α − DN (cid:88) n = | β n | + | N − . − α n | n σ − Z . As it turns out, this bound is not quite strong enough to be satisfactory for the numerical rangesof parameters we need. To do better we need to exploit the fact that when the sum (cid:80)
DNn = β n n σ + iT exhibits significant cancellation, then the sum (cid:80) DNn = N − . α n n σ + iT will also. The key tool here is Lemma 8.5 (Improved triangle inequality) . We have dist( E t , ( x + iy ) f t ( x + iy ) , ( −∞ , ≥ − N − . − α − DN (cid:88) n = max( | β n − N − . − α n | , − N − . − α + N − . − α | β n + N − . − α n | ) n σ − Z . Proof.
Write Y (cid:66) DN (cid:88) n = max( | β n − N − . − α n | , − N − . − α + N − . − α | β n + N − . − α n | ) n σ . We may assume that N − . − α + Y <
1, otherwise the claim is trivial. By (97) and convexity, itsu ffi ces to show that dist( DN (cid:88) n = β n + e i θ N − . − α n n σ + iT , ( −∞ , ≥ − N − . − α − Y for all phases θ ∈ R . We may write DN (cid:88) n = β n + e i θ N − . − α n n σ + iT = + e i θ N − . − α + O ≤ DN (cid:88) n = | β n + e i θ N − . − α n | n σ = (1 + e i θ N − . − α ) + O ≤ DN (cid:88) n = | β n + e i θ N − . − α n | / | + e i θ N − . − α | n σ . By the cosine rule, we have (cid:16) | β n + e i θ N − . − α n | / | + e i θ N − . − α | (cid:17) = β n + N − . − α n + N − . − α n β n cos θ + N − . − α + N − . − α cos θ . This is a fractional linear function of cos θ with no poles in the range [ − ,
1] of cos θ . Thus thisfunction is monotone on this range and attains its maximum at either cos θ = + θ = − | β n + e i θ N − . − α n || + e i θ N − . − α | ≤ max (cid:32) | β n − N − . − α n | − N − . − α , | β n + N − . − α n | + N − . − α (cid:33) PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 55 and thus DN (cid:88) n = | β n + e i θ N − . − α n | / | + e i θ N − . − α | n σ ≤ − N − . − α Y . We conclude from the triangle inequality that DN (cid:88) n = β n + e i θ N − . − α n n σ + iT = + e i θ N − . − α + O ≤ (cid:32) | + e i θ N − . − α | − N − . − α Y (cid:33) . By further application of the triangle inequalitydist( DN (cid:88) n = β n + e i θ N − . − α n n σ + iT , ( −∞ , ≥ dist(1 + e i θ N − . − α , ( −∞ , − | + e i θ N − . − α | − N − . − α Y = | + e i θ N − . − α | − | + e i θ N − . − α | − N − . − α Y = | + e i θ N − . − α | (cid:32) − Y − N − . − α (cid:33) ≥ (1 − N − . − α ) (cid:32) − Y − N − . − α (cid:33) = − N − . − α − Y as desired, where we have used the fact that 1 + e i θ N − . − α lies to the right of the imaginary axis(so that the closest element of ( −∞ ,
0] is the origin). (cid:3)
Bounding DN ≤ DN + , we see that in order to establish (93) in the range N ∈ [ N − , N + ], itsu ffi ces to verify the inequality1 − N − . − α − DN + (cid:88) n = max( | β n − N − . − α n | , − N − . − α + N − . − α | β n + N − . α n | ) n σ − Z ≥ . × − . From (21), (80) one has σ ≥ + y + t x N π − t x N − y + y (1 + y ) x N + = . +
120 log x N π − x N . + . x N = . +
120 log( N −
180 ) − x N . + . x N ≥ σ N − where σ N − (cid:66) . +
110 log N − while from (22), (80) one has | κ | ≤ ty x N − ≤ . x N − ≤ × −
136 D.H.J. POLYMATH and hence it su ffi ces to show that F N − , N + − Z N − , N ≥ . × − where F N − , N + (cid:66) − N − . − α − DN + (cid:88) n = max( | β n − N − . − α n | , − N − . − α + N − . − α | β n + N − . − α n | ) n σ N − and Z N − , N (cid:66) . N (cid:88) n = ( n / N ) . b tn n σ N − ( n × − − . Since σ N − ≥ . Z N − , N ≤ . N (cid:88) n = b tn n . ( n × − − ≤ − so it will su ffi ce to show that F N − , N + ≥ . × − for a collection of intervals [ N − , N + ] covering [ N , N ]. This can be done by ad hoc numericalexperimentation; for instance, one can calculate that F , × = . . . . F × , . × = . . . . F . × , . × = . . . . F . × , . × = . . . . .
9. A symptotic results
In this section we use the e ff ective estimates from Theorem 1.3 to obtain asymptotic informa-tion about the function H t , which improves (and makes more e ff ective) the results of Ki, Kim,and Lee [10], by establishing Theorem 1.5.We begin with Proposition 9.1 (Preliminary asymptotics) . Let < t ≤ / , x ≥ , and − ≤ y ≤ . (i) If x ≥ exp( Ct ) for a su ffi ciently large absolute constant C, thenH t ( x + iy ) = (1 + O ( x − ct )) M t (cid:32) + y − ix (cid:33) + (1 + O ( x − ct )) M t (cid:32) − y + ix (cid:33) for an absolute constant c > , where M t is defined in (10) . (ii) If instead we have ≤ y ≤ and x ≥ C for a su ffi ciently large absolute constant C, thenH t ( x + iy ) = (1 + O ≤ (0 . M t (cid:32) + y − ix (cid:33) . (iii) If x = x + O (1) for some x ≥ , thenH t ( x + iy ) = O (cid:32) x O (1)0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M t (cid:32) + y − ix (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) = O (cid:18) x O (1)0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M t (cid:18) ix (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 57
Proof.
We begin with (i). Since H t = H ∗ t and M t = M ∗ t , we may assume without loss ofgenerality that y ≥
0. Using (16), (11) we may write the desired estimate as H t ( x + iy ) B t ( x + iy ) = + O ( x − ct ) + γ. We apply Theorem 1.3. (Strictly speaking, the estimates there required y ≤ y ≤ H t ( x + iy ) B t ( x + iy ) = N (cid:88) n = b tn n s ∗ + γ N (cid:88) n = n y b tn n s ∗ + κ + O ≤ (cid:0) e A + e B + e C , (cid:1) where γ = O ( x − y / ) κ = O ( x − )Re s ∗ ≥ + y + t x π − O ( x − ) e A = O x − y / N (cid:88) n = b tn n − − y − t log x π − O ( x − ) log xx e B = O N (cid:88) n = b tn n − + y − t log x π + O ( x − ) log xx e C , = O (cid:18) x − + y (cid:19) Since N = O ( x / ), we have x − y / n y = O (1) and n O ( x − ) = O (1) for all 1 ≤ n ≤ N . We concludethat H t ( x + iy ) B t ( x + iy ) = + γ + O log xx + N (cid:88) n = b tn n + y + t log x π + x − + y so it will su ffi ce (for c small enough) to show that N (cid:88) n = b tn n + y + t log x π = O ( x − ct ) . By (15) we can write the left-hand side as N (cid:88) n = n + y + t log x π √ n = O ( x − ct ) . For 2 ≤ n ≤ N , we have 1 + y + t x π √ n ≥ ct log x for some absolute constant c >
0. By the integral test, the left-hand side is then bounded by12 ct log x + (cid:90) ∞ u ct log x du which, for x ≥ exp( C / t ) and C large, is bounded by O (2 − ct log x ). The claim then follows afteradjusting c appropriately.Now we prove (ii). As before we have the expansion (99). We have γ N (cid:88) n = n y b tn n s ∗ + κ = O x − y / N (cid:88) n = b tn n − y + t log x π = O x − y / N (cid:88) n = n y − = O ( x − y − );similar arguments give e A = O ( log xx x − y ), while e B = O log xx N (cid:88) n = b tn n − + y − t log x π = O log xx N (cid:88) n = n − = O (cid:32) log xx (cid:33) . We conclude that H t ( x + yi ) B t ( x + yi ) = N (cid:88) n = b tn n s ∗ + O ( x − y − ) = + O ≤ N (cid:88) n = n − + y − t log x π − O ( x − ) + O ( x − y − ) = + O ≤ N (cid:88) n = n − + O ( x − / ) = + O ≤ (cid:32) π − (cid:33) + O ( x − / ) = + O ≤ (0 . x ≥ C for C large enough.Finally, we prove (iii). Again our starting point is (99). The right-hand side can be boundedcrudely by O ( x O (1) ) = O ( x O (1)0 ), hence H t ( x + iy ) = O (cid:32) x O (1)0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M t (cid:32) + y + ix (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) . However, from (10), (6), (9) it is not hard to see that the log-derivative of M t ( s ) is of size O (log x ) in the region s = ix + O (1). Thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M t (cid:32) + y + ix (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) x O (1)0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M t (cid:18) ix (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 59 giving the claim. (cid:3)
To understand the behavior of M t ( x + iy ), we make the following simple observations: Lemma 9.2 (Behavior of M t ) . Let < t ≤ / , let x ∗ > be su ffi ciently large, and let x + iy = x ∗ + O (1) . ThenM t (cid:32) + y + ix (cid:33) = M t (cid:32) + ix ∗ (cid:33) exp (cid:32) ( i ( x − x ∗ ) + y ) (cid:32)
14 log x ∗ π + π i (cid:33) + O (cid:32) log x ∗ x ∗ (cid:33)(cid:33) . Also, there is a continuous branch of arg M t (cid:16) + ix ∗ (cid:17) for all large real x ∗ such that arg M t (cid:32) + ix ∗ (cid:33) = t π
16 log x ∗ π + π + x ∗ x ∗ π − x ∗ + O ( log x ∗ x ∗ ) . Proof.
By (10), (8), the log-derivative of M t is given by(100) M (cid:48) t M t = α + t αα (cid:48) . For s = ix ∗ + O (1), we have from (9) that(101) α ( s ) =
12 log x ∗ π + π i + O (cid:32) x ∗ (cid:33) and from this and (43) we conclude that M (cid:48) t ( s ) M t ( s ) =
12 log x ∗ π + π i + O (cid:32) log x ∗ x ∗ (cid:33) whenever s = ix ∗ + O (1). The first claim then follows by applying the fundamental theorem ofcalculus to a branch of log M t .For the second claim, we calculatearg M t (cid:32) + ix ∗ (cid:33) = t α (cid:32) + ix ∗ (cid:33) ) + π − x ∗ π + Im (cid:32) − + ix ∗ + ix ∗ − + ix ∗ (cid:33) = t (cid:32) π x ∗ π + O ( log x ∗ x ∗ ) (cid:33) + π − x ∗ π + Im (cid:32) − + ix ∗ (cid:32) log x ∗ + i π − ix ∗ + O (cid:32) x ∗ (cid:33)(cid:33)(cid:33) − x ∗ = t π
16 log x ∗ π + π − x ∗ π − x ∗ + x ∗ x ∗ − π + O (cid:32) log x ∗ x ∗ (cid:33) = t π
16 log x ∗ π + π + x ∗ x ∗ π − x ∗ + O (cid:32) log x ∗ x ∗ (cid:33) as desired. (cid:3) Now we can prove Theorem 1.5. We begin with (ii). Let n ≥ exp( Ct ), and suppose that x + iy = x n + O (1). By Proposition 9.1(i) and Lemma 9.2 we have H t ( x + iy ) = M t (cid:32) + ix n (cid:33) exp (cid:32) ( − i ( x − x n ) + y ) (cid:32)
14 log x n π − π i (cid:33) + O ( x − ctn ) (cid:33) + M t (cid:32) + ix n (cid:33) exp (cid:32) ( i ( x − x n ) − y ) (cid:32)
14 log x n π + π i (cid:33) + O ( x − ctn ) (cid:33) . (102) From Lemma 9.2 and (26) one hasarg M t (cid:32) + ix n (cid:33) = − π + O (cid:32) log x n x n (cid:33) mod π and hence(103) M t (cid:32) + ix n (cid:33) = − exp (cid:32) O ( log x n x n ) (cid:33) M t (cid:32) + ix n (cid:33) . If we now make the further assumption y = O (cid:16) x n (cid:17) , we can thus simplify the above approxi-mation as H t ( x + iy ) = − M t (cid:32) + ix n (cid:33) e − π ( x − x n ) / exp (cid:32) ( − i ( x − x n ) + y ) 14 log x n π + O ( | y | log x n + x − ctn ) (cid:33) + M t (cid:32) + ix n (cid:33) e − π ( x − x n ) / exp (cid:32) ( i ( x − x n ) − y ) 14 log x n π + O ( | y | log x n + x − ctn ) (cid:33) = iM t (cid:32) + ix n (cid:33) e − π ( x − x n ) / (cid:18) sin (cid:18) x + iy − x n x n π (cid:19) + O ( | y | log x n + x − ctn ) (cid:19) . (104)In particular, if x + iy traverses the circle { x n + c log n e i θ : 0 ≤ θ ≤ π } once anti-clockwise and c is small enough, the quantity H t ( x + iy ) will wind exactly once around the origin, and hence bythe argument principle there is precisely one zero of H t inside this circle. As the zeroes of H t are symmetric around the real axis, this zero must be real. This proves (ii).Now we prove (i). Suppose that H t ( x + iy ) = x ≥ exp( Ct ). We can assume | y | ≤ | y | > n be a natural number that minimises | x − x n | , then x = x n + O (cid:16) x n (cid:17) since the derivativeof the left-hand side of (26) in x n is comparable to log x n . From (102) we have0 = M t (cid:32) + ix n (cid:33) exp (cid:32) ( − i ( x − x n ) + y ) (cid:32)
14 log x n π − π i (cid:33) + O ( x − ctn ) (cid:33) + M t ( 1 + ix n (cid:32) ( i ( x − x n ) − y ) (cid:32)
14 log x n π + π i (cid:33) + O ( x − ctn ) (cid:33) . Thus both summands on the right-hand side have the same magnitude, which on taking loga-rithms and canceling like terms implies that y
14 log x n π + O ( x − ctn ) = − y
14 log x n π + O ( x − ctn )and hence y = O (cid:18) x − ctn log x n (cid:19) . We can now apply (104) to conclude thatsin (cid:18) x + iy − x n x n π (cid:19) + O ( x − ctn ) = | x − x n | is minimal) forces x − x n = O (cid:18) x − ctn log x n (cid:19) .This gives the claim.Next, we prove (iii). In view of parts (i) and (ii), and adjusting C if necessary, we may assumethat X takes the form X = x n + c log x n for some n ≥ exp( Ct ). By the argument principle, N t ( X ) is PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 61 equal to − π times the variation in the argument of H t on the boundary of the rectangle { x + iy :0 ≤ x ≤ X ; − ≤ y ≤ } traversed clockwise, since there are no zeroes with imaginary part ofmagnitude greater than one. By compactness, the variation on the left edge { iy : − ≤ y ≤ } is O (1), and similarly for any fixed portion { x + i : 0 ≤ x ≤ C } of the upper edge. FromProposition 9.1 (and (102)), we see that the variation of H t ( x + iy ) / M t ( + y − ix ) on the remainingupper edge { x + i : C ≤ x ≤ X } and on the top half { X + iy : 0 ≤ y ≤ } of the right edge areboth equal to O (1). Since H t = H ∗ t , the variation on the lower half of the rectangle is equal tothat of the upper half. We thus conclude that N t ( X ) = − π arg M t (cid:32) − iX (cid:33) + O (1)where we use a continuous branch of the argument of M t (cid:16) − iX (cid:17) that is bounded at 3 i . The claimnow follows from Lemma 9.2 (using M t = M ∗ t to work with + iX instead of − iX ).Finally, we prove (iv). From the Hadamard factorization theorem as in the proof of Proposi-tion 3.1 we have(105) H (cid:48) t ( z ) H t ( z ) = (cid:88) n > (cid:32) z − z n + z + z n (cid:33) where the zeroes of H t are indexed in pairs ± z n . Setting z = X + i , we see from Proposi-tion 9.1 and the generalized Cauchy integral formula that the logarithmic derivative of H t ( x + iy ) / M t (cid:16) + y − ix (cid:17) is equal to O (1) at X + i for all su ffi ciently large X , and hence for all X by sym-metry and compactness. On the other hand, from Stirling’s formula (or the logarithmic growth ofthe digamma function) one easily verifies that the logarithmic derivative of M t (cid:16) + y − ix (cid:17) is equalto O (log(2 + X )) at X + i . Hence H (cid:48) t ( X + i ) H t ( X + i ) = O (log(2 + X )). Taking imaginary parts, we concludethat (cid:88) n > − − y n ( X − x n ) + (4 − y n ) − + y n ( X + x n ) + (4 + y n ) = O (log(2 + X ))where we write z n = x n + iy n ; equivalently one has (cid:88) n (4 − y n )( X − x n ) + (4 − y n ) = O (log(2 + X ))where the sum now ranges over all zeroes, including any at the origin. Since | y n | ≤
1, everyzero in [ X , X +
1] makes a contribution of at least (say). As the summands are all positive,the first part of claim (iv) follows. To prove the second part, we may assume by compactnessthat x ≥ C . Repeating the proof of (iii), and reduce to showing that the variation of arg H t onthe short vertical interval { X + iy : 0 ≤ y ≤ } is O (log X ). If we let θ be a phase such that e i θ H t ( X + i ) is real and positive, we see that this variation is at most π ( m + m is thenumber of zeroes of Re ( e i θ H t ( X + yi )) for 0 ≤ y ≤
3, since every increment of π in arg e i θ H t must be accompanied by at least one such zero. As H t = H ∗ t , this is also the number of zeroesof e i θ H t ( X + yi ) + e − i θ H t (2 X − ( X + yi )). On the other hand, from Proposition 9.1(ii), (iii) andJensen’s formula we see that the number of such zeroes is O (log X ), and the claim follows. Remark 9.3.
Theorem 1.5 gives good control on H t ( x + iy ) whenever x ≥ exp( C / t ) . As aconsequence (and assuming for sake of argument that the Riemann hypothesis holds), then forany Λ > , the bound Λ ≤ Λ should be numerically verifiable in time O (exp( O (1 / Λ ))) , by applying the arguments of previous sections with t and y set equal to small multiples of Λ . Weleave the details to the interested reader. Remark 9.4.
Our discussion here will be informal. In view of the results of [9] , it is expectedthat the zeroes z j ( t ) of H t ( x + iy ) should evolve according to the system of ordinary di ff erentialequations ddt z k ( t ) = (cid:48) (cid:88) j (cid:44) k z k ( t ) − z j ( t ) where the sum is evaluated in a suitable principal value sense, and one avoids those times wherethe zero z k ( t ) fails to be simple; see [9, Lemma 2.4] for a verification of this in the regime t > Λ .In view of the Riemann-von Mangoldt formula (as well as variants such Corollary 1.5, it isexpected that the number of zeroes in any region of the form { x + iy : x + iy = x ∗ + O (1) } forlarge x ∗ should be of the order of log x ∗ . As a consequence, we expect a typical zero z k ( t ) tomove with speed O (log | z k ( t ) | ) , although one may occasionally move much faster than this if twozeroes are exceptionally close together, or less than this if the zeroes are close to being evenlyspaced. As a consequence, if the Riemann hypothesis fails and there is a zero x + iy of H withy comparable to , it should take time comparable to x for this zero to move towards the realaxis, leading to the heuristic lower bound Λ (cid:29) x . Thus, in order to obtain an upper bound Λ ≤ Λ , it will probably be necessary to verify that there are no zeroes x + iy of H with ycomparable to and | x | ≤ c log Λ for some small absolute constant c > . This suggests that thetime complexity bound in Remark 9.3 is likely to be best possible (unless one is able to prove theRiemann hypothesis, of course).In [9, Lemma 2.1] it is also shown that the velocity of a given zero z ( t ) is given by the formuladdt z ( t ) = H (cid:48)(cid:48) t ( z ( t )) H (cid:48) t ( z ( t )) assuming that the zero is simple. By using the asymptotics in Proposition 9.1 and Corollary 1.5together with the generalized Cauchy integral formula to then obtain asymptotics for H (cid:48) t andH (cid:48)(cid:48) t , it is possible to show that for the zeroes x ( t ) that are real and larger than exp( C / t ) , andmove leftwards with velocity ddt x ( t ) = − π + O ( x − ct ); we leave the details to the interested reader.
10. F urther numerical results
By Theorem 1.5, one can verify the second hypothesis of Theorem 1.2 when X ≥ exp( C / t )for a large constant C . If we ignore for sake of discussion the third hypothesis of Theorem 1.2(which turns out to be relatively easy to verify numerically in practice), this suggests that onecan obtain a bound of the form Λ ≤ O ( t ) provided that one can verify the Riemann hypothesisup to a height exp( C / t ). In other words, if one has numerically verified the Riemann hypothesisup to a large height T , this should soon lead to a bound of the form Λ ≤ O (cid:16) T (cid:17) .Aside from improving the implied constant in this bound, it does not seem easy to improvethis sort of implication without a major breakthrough on the Riemann hypothesis (such as amassive expansion of the known zero-free regions for the zeta function inside the critical strip).We shall justify this claim heuristically as follows. Suppose that there was a counterexample to PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 63 F igure
18. Real zeros moving up leftwards and getting ‘solidified’.the Riemann hypothesis at a large height T , so that H (2 T + iy ) = y , whichfor this discussion we will take to be comparable to 1. The Riemann von Mangoldt formulaindicates that the number of zeroes of H within a bounded distance of this zero should becomparable to log T ; the majority of these zeroes should obey the Riemann hypothesis and thusstay at roughly unit distance from our initial zero 2 T + iy . Proposition 3.1 then suggests thatas time t advances, this zero should move at speed comparable to log T . Thus one should notexpect this zero to reach the real axis until a time comparable to T . This heuristic analysistherefore indicates that it is unlikely that one can significantly improve the bound Λ ≤ O (cid:16) T (cid:17) without being able to exclude significant violations of the Riemann hypothesis at height T .The table below collects some numerical results verifying the second two hypotheses of The-orem 1.2 for larger values of X , and smaller values of t , y , than were considered in Section 8.This leads to improvements to the bound Λ ≤ .
22 conditional on the assumption that the Rie-mann Hypothesis can be numerically verified beyond the height T ≈ . × used in Section8. For instance, the final row of the table implies that one has the bound Λ ≤ . T ≈ . × . Note that this is broadlyconsistent with the previous heuristic that the upper bound on Λ is proportional to T .The selection of parameters in this table proceeded as follows. One first located parameters t , y , N (with the quantity Λ = t + y as small as possible) for which one could obtain agood lower bound for f t ( x + iy ) when x = N ; we arbitrarily chose a target lower bound of F igure
19. Actual trajectories of some real and complex zeros.T able
1. Conditional Λ Results
X t y Λ Winding Number N | f t ( x + iy ) | lower bound2 × + × + × + × + × + × + × + × + × + × + × + × + | f t ( x + iy ) | ≥ .
03 to provide an adequate safety margin. From (96) one had(106) f t ( x + iy ) = N (cid:88) n = β n + O ≤ ( | γ || N (cid:88) n = α n | ) + O ≤ | γ | N (cid:88) n = n y b t n n σ ( n | κ | − PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 65 where β n (cid:66) b t n n σ + iT and α n (cid:66) n y b t n n σ + iT . The final term on the right-hand side of (106) can be estimated as in Section 8.5 and is negligiblein practice. To control the other two terms, we use the following lemma (which roughly speakingcorresponds to a simplified version of the “Euler 2 mollifier” version of the “Euler 5 mollifier”analysis in Section 8.5):
Lemma 10.1.
Let α , . . . , α N be complex numbers, and let β be a number such that whenever ≤ n ≤ N is even, β α n / lies on the line segment { θα n : 0 ≤ θ ≤ } connecting 0 with α n . Thenwe have the lower bound | − β | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = α n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | − (1 − | β | ) N (cid:88) n = | α n | − | β | (cid:88) N / < n ≤ N | α n | and the upper bound | − β | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = α n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 − | β | ) N (cid:88) n = | α n | + | β | (cid:88) N / < n ≤ N | α n | . Proof.
The quantity | − β || (cid:80) Nn = α n | can be written as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = (1 n ≤ N α n − | n α n / β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By the triangle inequality, this is bounded above by N (cid:88) n = | n ≤ N α n − | n α n / β | and below by 2 | α | − N (cid:88) n = | n ≤ N α n − | n α n / β | . We have N (cid:88) n = | n ≤ N α n − | n α n / β | = N (cid:88) n = | α n | − | n | α n / || β | + N (cid:88) n = N + | n | α n / || β | which we can rearrange as (1 − | β | ) N (cid:88) n = | α n | + | β | (cid:88) N / < n ≤ N | α n | and the claim follows. (cid:3) Using this lemma to lower bound | (cid:80) Nn = β n | and upper bound | (cid:80) Nn = α n | , and then using thetriangle inequality, yields a lower bound on | f t ( x + iy ) | when N = N . These quantities can bereadily computed for many values of t , y , N , leading to an envelope for Λ and x ≈ π N thatis depicted in Figure 20.F igure
20. The envelope of potential choices of x , Λ . Note the approximateinverse relationship between Λ and x .By working in intervals N ∈ [ N − , N + ] for some finite number of intervals [ N − , N + ] covering[ N , N ] for some large N as in Section 8.5, and then using a crude triangle inequality bound for N ≥ N as in Section 8.3, we thus (in view of the conservative safety margin in our lower boundsfor | f t ( x + iy ) | ) expect to be able to verify the hypothesis in Theorem 1.2(iii) for any choiceof parameters t , y , N as above. The main remaining di ffi culty is then to verify the barrierhypothesis (Theorem 1.2(ii)). This is by far the most numerically intensive step, and we proceedas in Section 8.4, after using the ad hoc procedure in Section 8.1 to select X . The graphs infigure 21 illustrate that for increasing x , the number of xy-rectangles to be evaluated within thebarrier, as well as the number of mesh points required per rectangle (measured at t = PPER BOUND FOR DE BRUIJN-NEWMAN CONSTANT 67 F igure
21. The left graph shows how the number mesh points of the xy-rectangle at t = x for each barrier. The graph on the rightdoes the same, but now for the total number of xy-rectangles that need to beevaluated per barrier.used (to be able to compute it within a reasonable time). Checks where made before each formalrun to assure the target accuracy would be achieved.The computations for X = × + X = × + anthgrid.com/dbnupperbound .11. C onflict of interest statement There are no conflicts of interest for this paper.R eferences [1] D. P. Anderson,
BOINC: A System for Public-Resource Computing and Storage , In GRID ’04: Proceedings ofthe Fifth IEEE / ACM International Workshop on Grid Computing (2004), 4–10.[2] J. Arias de Reyna,
High-precision computation of Riemann’s zeta function by the Riemann-Siegel asymptoticformula, I , Mathematics of Computation, (2011), 995–1009.[3] J. Arias de Reyna, J. Van de lune, On the exact location of the non-trivial zeroes of Riemann’s zeta function ,Acta Arith. (2014), 215–245.[4] W. G. C. Boyd,
Gamma Function Asymptotics by an Extension of the Method of Steepest Descents , Proceedings:Mathematical and Physical Sciences, Vol. 447, No. 1931 (Dec. 8, 1994), 609–630.[5] N. C. de Bruijn,
The roots of trigonometric integrals , Duke J. Math. (1950), 197–226.[6] G. Csordas, T. S. Norfolk, R. S. Varga, A lower bound for the de Bruijn-Newman constant Λ , Numer. Math. (1988), 483–497.[7] G. Csordas, A. M. Odlyzko, W. Smith, R. S. Varga, A new Lehmer pair of zeros and a new lower bound for theDe Bruijn-Newman constant Lambda , Electronic Transactions on Numerical Analysis. (1993), 104–111.[8] G. Csordas, A. Ruttan, R.S. Varga, The Laguerre inequalities with applications to a problem associated withthe Riemann hypothesis , Numer. Algorithms, (1991), 305–329.[9] G. Csordas, W. Smith, R. S. Varga, Lehmer pairs of zeros, the de Bruijn-Newman constant Λ , and the Riemannhypothesis , Constr. Approx. (1994), no. 1, 107–129.[10] H. Ki, Y. O. Kim, and J. Lee, On the de Bruijn-Newman constant , Advances in Mathematics, (2009), 281–306.[11] D. H. Lehmer, On the roots of the Riemann zeta-function , Acta Math. (1956), 291–298. [12] H. L. Montgomery, The pair correlation of zeros of the zeta function , Analytic number theory (Proc. Sympos.Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), 181–193. Amer. Math. Soc., Providence, R.I.,1973.[13] H. L. Montgomery, R. C. Vaughan, Multiplicative number theory. I. Classical theory. Cambridge Studies inAdvanced Mathematics, 97. Cambridge University Press, Cambridge, 2007.[14] C. M. Newman,
Fourier transforms with only real zeroes , Proc. Amer. Math. Soc. (1976), 246–251.[15] T. S. Norfolk, A. Ruttan, R. S. Varga, A lower bound for the de Bruijn-Newman constant Λ II. , in A. A. Goncharand E. B. Sa ff , editors, Progress in Approximation Theory, 403–418. Springer-Verlag, 1992.[16] A. M. Odlyzko, An improved bound for the de Bruijn-Newman constant , Numerical Algorithms (2000),293–303.[17] T. K. Petersen, Eulerian numbers. With a foreword by Richard Stanley. Birkh¨auser Advanced Texts: BaslerLehrb¨ucher. Birkh¨auser / Springer, New York, 2015.[18] D. J. Platt,
Isolating some non-trivial zeros of zeta , Math. Comp. 86 (2017), 2449–2467.[19] G. Polya, ¨Uber trigonometrische Integrale mit nur reelen Nullstellen, J. Reine Angew. Math. (1927), 6–18.[20] D. H. J. Polymath, github.com/km-git-acc/dbn upper bound [21] D. H. J. Polymath, Zeroes of the heat flow evolution of the Riemann ξ func-tion at negative times: numerical experiments and heuristic justifications , github.com/km-git-acc/dbn upper bound/blob/master/Writeup/Sharkfin/sharkfin.pdf [22] B. Rodgers, T. Tao, The De Bruijn-Newman constant is nonnegative , preprint. arXiv:1801.05914 [23] Y. Saouter, X. Gourdon, P. Demichel,
An improved lower bound for the de Bruijn-Newman constant , Mathe-matics of Computation. (2011), 2281–2287.[24] J. Stopple, Notes on Low discriminants and the generalized Newman conjecture , Funct. Approx. Comment.Math., vol. 51, no. 1 (2014), 23–41.[25] J. Stopple,
Lehmer pairs revisited , Exp. Math. (2017), no. 1, 45–53.[26] H. J. J. te Riele, A new lower bound for the de Bruijn-Newman constant , Numer. Math., (1991), 661–667.[27] E. C. Titchmarsh, The Theory of the Riemann Zeta-function, Second ed. (revised by D. R. Heath-Brown),Oxford University Press, Oxford, 1986.(1991), 661–667.[27] E. C. Titchmarsh, The Theory of the Riemann Zeta-function, Second ed. (revised by D. R. Heath-Brown),Oxford University Press, Oxford, 1986.