Accurate estimation of sums over zeros of the Riemann zeta-function
AAccurate estimation of sums over zeros of theRiemann zeta-function ∗ Richard P. Brent † , David J. Platt ‡ and Timothy S. Trudgian § September 30, 2020
Abstract
We consider sums of the form (cid:80) φ ( γ ), where φ is a given function,and γ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation ofsuch sums can be accelerated by a simple device, and give examplesinvolving both convergent and divergent infinite sums. Let the nontrivial zeros of the Riemann zeta-function ζ ( s ) be denoted by ρ = β + iγ . In order of increasing height, the ordinates of the zeros in theupper half-plane are γ ≈ . < γ < γ < · · · .Let φ : [ T , ∞ ) (cid:55)→ [0 , ∞ ) be a non-negative function on the interval[ T , ∞ ), for some T (cid:62)
1. Throughout this paper we assume that φ ( t ) istwice continuously differentiable and satisfies the conditions φ (cid:48) ( t ) (cid:54) φ (cid:48)(cid:48) ( t ) (cid:62) T , ∞ ). These conditions imply that φ ( t ) is convex on [ T , ∞ ].We are interested in sums of the form (cid:80) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) and (cid:80) (cid:48) T (cid:54) γ φ ( γ ) , where T (cid:54) T (cid:54) T . Here the prime symbol ( (cid:48) ) indicates that if γ = T or γ = T then the term φ ( γ ) is given weight . If multiple zeros exist, thenterms involving such zeros are weighted by their multiplicities. Sums of thisform can be bounded using a lemma of Lehman [9, Lem. 1] that we statefor reference. We have changed Lehman’s wording slightly, but the proof isthe same. In the lemma and elsewhere, ϑ denotes a real number in [ − , ∗ . Primary 11M06; Secondary 11M26. † Australian National University, Canberra, Australia
0. Lemma 3 is stated and proved in §
3. For simplicity we outline here the case T → ∞ , since this case has onefewer parameter and is of interest in many applications.If the infinite sum (cid:80) (cid:48) T (cid:54) γ φ ( γ ) converges, then the error term in Lemma 1is (cid:29) φ ( T ) log T . In Theorem 1 we express the error as − φ ( T ) Q ( T ) + E ( T ),where Q ( T ) (cid:28) log T can be computed from (4)–(5), and E ( T ) is generallyof lower order than φ ( T ) log T . We state Theorem 1 here; the proof is givenin §
4. Note that the lower bound on T is 2 π , not 2 πe as in Lehman’s lemma.This is convenient in applications because 2 π < γ < πe . Theorem 1.
Suppose that π (cid:54) T (cid:54) T and (cid:82) ∞ T φ ( t ) log( t/ π ) dt < ∞ . Let E ( T ) := (cid:88) (cid:48) T (cid:54) γ φ ( γ ) − π (cid:90) ∞ T φ ( t ) log( t/ π ) dt . (1) Then E ( T ) = − φ ( T ) Q ( T ) + E ( T ) , where E ( T ) = − (cid:90) ∞ T φ (cid:48) ( t ) Q ( t ) dt , (2) and Q ( T ) = N ( T ) − L ( T ) is defined by (4) – (5) . Also, | E ( T ) | (cid:54) A + A log T ) | φ (cid:48) ( T ) | + ( A + A ) φ ( T ) /T. (3)Here A and A are constants satisfying condition (10) below, and A isa small constant which, from Lemma 2, we can take as A = 1 / E ( T ) is a continuous function of T , as can be seen from (2), whereas E ( T ) has jumps at the ordinates of nontrivial zeros of ζ ( s ).Disregarding the constant factors, Theorem 1 shows that E ( T ) (cid:28) | φ (cid:48) ( T ) | log T + φ ( T ) /T. In Lemma 1, A is a constant such that | Q ( T ) | (cid:54) A log T for all T (cid:62) πe , where Q ( T )is as in (4). From [3, Cor. 1], we may take A = 0 . φ ( t ) = t − c for some c >
1, then E ( T ) (cid:28) T − c log T , and E ( T ) (cid:28) T − ( c +1) log T is smaller by a factor of order T .As well as convergent sums, we also consider certain divergent sums.Theorem 2 shows that, if (cid:82) ∞ T t − φ ( t ) dt < ∞ , then there exists F ( T ) := lim T →∞ (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) TT φ ( t ) log( t/ π ) dt . In Theorem 3 we consider approximating F ( T ) by computing a finite sum(over γ (cid:54) T ), with error term E ( T ) the same as in Theorem 1.For example, if φ ( t ) = 1 /t and T = 2 π , we have E ( T ) (cid:28) T − log T and E ( T ) (cid:28) T − log T . The latter bound allows us to obtain an accurateapproximation to the constant H = F (2 π ) that can equally well be defined,in analogy to Euler’s constant, by H := lim T →∞ (cid:88) <γ (cid:54) T γ − π log ( T / π ) . This example is considered in detail in [4], where it is shown that H = − . ϑ (10 − ) . The motivation for this paper was an attempt to generalise the results of [4].In § Q ( t ) − S ( t ). Lemma 3in § § § The
Riemann-Siegel theta function θ ( t ) is defined for real t by θ ( t ) := arg Γ (cid:18)
14 + it (cid:19) − t π, see for example [6, § θ ( t ) is continuouson R , and θ (0) = 0. 3et F denote the set of positive ordinates of zeros of ζ ( s ). FollowingTitchmarsh [11, § § < T (cid:54)∈ F , then we let N ( T ) denote thenumber of zeros β + iγ of ζ ( s ) with 0 < γ (cid:54) T , and S ( T ) denote the value of π − arg ζ ( + iT ) obtained by continuous variation along the straight linesjoining 2, 2 + iT , and + iT , starting with the value 0. If 0 < T ∈ F , wetake S ( T ) = lim δ → [ S ( T − δ ) + S ( T + δ )] /
2, and similarly for N ( T ). Thisconvention is the reason why we consider sums of the form (cid:80) (cid:48) T (cid:54) γ (cid:54) T φ ( t )instead of (cid:80) T (cid:54) γ (cid:54) T φ ( t ).By [11, Thm. 9.3], we have N ( T ) = L ( T ) + Q ( T ) , (4) L ( T ) = T π (cid:18) log (cid:18) T π (cid:19) − (cid:19) + 78 , and (5) S ( T ) = Q ( T ) + O (1 /T ) . (6)From [11, Thm. 9.4]), S ( T ) (cid:28) log T . Thus, from (6), Q ( T ) (cid:28) log T .Trudgian [13, Cor. 1] gives the explicit bound | Q ( T ) − S ( T ) | (cid:54) . /T for all T (cid:62) e . In Lemma 2 we obtain a sharper constant, assuming that T (cid:62) π . The result of Lemma 2 is close to optimal, since the proof showsthat the constant 150 could at best be replaced by 48 π ≈ . Lemma 2. If Q ( t ) and S ( t ) are defined as above then, for all t (cid:62) π , | Q ( t ) − S ( t ) | (cid:54) t . Proof.
We shall assume that t (cid:54)∈ F , since otherwise the result follows bycontinuity of Q ( t ) − S ( t ). The Riemann-von Mangoldt formula states, in itsmost precise form, N ( t ) = θ ( t ) /π + 1 + S ( t ) . From (4), this implies that Q ( t ) − S ( t ) = θ ( t ) π + 1 − L ( t ) . Now θ ( t ) has a well-known asymptotic expansion [7, Satz 4.2.3(c)] θ ( t ) ∼ t (cid:18) log (cid:18) t π (cid:19) − (cid:19) − π (cid:88) j (cid:62) (1 − − j ) | B j | j (2 j − t j − , (7)where B = , B = − , . . . are Bernoulli numbers. Thus, using (5), Q ( t ) − S ( t ) has an asymptotic expansion Q ( t ) − S ( t ) ∼ π (cid:88) j (cid:62) (1 − − j ) | B j | j (2 j − t j − . (8)4n order to give an explicit bound on Q ( t ) − S ( t ), we use an explicit boundon the error incurred by taking the first k terms (cid:101) T j ( t ), j = 1 , . . . , k in (7).From [2, (47)], for all t >
0, this error is | (cid:101) R k +1 ( t ) | < (1 − − k ) − ( πk ) / (cid:101) T k ( t ) + e − πt . (9)Substituting the expression for (cid:101) T k ( t ) into (9) gives a bound | (cid:101) R k +1 ( t ) | π < | B k | πk ) / (2 k − t k − + e − πt π for the error incurred by taking the first k terms in (8). Thus, for all k (cid:62) t > Q ( t ) − S ( t ) = 1 π k (cid:88) j =1 (1 − − j ) | B j | j (2 j − t j − + ϑ | B k | πk ) / (2 k − t k − + ϑe − πt π . Taking k = 3 and using the assumption t (cid:62) π , we obtain the result.Define S ( T ) := (cid:82) T S ( t ) dt . We know that S ( T ) (cid:28) log T , and that S ( T ) = o (log T ) if and only if the Lindel¨of Hypothesis is true — see Titch-marsh [11, Thm. 9.9(A), Thm. 13.6(B), and Note 13.8].Explicit bounds on S ( T ) are known [6, 12, 14, 15]. From [12, Thm 2.2], | S ( T ) − c | (cid:54) A + A log T for all T (cid:62) π, (10)where c = S (168 π ), A = 2 . A = 0 . T ∈ [2 π, π ]. Hence, from nowon we assume that T (cid:62) π and that (10) holds for T (cid:62) T . In this section we prove Lemma 3, which may be seen as a refinement ofLemma 1 if the conditions φ (cid:48) ( t ) (cid:54) φ (cid:48)(cid:48) ( t ) (cid:62) (cid:82) T T φ (cid:48) ( t ) Q ( t ) dt is bounded.From the discussion in §
2, we may assume that the constants A , A , A occurring in Lemma 3 are A = 2 . A = 0 . A = 1 / < . emma 3. If π (cid:54) T (cid:54) T (cid:54) T and E ( T , T ) := (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) T T φ ( t ) log( t/ π ) dt , then E ( T , T ) = φ ( T ) Q ( T ) − φ ( T ) Q ( T ) + E ( T , T ) , where E ( T , T ) = − (cid:90) T T φ (cid:48) ( t ) Q ( t ) dt , (11) and | E ( T , T ) | (cid:54) A + A log T ) | φ (cid:48) ( T ) | + ( A + A ) φ ( T ) /T . (12) Proof.
Assume initially that T (cid:54)∈ F , T (cid:54)∈ F . Using Stieltjes integrals, wesee that (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) = (cid:90) T T φ ( t ) dN ( t ) = (cid:90) T T φ ( t ) dL ( t ) + (cid:90) T T φ ( t ) dQ ( t )= 12 π (cid:90) T T φ ( t ) log( t/ π ) dt + (cid:90) T T φ ( t ) dQ ( t ) , so E ( T , T ) = (cid:90) T T φ ( t ) dQ ( t ) = (cid:20) φ ( t ) Q ( t ) − (cid:90) φ (cid:48) ( t ) Q ( t ) dt (cid:21) T T = φ ( T ) Q ( T ) − φ ( T ) Q ( T ) − (cid:90) T T φ (cid:48) ( t ) Q ( t ) dt . (13)This proves (11). To prove (12), note that, from (6) and Lemma 2, (cid:90) T T φ (cid:48) ( t ) Q ( t ) dt = (cid:90) T T φ (cid:48) ( t ) S ( t ) dt + ϑA (cid:90) T T φ (cid:48) ( t ) t dt, (14)and the last integral can be bounded using (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T T φ (cid:48) ( t ) t dt (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) T (cid:90) T T | φ (cid:48) ( t ) | dt = φ ( T ) − φ ( T ) T (cid:54) φ ( T ) T . (15)Also, (cid:90) T T φ (cid:48) ( t ) S ( t ) dt = (cid:20) φ (cid:48) ( t )( S ( t ) − c ) − (cid:90) φ (cid:48)(cid:48) ( t )( S ( t ) − c ) dt (cid:21) T T = φ (cid:48) ( T )( S ( T ) − c ) − φ (cid:48) ( T )( S ( T ) − c ) − (cid:90) T T φ (cid:48)(cid:48) ( t )( S ( t ) − c ) dt. φ (cid:48) ( t ) (cid:54) | S ( t ) − c | (cid:54) A + A log t , we have | φ (cid:48) ( t )( S ( t ) − c ) | (cid:54) − ( A + A log t ) φ (cid:48) ( t )for t = T , T . Thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T T φ (cid:48) ( t ) S ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) − (cid:88) j =1 ( A + A log T j ) φ (cid:48) ( T j ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T T φ (cid:48)(cid:48) ( t )( S ( t ) − c ) dt (cid:12)(cid:12)(cid:12)(cid:12) . (16)Also, using φ (cid:48)(cid:48) ( t ) (cid:62)
0, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T T φ (cid:48)(cid:48) ( t )( S ( t ) − c ) dt (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) A (cid:90) T T φ (cid:48)(cid:48) ( t ) dt + A (cid:90) T T φ (cid:48)(cid:48) ( t ) log t dt = A ( φ (cid:48) ( T ) − φ (cid:48) ( T )) + A (cid:20) φ (cid:48) ( t ) log t − (cid:90) φ (cid:48) ( t ) t dt (cid:21) T T = ( A + A log T ) φ (cid:48) ( T ) − ( A + A log T ) φ (cid:48) ( T ) − A (cid:90) T T φ (cid:48) ( t ) t dt. (17)Inserting (17) in (16) and simplifying, terms involving T cancel, giving (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T T φ (cid:48) ( t ) S ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) − A + A log T ) φ (cid:48) ( T ) − A (cid:90) T T φ (cid:48) ( t ) t dt. (18)Combining (11) with (14), (15), and (18), gives (12). Finally, we notethat (11)–(12) hold even if T ∈ F and/or T ∈ F , because of the waythat we defined N ( T ) (and hence Q ( T ) = N ( T ) − L ( T )) for T ∈ F . Remark 1.
With the assumptions and notation of Lemma 3, Lemma 1gives the bound | E ( T , T ) | (cid:54) A (cid:18) φ ( T ) log T + (cid:90) T T φ ( t ) t dt (cid:19) . (19)Our bound (12) on E ( T , T ) is often smaller than the bound (19) on E ( T , T ). We can take advantage of this if the terms φ ( T j ) Q ( T j ) ( j = 1 , §§ Convergent sums
In this section we assume that (cid:80) T (cid:54) γ φ ( γ ) < ∞ , or equivalently (given ourconditions on φ ), that (cid:82) ∞ T φ ( t ) log( t/ π ) dt < ∞ . We first state an easylemma, and then prove Theorem 1.
Lemma 4.
Suppose that π (cid:54) T (cid:54) T and (cid:82) ∞ T φ ( t ) log( t/ π ) dt < ∞ . Then φ ( t ) log t = o (1) as t → ∞ , (20) φ (cid:48) ( t ) log t = o (1) as t → ∞ , and (21) (cid:82) ∞ T | φ (cid:48) ( t ) | log t dt < ∞ . (22) Proof.
For u (cid:62) T , (cid:90) u +1 u φ ( t ) log( t/ π ) dt (cid:62) φ ( u + 1) log( u/ π ) . Thus φ ( u + 1) log( u/ π ) = o (1) as u → ∞ , and φ ( t ) log(( t − / π ) = o (1).Since log(( t − / π ) ∼ log t , (20) follows.For (21), we have φ ( u ) (cid:62) φ ( u ) − φ ( u + 1) = (cid:90) u +1 u | φ (cid:48) ( t ) | dt (cid:62) | φ (cid:48) ( u + 1) | , (23)so (20) implies that φ (cid:48) ( u + 1) log u = o (1). Taking t = u + 1, we have φ (cid:48) ( t ) log( t −
1) = o (1). Since log( t − ∼ log t , (21) follows.Finally, from (23), | φ (cid:48) ( t ) | (cid:54) φ ( t −
1) for t (cid:62) T + 1, so (cid:90) ∞ T +1 | φ (cid:48) ( t ) | log t dt (cid:54) (cid:90) ∞ T +1 φ ( t −
1) log t dt (cid:28) (cid:90) ∞ T φ ( t ) log( t/ π ) dt < ∞ . and (22) follows. Proof of Theorem . We have φ ( t ) log t = o (1) by Lemma 4 and convergenceof the integral in (1). Also, from Lemma 4 we have (cid:82) ∞ T | φ (cid:48) ( t ) | log t dt < ∞ , but Q ( t ) (cid:28) log t , so (cid:82) ∞ T φ (cid:48) ( t ) Q ( t ) dt converges absolutely. Now, Lemma 3gives (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) T T φ ( t ) log( t/ π ) dt = φ ( T ) Q ( T ) − φ ( T ) Q ( T ) − (cid:90) T T φ (cid:48) ( t ) Q ( t ) dt. (24)8f we let T → ∞ in (24), φ ( T ) Q ( T ) → (cid:82) T T φ (cid:48) ( t ) Q ( t ) dt tends to afinite limit. Thus, the right side of (24) tends to a finite limit, and the leftside must tend to the same limit. This gives (cid:88) (cid:48) T (cid:54) γ φ ( γ ) − π (cid:90) ∞ T φ ( t ) log( t/ π ) dt = − φ ( T ) Q ( T ) − (cid:90) ∞ T φ (cid:48) ( t ) Q ( t ) dt. We have proved (1)–(2) of Theorem 1. The bound (3) follows by observingthat the bound (12) of Lemma 3 is independent of T , so (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ T φ (cid:48) ( t ) Q ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) A + A log T ) | φ (cid:48) ( T ) | + ( A + A ) φ ( T ) /T. This completes the proof of Theorem 1.
Example 1.
We consider computation of the constant c := (cid:88) γ> γ = 0 . . . . . The approximation 0 . φ ( t ) = 1 /t in Lemma 1 gives an error term | E ( T ) | (cid:54) A (cid:32) + 2 log TT (cid:33) = 0 .
14 + 0 .
56 log TT , using the value A = 0 .
28 mentioned above. The corresponding error termgiven by Theorem 1 is | E ( T ) | (cid:54) (4 A + A + A ) + 4 A log TT (cid:54) .
334 + 0 .
236 log TT , using the values of A , A , A above. For example, taking T = 1000 (cor-responding to the first 649 nontrivial zeros), we get | E ( T ) | (cid:54) . × − and | E ( T ) | (cid:54) . × − , an improvement by a factor of 400. If we use10 zeros, as in Corollary 1, the improvement is by a factor of 3 × . Corollary 1.
We have c = (cid:88) γ> γ = 0 . ϑ (5 × − ) . roof. This follows from Theorem 1 by an interval-arithmetic computationusing the first n = 10 zeros, with T = 3293531632 . · · · ∈ ( γ n , γ n +1 ). Remark 2.
Assuming the Riemann Hypothesis (RH), there is an equivalentexpression: c = d log ζ ( s ) /ds | s =1 / π G − , (25)where G = β (2) is Catalan’s constant 0 . · · · . This enables us toconfirm Corollary 1 without summing over any zeros of ζ ( s ), but assumingRH. It is only rarely that such a closed form is known. One other exampleis the following — see, e.g., [5, Ch. 12]. Assuming RH, we have (cid:88) γ> γ + = (cid:88) ρ (cid:60) (cid:18) ρ (cid:19) = 1 + C − log 4 π . . . . , where C = 0 . . . . is Euler’s constant. In this section we give two theorems that apply, subject to a mild condi-tion (30) on φ ( t ), even if (cid:80) T (cid:54) γ φ ( γ ) diverges. Theorem 2 shows the existenceof a limit for the difference between a sum and the corresponding integral.Theorem 3 shows how we can accurately approximate the limit.First we prove two lemmas that strengthen the first and third parts ofLemma 4. In Lemma 5, f is non-increasing but need not be differentiable. Lemma 5.
Suppose that, for some T (cid:62) , f : [ T, ∞ ] (cid:55)→ [0 , ∞ ) is non-negative and non-increasing on [ T, ∞ ) . If (cid:90) ∞ T f ( t ) t dt < ∞ , (26) then f ( t ) log t = o (1) .Proof. Assume, by way of contradiction, that f ( t ) log t (cid:54) = o (1). Thus, thereexists a constant c > t n ) n (cid:62) suchthat t > T and f n := f ( t n ) (cid:62) c log t n . (27) The formula (25) is stated in [8, (21)] and is proved in [1, p. 13]. An almost indeci-pherable sketch of this result may be found in Riemann’s Nachlass. t n ) n (cid:62) if necessary, we can assumethat t n +1 (cid:62) t n for all n (cid:62)
1. Thuslog (cid:18) t n +1 t n (cid:19) (cid:62) log t n +1 . (28)Since f ( t ) is non-increasing, we have f ( t ) (cid:62) f n +1 on [ t n , t n +1 ], and (cid:90) t n +1 t n f ( t ) t dt (cid:62) (cid:90) t n +1 t n f n +1 t dt = f n +1 log (cid:18) t n +1 t n (cid:19) . Using (27)–(28), this gives (cid:90) t n +1 t f ( t ) t dt (cid:62) n (cid:88) k =1 f k +1 log t k +1 (cid:62) c n (cid:88) k =1 cn → ∞ . This contradicts the condition (26). Thus, our assumption is false, and wemust have f ( t ) log t = o (1). Lemma 6. If (cid:82) ∞ T φ ( t ) t dt < ∞ , then (cid:82) ∞ T φ (cid:48) ( t ) log t dt is absolutely convergent.Proof. For T (cid:62) T we have (cid:90) TT φ (cid:48) ( t ) log t dt = φ ( T ) log T − φ ( T ) log T − (cid:90) TT φ ( t ) t dt. (29)As T → ∞ in (29), the term φ ( T ) log T → φ (cid:48) ( t ) log t (cid:54) T , ∞ ), the integral is absolutely convergent. Theorem 2.
Suppose that T (cid:62) π , and (cid:90) ∞ T φ ( t ) t dt < ∞ . (30) Then there exists F ( T ) := lim T →∞ (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) TT φ ( t ) log( t/ π ) dt , and F ( T ) = − φ ( T ) Q ( T ) − (cid:90) ∞ T φ (cid:48) ( t ) Q ( t ) dt. (31)11 roof. Suppose that T (cid:62) T . Applying Lemma 3, we have (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) TT φ ( t ) log( t/ π ) dt = φ ( T ) Q ( T ) − φ ( T ) Q ( T ) − (cid:90) TT φ (cid:48) ( t ) Q ( t ) dt. (32)Let T → ∞ in (32). On the right-hand side, φ ( T ) Q ( T ) → Q ( t ) (cid:28) log t .Thus the left-hand side tends to a finite limit F ( T ). This gives (31).The identity (31) is not convenient for accurately approximating F ( T )when T is small, because (cid:82) ∞ T φ (cid:48) ( t ) Q ( t ) dt is not necessarily small. In Theo-rem 3 we use a finite sum (over γ (cid:54) T ) and integral to approximate F ( T ).Theorem 3 has the same expression for the error term E as Theorem 1,essentially because the bounds in both theorems are proved using Lemma 3. Theorem 3.
Suppose that π (cid:54) T (cid:54) T and φ ( t ) satisfies (30) . Let F ( T ) := lim T →∞ (cid:32) (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) TT φ ( t ) log( t/ π ) dt (cid:33) . Then F ( T ) = (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) T T φ ( t ) log( t/ π ) dt − φ ( T ) Q ( T ) + E ( T ) , where E ( T ) = − (cid:82) ∞ T φ (cid:48) ( t ) Q ( t ) dt , and | E ( T ) | (cid:54) A + A log T ) | φ (cid:48) ( T ) | + ( A + A ) φ ( T ) /T . Proof.
We note that, from Theorem 2, the limit defining F ( T ) exists. Also,from Lemmas 5–6, φ ( T ) Q ( T ) = o (1) and (cid:82) ∞ T φ (cid:48) ( t ) Q ( t ) dt < ∞ . Thus, usingLemma 3 as in the proof of Theorem 1, we see thatlim T →∞ (cid:32) (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) T T φ ( t ) log( t/ π ) dt (cid:33) = − φ ( T ) Q ( T ) − (cid:90) ∞ T φ (cid:48) ( t ) Q ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ T φ (cid:48) ( t ) Q ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) A + A log T ) | φ (cid:48) ( T ) | + ( A + A ) φ ( T ) /T . Since F ( T ) = lim T →∞ (cid:32) (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) T T φ ( t ) log( t/ π ) dt (cid:33) + (cid:88) (cid:48) T (cid:54) γ (cid:54) T φ ( γ ) − π (cid:90) T T φ ( t ) log( t/ π ) dt, the result follows. Example 2.
To illustrate the divergent case, we consider the example φ ( t ) = 1 / (log( t/ π )) . The constant 2 π here is unimportant, but this choicesimplifies some of the expressions below.From Lemma 1, the asymptotic behaviour of (cid:80) <γ (cid:54) T φ ( γ ) is given by12 π (cid:90) Tc φ ( t ) log( t/ π ) dt = li( T / π ) − li( c/ π ) ∼ T π log T , where c (cid:62) πe is an arbitrary constant, and li( x ) is the logarithmic integral,defined in the usual way by a principal value integral. This motivates thedefinition of a constant c by c := lim T →∞ (cid:32) (cid:88) (cid:48) <γ (cid:54) T φ ( γ ) − li( T / π ) (cid:33) , (33)where the limit exists by Theorem 2.If we use (33) to estimate c then, by Theorem 3, the error is E ( T ) = − φ ( T ) Q ( T ) + O ( | φ (cid:48) ( T ) | log T ) + O ( φ ( T ) /T ) (cid:28) T .
Convergence is so slow that it is difficult to obtain more than two correctdecimal digits. On the other hand, if we estimate c using the approximation (cid:88) (cid:48) <γ (cid:54) T φ ( γ ) − li( T / π ) − φ ( T ) Q ( T ) (34)suggested by Theorem 3, then the error is E ( T ) (cid:28) ( T log T ) − , smallerby a factor of order T log T . More precisely, from Theorem 3 we have | E ( T ) | (cid:54) A + A log T ) T log ( T / π ) + A + A T log ( T / π ) (cid:54) .
302 log(
T / π ) + 8 . T log ( T / π ) . (35)13 orollary 2. If c is defined by (33) , then c = − . ϑ (10 − ) . Proof.
Using the first n = 10 nontrivial zeros with T ≈ ( γ n + γ n +1 ) / c obtained from (33) and (34) by summing over the first n nontrivialzeros, and the error bound (35), with T = ( γ n + γ n +1 ) /
2. The first incorrectdigit in each entry is underlined. n estimate via (33) estimate via (34) | E | bound (35)10 -0.49986259 -0.52733908 1 . × − -0.54054724 -0.52767238 8 . × − -0.52244974 -0.52767173 4 . × − -0.53117846 -0.52766980 2 . × − -0.53026260 -0.52766977 1 . × − Table 1: Numerical estimation of c . Acknowledgements
We are indebted to Juan Arias de Reyna for information on the identity (25),and for his translation of the relevant page from Riemann’s Nachlass. DJP issupported by ARC Grant DP160100932 and EPSRC Grant EP/K034383/1;TST is supported by ARC Grants DP160100932 and FT160100094.
References [1] J. Arias de Reyna, 130802-report, unpublished, 2 August 2013. Avail-able from the author via https://personal.us.es/arias/ .[2] R. P. Brent, On asymptotic approximations to the log-Gamma andRiemann-Siegel theta functions,
J. Austral. Math. Soc. :319–337,2019.[3] R. P. Brent, D. J. Platt, and T. S. Trudgian,
The mean squareof the error term in the prime number theorem , submitted. AlsoarXiv:2008.06140, 13 Aug. 2020.144] R. P. Brent, D. J. Platt, and T. S. Trudgian, A harmonic sum over theordinates of nontrivial zeros of the Riemann zeta-function,
Bull. Aust.Math. Soc. , to appear. Also arXiv:2009.05251, 11 Sept. 2020.[5] H. Davenport,
Multiplicative Number Theory , 3rd ed., Grad. Texts inMath., vol. 74, Springer, New York, 2000.[6] H. M. Edwards,
Riemann’s Zeta Function , Academic Press, New York,1974.[7] W. Gabcke,
Neue Herleitung und Explizite Restabsch¨atzung derRiemann-Siegel-Formel , Dissertation, Mathematisch-Naturwissen-schaftlichen, G¨ottingen, 1979. Online version (revised 2015), availablefrom http://ediss.uni-goettingen.de/ .[8] J. Guillera,
Some sums over the non-trivial zeros of the Riemann zetafunction , arXiv:1307.5723v7, 19 June 2014.[9] R. S. Lehman, On the difference π ( x ) − li( x ), Acta Arith. :397–410,1966.[10] Y. Saouter, T. Trudgian, and P. Demichel, A still sharper region where π ( x ) − li( x ) is positive, Math. Comp. (295):2433–2446, 2015.[11] E. C. Titchmarsh, The Theory of the Riemann Zeta-function , 2nd ed.(edited and with a preface by D. R. Heath-Brown), Oxford, 1986.[12] T. S. Trudgian, Improvements to Turing’s method,
Math. Comp. (276):2259–2279, 2011.[13] T. S. Trudgian, An improved upper bound for the argument of the Rie-mann zeta-function on the critical line II, J. Number Theory :280–292, 2014.[14] T. S. Trudgian, Improvements to Turing’s method II,
Rocky MountainJ. Math. :325–332, 2016.[15] A. M. Turing, Some calculations of the Riemann zeta-function, Proc.Lond. Math. Soc.3