Pair correlation for Dedekind zeta functions of abelian extensions
aa r X i v : . [ m a t h . N T ] A ug PAIR CORRELATION FOR DEDEKIND ZETA FUNCTIONSOF ABELIAN EXTENSIONS
DAVID DE LAAT, LARRY ROLEN, ZACK TRIPP, IAN WAGNER
Abstract.
Here we study problems related to the proportions of zeros, especially simpleand distinct zeros on the critical line, of Dedekind zeta functions. We obtain new boundson a counting function that measures the discrepancy of the zeta functions from having allzeros simple. In particular, for quadratic number fields, we deduce that more than 45% ofthe zeros are distinct. This extends work based on Montgomery’s pair correlation approachfor the Riemann zeta function. Our optimization problems can be interpreted as interpolantsbetween the pair correlation bound for the Riemann zeta function and the Cohn-Elkies spherepacking bound in dimension 1. We compute the bounds through optimization over Schwartzfunctions using semidefinite programming and also show how semidefinite programming canbe used to optimize over functions with bounded support. Introduction
In this paper, we study the zeros of Dedekind zeta functions on the critical line. To motivatethe results and setup our main questions, we first recall the Riemann zeta function, defined as ζ ( s ) := P ∞ n =1 n − s for Re( s ) >
1. The Riemann Hypothesis (RH) states that the non-trivialzeros of ζ ( s ) are all on the line Re( s ) = . Furthermore, an important conjecture states thatthat all of its zeros are simple. Although this conjecture is far out of current reach, progresshas been made on bounding measures of the discrepancy of these zeros from being simple.As usual, we denote by N ( T ) the number of zeros ρ = β + iγ of ζ ( s ) in the critical strip(counting multiplicity) with 0 < γ ≤ T . That is, N ( T ) := X <γ ≤ T . We also define the counting functions N s ( T ) := X <γ ≤ Tm ρ =1 , N d ( T ) := X <γ ≤ T m ρ , N ∗ ( T ) := X <γ ≤ T m ρ , which count the number of simple zeros, distinct zeros, and multiplicities of zeros respectively,where m ρ is the multiplicity of the zero ρ = β + iγ . Of course, assuming the simplicityconjecture one would have N ( T ) ? = N s ( T ) ? = N d ( T ) ? = N ∗ ( T ) . Recently, Chirre, Gon¸calves, and the first author [10] used semidefinite programming toobtain improved estimates of these quantities. This provides a method to bootstrap theasymptotic on the pair correlation function of the zeros of ζ ( s ) using numerical optimization.The study of the pair correlation function of the (normalized) zeros of ζ ( s ) was pioneeredby Montgomery [25], which has subsequently led to deep conjectural insights connecting thezeros of ζ ( s ) to random matrix theory and has led to a broad framework of n -point correlationfunctions and random matrix model predictions for L -functions by [22], [23], [28], and [30], among many others. Utilizing Montgomery’s asymptotic, the following new bounds on N d and N ∗ were obtained in [10], conditional on RH: N d ( T ) ≥ (0 . o (1)) N ( T ) and N ∗ ( T ) ≤ (1 . o (1)) N ( T ) . This improved the previously known bounds of 0 . . L -functions instead of ζ ( s ). For example, [32] gives a lower boundof 0 . q -aspect of the entire family of Dirichlet L -functions being distinct.That paper makes use of the important Asymptotic Large Sieve, originating from the workof Conrey, Iwaniec, and Soundararajan (see [12, 13]).Given these results, it is natural to ask about analogous results for other zeta functions.Here we consider Dedekind zeta functions for abelian extensions, where we extend Mont-gomery’s pair correlation method to give bounds for families of Dedekind of zeta functions.To be precise, if K/ Q is a number field of degree n with ring of integers O K , the Dedekindzeta function is ζ K ( s ) := X a ⊆O K N ( a ) s , where the sum is over ideals a of O K and N ( a ) is the norm of a . For K = Q , ζ K ( s ) is simplythe Riemann zeta function, while if K = Q ( √ D ) is a quadratic extension, ζ K ( s ) factors as ζ K ( s ) = ζ ( s ) L ( χ, s ), where χ is the quadratic character of K , and where L ( χ, s ) := ∞ X n =1 χ ( n ) n − s for Re( s ) >
1. More generally, by class field theory, for any abelian number field K , ζ K ( s )factors as ζ K ( s ) = ζ ( s ) · Q n − i =1 L ( χ i , s ) for some Dirichlet characters χ i . For any finite extension K/ Q , the Grand Riemann Hypothesis (GRH) predicts that the zeros of ζ K ( s ) also lie on theline Re( s ) = . Finally, we let N K ( T ), N K,d ( T ), and N ∗ K ( T ) be defined exactly as N ( T ), N d ( T ), and N ∗ ( T ) above, but with the sums now being over the zeros of ζ K ( s ).In this paper, we will restrict our focus to the case where K/ Q is an abelian extension.Similarly to Montgomery [25], we define the pair correlation function of the zeros of ζ K ( s ) tobe F K ( α ) := (cid:18) nT π log T (cid:19) − X <γ,γ ′ ≤ T T iα ( γ − γ ′ ) w ( γ − γ ′ ) , where α and T ≥ w ( u ) = 4 / (4 + u ). Our first result to prove our main boundsis the following asymptotic formula. Theorem 1.1.
Let K be an abelian number field of degree n , and assume GRH. Then wehave F K ( α ) = ( n + o K (1)) T − | α | log T + | α | + o K (1) uniformly for | α | ≤ as T → ∞ . Using semidefinite programming, we are then able to obtain results analogous to thoseof [10] for all abelian extensions.
Theorem 1.2.
Assume GRH and the notation above. Then we have N ∗ K ( T ) ≤ ( c n + o K (1)) N K ( T ) , AIR CORRELATION FOR DEDEKIND ZETA FUNCTIONS OF ABELIAN EXTENSIONS 3 where c n = . for n = 2 , . for n = 3 , . for n = 4 , (1 + 10 − ) n + 0 . for n ≥ ,n + 1 / for n ≥ . The use of optimization techniques have proved to be useful elsewhere in number theory,such as for the study of prime gaps [5] and spacings between zeros of ζ ( s ) [4]. While wewere unable to find previous results on N ∗ K for general abelian extensions, work of Conrey,Ghosh, and Gonek [11] gave that on GRH, at least 1 /
27 of the zeros of ζ K ( s ) are simple forquadratic extensions K/ Q . As a corollary we get new results on the proportion of zeros thatare distinct. Corollary 1.3.
Assuming GRH, we have the following.i). If K/ Q is a degree 2 extension, then N K,d ( T ) ≥ (0 . o K (1)) N K ( T ) . ii). If K/ Q is a degree 3 extension, then N K,d ( T ) ≥ (0 . o K (1)) N K ( T ) . iii). If K/ Q is a degree 4 extension, then N K,d ( T ) ≥ (0 . o K (1)) N K ( T ) . Remark.
While there did not appear to be explicit results of this type in the literature previ-ously, bounds of [3], [10], and [11] could directly be combined to give an elementary estimateof about . for quadratic extensions. For higher degrees, our bounds on N ∗ K versus n failto produce anything new concerning distinct zeros. The paper is organized as follows. In Section 2.1 we establish a few basic definitionsand lemmas required for the proof of the pair correlation asymptotic of Theorem 1.1, andin Section 2.2 we give a general description of the semidefinite programming techniques andhow they can be used to obtain bounds on quantities in analytic number theory. We concludewith the proofs of the theorems in Section 3.
Acknowledgments
The authors thank Ken Ono for useful discussions related to this work.2.
Preliminaries
In this section, we review the basic definitions and notations for the proof of Theorem 1.1.Since almost every estimate depends on the field K in some way, we will drop it from thesubscript for our big- O estimates in this section and throughout the rest of the paper. DAVID DE LAAT, LARRY ROLEN, ZACK TRIPP, IAN WAGNER
Ingredients for the proof of Theroem 1.1.
For ζ K ( s ), the analogue of the Riemann ξ function is(2.1) ξ K ( s ) := 12 s ( s − | ∆ K | s π − ns (1 − s ) r Γ (cid:16) s (cid:17) r Γ( s ) r ζ K ( s ) , where r and r are the number of real embeddings and the number of pairs of complexembeddings of K respectively, yielding the relation r + 2 r = n , and ∆ K is the discriminantof K . As in the Riemann case, ξ K is entire, shares the same non-trivial zeros as ζ K , and hasthe functional equation ξ K ( s ) = ξ K (1 − s ) (see [27]). Because of this functional equation, wecall the region 0 < Re( s ) < critical strip . Using the functional equation and propertiesof the Γ function, it is easy to check that there are trivial zeros of ζ K ; namely, there are zerosof order r + r at negative even integers, of order r at negative odd integers, and of order r + r − K ( a ) = (cid:26) log N ( p ) if a = p k for p prime0 otherwise . Note that Λ Q = Λ is the classical von Mangoldt function. Similar to how the von Mangoldtfunction gives the coefficients of − ζ ′ /ζ , one can easily see that(2.2) − ζ ′ K ζ K ( s ) = X a Λ K ( a ) N ( a ) s , where the sum is over ideals a of O K . Using this fact, we can obtain the following formula. Lemma 2.1.
For x > and s = 1 , , − m, ρ , we have X ′ N ( a ) ≤ x Λ K ( a ) N ( a ) s = − ζ ′ K ζ K ( s ) + x − s − s − X ρ x ρ − s ρ − s + r ∞ X m =0 x − m − s m + s + r ∞ X m =0 x − m − s m + s − x − s s , where the sum is over the zeros ρ of the ζ K . In this lemma and throughout the rest of the paper, X ′ N ( a ) ≤ x will indicate that termswith N ( a ) = x are multiplied by 1 / ζ ( s )and L ( χ, s ), which can be found in [24] and [33] respectively. We will omit the proof since itis easily adapted from previous proofs. For example, by replacing J ( x, T ) by J ( x, s, T ) = 12 πi c + iT Z c − iT (cid:20) − ζ ′ K ( s ) ζ K ( s ) (cid:21) x z − s z − s dz for 0 < s < a K ( m ) := { a : N ( a ) = m } ≪ m , all of the details work essentially the same way as in Davenport, and byuniqueness of analytic continuation, the lemma then holds for the desired s . The only otherpiece of information we need for Davenport’s proof to work is that | ζ ′ K ( s ) /ζ K ( s ) |≪ log(2 | s | )for σ ≤ − ζ K ( s ). We will prove this now and use this version of the functionalequation for the proof of the next lemma as well.Using the functional equation ξ K ( s ) = ξ K (1 − s ) and the definition of ξ K given in (2.1), we AIR CORRELATION FOR DEDEKIND ZETA FUNCTIONS OF ABELIAN EXTENSIONS 5 can solve for ζ K (1 − s ) and see that(2.3) ζ K (1 − s ) = | ∆ K | s − π n ( − s ) (1 − s ) r Γ( s ) r Γ( s ) r Γ( − s ) r Γ(1 − s ) r ζ K ( s ) . From Γ( s )Γ( − s ) = π − − s cos sπ s ) and Γ( s )Γ(1 − s ) = Γ( s ) sin πsπ , (see Chapter 10 of [14]) we can write the logarithmic derivative of (2.3) as(2.4) − ζ ′ K ζ K (1 − s ) = O (1) − πr sπ πr cot πs + n Γ ′ Γ ( s ) + ζ ′ K ζ K ( s ) . For σ ≥ − σ ≤ − ζ ′ K ( s ) /ζ K ( s ) is bounded, and bounded awayfrom their poles (which occur at integer values), tan( πs/
2) and cot( πs ) are bounded as well.From Stirling’s asymptotic formula, we conclude that ζ ′ K ( s ) /ζ K ( s ) ≪ log(2 | s | ) as desired.From Lemma 2.1 and the (2.4), we will be able to prove the following lemma, which isanalogous to a lemma of Montgomery [25]. Lemma 2.2.
Assume GRH. For x ≥ and < σ < fixed, (2 σ − X γ x iγ ( σ − ) + ( t − γ ) = − x − X N ( a ) ≤ x Λ K ( a ) (cid:18) xN ( a ) (cid:19) − σ + it + X N ( a ) >x Λ K ( a ) (cid:18) xN ( a ) (cid:19) σ + it + x − σ + it ( n log τ + O σ (1)) + O σ ( x τ − ) + O σ ( x − τ − ) , where τ = | t | +2 and the sum is over the ordinates γ of the zeros of ζ K . Since the proof of this is also a direct adaptation of that of Montgomery’s, we will simplyrecall the outline of the proof. Subtract the formula of Lemma 2.1 at s = 1 − σ + it from thesame formula at s = σ + it , multiply both sides by x it , and use (2.2). This will yield all ofthe terms above except for the x − σ + it . This term comes from the nonsymmetric functionalequation (2.4), Stirling’s approximation again, and the fact that the remaining terms in (2.4)are bounded for a fixed 1 < σ < − ζ ′ K ( s ) /ζ K ( s ). From (2.2), it isclear that they are given by c K ( m ) := P N ( a )= m Λ K ( a ). We will use the Euler product for ζ K ( s )and for Dirichlet L -functions to come up with a more explicit formula for c K ( m ).The Eulerproduct for the Dedekind zeta function ζ K ( s ) = Y p (cid:18) − N ( p ) s (cid:19) − , where the product is over the prime ideals p of O K . If we assume that K/ Q is a Galoisextension, then we can rewrite this as a product over primes p of Z . If K/ Q is Galois, weknow that every prime p in Z has a factorization of the form p O K = p e . . . p eg , DAVID DE LAAT, LARRY ROLEN, ZACK TRIPP, IAN WAGNER where e ≥
1, the p i are distinct primes of norm p f in O K , and ef g = n . Using this, the Eulerproduct becomes ζ K ( s ) = Y p (1 − p − fs ) − g . Taking the logarithmic derivative of both sides and utilizing the Taylor series for log(1 − x )gives(2.5) ζ ′ K ζ K ( s ) = − X p ne log p ∞ X k =1 (cid:0) p kf (cid:1) − s . From this, we could explicitly write down the c K ( m ), but we can obtain a more usefuldescription in the abelian case. Recall that for an abelian extension K/ Q , we can write ζ K as a product of Dirichlet L -functions(2.6) ζ K ( s ) = n − Y i =0 L ( s, χ i ) , where χ i is a Dirichlet character of conductor q i , χ is the trivial character, and the productof the conductors is ∆ K . Each Dirichlet L -function also has an Euler product, and throughthe same method as above, we can obtain a Dirichlet series for its logarithmic derivative: L ′ ( s, χ i ) L ( s, χ i ) = − X p log p ∞ X k =1 χ i ( p k ) p − ks . (This can also be found in [14].) We can now use this equation when taking the logarithmicderivative of (2.6) to find an another expression for the Dirichlet series(2.7) ζ ′ K ζ K ( s ) = n − X i =0 L ′ ( s, χ i ) L ( s, χ i ) = − X p log p ∞ X k =1 n − X i =0 χ i ( p k ) ! p − ks . From this, we will write down the necessary information about c K ( m ) in two separate cases.When ( m, ∆ K ) = 1, we wish to show that(2.8) c K ( m ) = ( n Λ( m ) if χ ( m ) = χ ( m ) = · · · = χ n − ( m ) = 1 , . If m is not a prime power, it is clear from (2.5) that c K ( m ) = 0 = n Λ( m ), so suppose m = p k .If χ i ( p k ) = 1 for all i , (2.7) clearly gives a coefficient of n log p = n Λ( m ). On the other hand,it suffices to show that if c K ( p k ) = 0, then χ i ( p k ) = 1 for all i . Note that p being relativelyprime to ∆ K means p is unramified, i.e. e = 1. Therefore, (2.5) tells us that any non-zerocoefficient of p − ks must be n log p . But from (2.7), this can only occur if χ i ( p k ) = 1 for all i ,proving (2.8).On the other hand, when ( m, ∆ K ) >
1, we will only need the fact that(2.9) 0 ≤ c K ( m ) ≤ n Λ( m ) , which clearly follows from (2.5).Finally, the last things we will need are sum estimates similar to those used by Montgomery. AIR CORRELATION FOR DEDEKIND ZETA FUNCTIONS OF ABELIAN EXTENSIONS 7
Lemma 2.3.
Let ( a, q ) = 1 . Then X m ≤ xm ≡ a (mod q ) m Λ( m ) = 12 ϕ ( q ) x log x + O ( x ) for x ≥ q . Lemma 2.4.
Let ( a, q ) = 1 . Then X m ≤ xm ≡ a (mod q ) m Λ( m ) = O ( x log x ) . for x ≥ q . We will simply state that these estimates come from the prime number theorem on arith-metic progressions (see Chapter 20 of [14]), where in the following ϕ ( q ) denotes Euler’s totientfunction. Lemma 2.5.
For s > real and ( a, q ) = 1 , we have X m>xm ≡ a (mod q ) Λ( m ) m s = 1 ϕ ( q ) x − s s − O s,q (cid:16) x − s log x (cid:17) . Lemma 2.6.
For s > real and ( a, q ) = 1 , we have X m>xm ≡ a (mod q ) Λ( m ) m s = 1 ϕ ( q ) x − s log xs − O s,q (cid:0) x − s (cid:1) . It is easy to prove Lemma 2.6 from Lemma 2.5. The proof of Lemma 2.5 is similar tothe proof of the same sum without the congruence conditions; however, the explicit formulafor ψ ( x ) = P n ≤ x Λ( n ) is replaced by the explicit formulas for ψ ( x, χ ) = P n ≤ x Λ( n ) χ ( n ) for χ (mod q ), which can be found in [33] as mentioned before. While Yildirim’s formula onlyholds for primitive characters, one can easily keep track of the constants in the differencebetween L ( s, χ ) and L ( s, χ ) when χ is a primitive character inducing χ . Combining all ofthese formulas and using them as in the proof of the same sum without congruence conditions,one can deduced Lemma 2.5 and hence Lemma 2.6 as well.2.2. Overview of semidefinite programming techniques.
In this paper we considerlinear programming problems of the forminf { L ( f ) : f ∈ L ( R ) even and continuous , (2.10) f ( x ) ≤ | x | ≥ , ˆ f (0) = 1 , ˆ f ≥ } , where L is a linear functional. For L ( f ) = f (0) this is the Cohn-Elkies sphere packing boundin dimension 1 [6]. The difficulty in solving this optimization problem is that is that wesimultaneously consider constraints on f and its Fourier transform.To find numerical approximations of the optimal solution, Cohn and Elkies parameterizethe functions as(2.11) f ( x ) = p ( x ) e − πx , where p is a polynomial of given degree 2 d . One approach is to define f and ˆ f by their realroots (assuming there are sufficiently many, and that we know their degrees) and find good DAVID DE LAAT, LARRY ROLEN, ZACK TRIPP, IAN WAGNER locations for the roots, which works very well for sphere packing problems where the correctroot locations are known [8]. Here instead we will use semidefinite programming to optimizeover functions of the form (2.11) as was also done for the zeta function in [10].Let T be the linear operator so that ( T p )( x ) e − πx is the Fourier transform of p ( x ) e − πx .Since e − πx is nonnegative, the constraints in (2.10) reduce to the constraints p ( x ) ≤ x ≥ T ( p )( x ) ≥ x ≥
0, and T ( p )(0) = 1. As explained in [29], the first two constraintsare equivalent to the condition that there exist sum-of-square polynomials s , . . . , s of degreeat most 2 d such that p ( x ) = − s ( x ) − ( x − s ( x ) and T ( p )( x ) = s ( x ) + xs ( x ) . A polynomial s ( x ) of degree 2 d is a sum-of-squares polynomial if and only if it can bewritten as s ( x ) = v ( x ) T Qv ( x ), where v ( x ) is a vector whose entries form as basis of thepolynomials up to degree d , and where Q is a positive semidefinite matrix (a symmetricmatrix with nonnegative eigenvalues).This shows that if we restrict to functions of the form (2.11) and set s i ( x ) = v ( x ) T Q i v ( x ),then (2.10) reduces to(2.12) inf n L ( f ) : Q , . . . , Q (cid:23) , T ( p )( x ) = s ( x ) + xs ( x ) , s (0) = 1 o , where we use the notation Q , . . . , Q (cid:23) T ( p )( x ) = s ( x ) + xs ( x ) and s (0) = 1 are linear in the entries of thematrices. If we can also write L ( f ) as a linear functional in the entries of the matrices, then(2.12) is a semidefinite program, which can be solved numerically using an interior pointsolver such as SDPA-GMP [26].3. Proofs of the main results
Proof of Theorem 1.1.
We will first follow the outline of Montgomery’s proof of thepair correlation asymptotics that allow us to obtain the formula for F K ( α ) for 0 ≤ α < ≤ α ≤
1, which will be necessary for the proof ofTheorem 1.2.Letting σ = 3 / L ( x, t ) and R ( x, t ) be the left-hand and right-hand sides of the equation respectively, we wish to estimate T R | L ( x, t ) | dt and T R | R ( x, t ) | dt toobtain the desired asymptotic. First, define F K ( x, T ) := X <γ,γ ′ ≤ T x i ( γ − γ ′ ) w ( γ − γ ′ )for x ≥ T ≥ F K ( α ) = (cid:18) nT π log T (cid:19) − F ( T α , T ) . Using residue calculus, one can see that ∞ Z −∞ dt (1 + ( t − γ ) )(1 + ( t − γ ′ ) ) = π w ( γ − γ ′ ) , AIR CORRELATION FOR DEDEKIND ZETA FUNCTIONS OF ABELIAN EXTENSIONS 9 so we can rewrite(3.2) F K ( x, T ) = 2 π ∞ Z −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X <γ ≤ T x iγ t − γ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt. As in Montgomery’s proof, we can take the absolute value squared, switch sum and integral,and bound the sum of the integrals from T to ∞ and the integrals from −∞ to 0 by O (log T )using the fact that there are at most ≪ log T zeros with ordinate T ≤ γ ≤ T + 1 (see Chapter5 of [20]). The same fact and same type of estimates allow us to bound the sums over γ and γ ′ outside of the interval (0 , T ] by O (log T ), so combining all of this yields(3.3) F K ( x, T ) = 2 π T Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X γ x iγ t − γ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt + O (log T ) = 12 π T Z | L ( x, t ) | dt + O (log T ) . Now, we use L ( x, t ) = R ( x, t ). In order to estimate T R | R ( x, t ) | dt , we can use Cauchy-Schwarzand Parseval’s identity:(3.4) T Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X m a m m − it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = X m | a m | ( T + O ( m )) . Using this, we find that1 x T Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X N ( a ) ≤ x Λ K ( a ) (cid:18) xN ( a ) (cid:19) − + it + X N ( a ) >x Λ K ( a ) (cid:18) xN ( a ) (cid:19) + it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = 1 x X m ≤ x c K ( m ) (cid:16) xm (cid:17) − ( T + O ( m )) + 1 x X m>x c K ( m ) (cid:16) xm (cid:17) ( T + O ( m )) . (3.5)First, we can break up the sum into sums over relatively prime congruence classes modulo∆ K . Let C K be the number of congruence classes m modulo ∆ K such that χ i ( m ) = 1 for i = 0 , . . . , n −
1. For this part of the sum, we combine equation (2.8), Lemma 2.3, Lemma2.4, and Lemma 2.6 to see that this last expression is equal to n Tx · C K (cid:20) ϕ (∆ K ) x log x + O ( x ) (cid:21) + O ( x log x )+ n T x · C K (cid:20) ϕ (∆ K ) log xx + O ( x − ) (cid:21) + O ( x log x )= T (cid:18) n C K ϕ (∆ K ) log x + O (1) (cid:19) + O ( x log x ) . (3.6)We can simplify this expression further by considering the proportion of primes that splitcompletely in K . By Chebotarev density theorem, we know these primes have density 1 /n in the set of all primes. On the other hand, since the set of these primes is the union of C K congruence classes modulo ∆ K , we see that they also have density C K /ϕ (∆ K ) in the set ofall primes, so C K /ϕ (∆ K ) = 1 /n . Therefore, (3.6) becomes(3.7) T ( n log x + O (1)) + O ( x log x ) =: M . Note that using (2.9), we can group the terms with ( m, ∆ K ) > R ( x, t ) aregiven by M := T Z | nx − it log τ | dt = n Tx (log T + O (log T )) ,M := T Z | O ( x − it ) | dt ≪ Tx ,M := T Z | O ( x τ − ) | dt ≪ x. For 1 ≤ x ≤ (log T ) / , M i = o ( M ) for i = 1 , ,
4. For (log T ) / < x ≤ (log T ) / , all of theterms are uniformly o ( T log T ). For (log T ) / < x ≤ T / log T , M i = o ( M ) for i = 2 , , x = T α , we get nF K ( α ) T log T + O (log T ) = T Z | R ( T α , t ) | dt = (cid:0) (1 + o (1)) n T − α log T + nα + o (1) (cid:1) T log T uniformly in 0 ≤ α ≤ − ε .To obtain the uniformity near 1, we refine the estimate (3.3). The sum inside the absolutevalue of the integrand of this equation is bounded by ≪ log τ (Chapter 5 of [20]), so theintegral from 0 to T / is seen to be bounded above by ≪ T / log T ≪ T . Then in orderto rewrite the integral from T / to T , we can repeat all of the steps above, namely to useLemma 2.2 with σ = 3 / T ε ≤ x ≤ T instead, in which case the error can be shown to bebounded above by O ( T ). In other words, using the same type of estimates as above yields(3.8) F K ( x, T ) = n πx T Z T / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X m ∈S Λ( m ) d m ( x ) m it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt + O ( T )for T ε ≤ x ≤ T , where d m ( x ) = min(( x/m ) − / , ( x/m ) / ) and S is the set of naturalnumbers m with χ i ( m ) = 1 for i = 0 , . . . , n −
1. As Goldston does, we define two newauxiliary functions in order to evaluate this integral: A ( x, T ) := 1 x T Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X m ∈S Λ( m ) d m ( x ) m it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dtB ( x, T ) := 1 x T Z − T (cid:18) − | t | T (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X m ∈S Λ( m ) d m ( x ) m it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt. AIR CORRELATION FOR DEDEKIND ZETA FUNCTIONS OF ABELIAN EXTENSIONS 11
While A ( x, T ) is the desired integral we wish to compute to further write down F K ( x, T ), wecan square the absolute value in the integral of B ( x, T ), which yields B ( x, T ) = Tx X m ∈S Λ ( m ) d m ( x ) + Tx X m,j ∈S m = j Λ( m )Λ( j ) d m ( x ) d j ( x ) sin (cid:16) T log mj (cid:17) T log mj =: S + S . Through the same calculations as above using the sum estimates of section 2.1, we determinethat S = ( T /n ) log x + O ( T ). For S , notice that regardless of the values of m and j , theterms of the sum are non-negative. Thus, S is at most the sum over all m = j , i.e. at mostthe sum without the character conditions. In Goldston’s thesis, he shows that this is O ( x )(see pages 48-53 of [18]). Combined, this yields B ( x, T ) = 1 n T log x + O ( T ) + O ( x ) . Now, in order to compute A ( x, T ), the following relations are easily obtained from writingdown the integral definition of B ( x, T ): T B ( x, T ) − ( T − δ ) B ( x, T − δ ) ≤ δA ( x, T ) ≤ ( T + δ ) B ( x, T + δ ) − T B ( x, T )for any δ >
0. From the formula we derived for B ( x, T ), this tells us that (cid:12)(cid:12)(cid:12)(cid:12) A ( x, T ) − n T log x (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ n log x + O ( x ) + O ( T ) + O ( δ ) + O (cid:18) Tδ ( x + T ) (cid:19) . With the appropriate choice of δ , namely δ = T for 1 ≤ x ≤ δ = T / (log x ) / for 2 < x ≤ T ,and δ = ( T x/ log x ) / for x > T , we can rewrite the error terms to obtain A ( x, T ) = 1 n T log x + O ( x ) + O ( T ) + O (cid:16) ( xT log x ) (cid:17) + O (cid:16) T (log x ) (cid:17) . By (3.8), we obtain F K ( x, T ) = n π (cid:16) A ( x, T ) − A ( x, T ) (cid:17) + O ( T )= n π T log x + O ( x ) + O (cid:16) ( xT log x ) (cid:17) + O (cid:16) T (log x ) (cid:17) for T ε ≤ x ≤ T . Plugging in T α and solving for F K ( α ) yields the desired result, noting thateach of the error terms is o (1) when ε ≤ α ≤ Proofs of Theorem 1.2 and Corollary 1.3.
Let A LP be the set of even continuousfunctions f ∈ L ( R ) that satisfy:(1) ˆ f (0) = 1,(2) ˆ f ≥ f ( x ) ≤ | x | ≥ N K ( T ) ∼ nT π log T. Using Fourier inversion on the definition of F K ( α ) and using this asymptotic, we obtain X <γ,γ ′ ≤ T g (cid:18) ( γ − γ ′ ) log T π (cid:19) w ( γ − γ ′ ) = (cid:18) nT π log T (cid:19) ∞ Z −∞ ˆ g ( α ) F K ( α ) dα (3.9) = N K ( T )(1 + o K (1)) ∞ Z −∞ ˆ g ( α ) F K ( α ) dα for suitable g . To state the lemma, we define the following linear functionals for f ∈ A LP : Z n ( f ) = nf (0) + 2 Z f ( x ) x dx. Lemma 3.1.
Let f ∈ A LP . Assuming GRH, we have N ∗ K ( T ) ≤ ( Z n ( f ) + o K (1)) N K ( T ) . Proof.
By our theorems above, we have that F K ( α, T ) = (cid:0) nT − | α | log T + | α | (cid:1) (1 + o K (1))uniformly for | α | ≤
1. Since f is continuous and T − | α | log T → δ ( x ) as T → ∞ in thedistributional sense, we can rewrite (3.9) as X <γ,γ ′ ≤ T ˆ f (cid:18) ( γ − γ ′ ) log T π (cid:19) w ( γ − γ ′ )= N K ( T ) nf (0) + Z − f ( α ) | α | dα + Z | α | > f ( α ) F K ( α ) dα + o K (1) . Since F K ( α ) is non-negative, f ( x ) ≤ | x | ≥
1, and f is even, we see that X <γ,γ ′ ≤ T ˆ f (cid:18) ( γ − γ ′ ) log T π (cid:19) w ( γ − γ ′ ) ≤ N K ( T ) nf (0) + 2 Z f ( α ) αdα + o K (1) = N K ( T ) [ Z n ( f ) + o K (1)] . On the other hand, X <γ,γ ′ ≤ T ˆ f (cid:18) ( γ − γ ′ ) log T π (cid:19) w ( γ − γ ′ ) ≥ ˆ f (0) X <γ ≤ T m ρ = N ∗ K ( T ) . Putting these inequalities together yields the result. (cid:3)
This lemma is very similar to [10, Theorem 8], but the setup here is a bit different: In [10]there is the constraint f (0) = 1, instead of f ( x ) ≤ | x | ≥ r ( f ) := inf { R : f ( x ) ≤ | x | ≥ R } is finite, and instead of Z , the functional is Z ( f ) = r ( f ) + 2 r ( f ) r ( f ) Z f ( x ) xdx. By rescaling and renormalizing, we see this gives the same result. However, since Z n ( · ) islinear, we now only have to solve a single semidefinite program for each n .As in [10], we use the identity(3.10) Z x m e − πx dx = − π m/ / Γ (cid:16) m + 12 , πx (cid:17) , AIR CORRELATION FOR DEDEKIND ZETA FUNCTIONS OF ABELIAN EXTENSIONS 13 where Γ is the upper incomplete gamma function, to model Z n ( f ) as a linear functional inthe matrix entries. This allows us to use the semidefinite programming approach as outlinedin Section 2.2 to find the bounds for n = 2 , , d = 40 forall computations. To get the bound c n ≤ (1 + 10 − ) n + 0 . f (0) and Z n ( f ) − nf (0) of a (near) optimal function f ∈ A LP for Z n ( f ) with n = 10 , also obtainedusing the same semidefinite programming approach. The proof of the final part of Theorem 1.2is mentioned in the next section. In order to obtain rigorous proofs we use interval arithmeticto verify the solver output, as is done in [10]. The Julia/Nemo/Arb [1,16,21] code to generatethe semidefinite programs, to solve them with SDPA-GMP [26], and to verify the output usinginterval arithmetic is included in the arXiv version of this paper.In order to obtain Corollary 1.3, notice that we have the following relationship between N K,s , N K,d , and N ∗ K :2 N K,s ( T ) = 2 X <γ ≤ Tm ρ =1 ≤ X <γ ≤ T ( m ρ − m ρ − m ρ = N ∗ K ( T ) − N K ( T ) + 6 N K,d ( T ) . Solving for N K,d ( T ), using the bounds that Theorem 1.2 gives, and using the bound of N K,s ( T ) ≥ ( + o K (1)) N K ( T ) for quadratic fields given by [11] or the trivial bound of N K,s ( T ) ≥ Optimizing over functions with bounded support.
As mentioned before, theCohn-Elkies bound in dimension 1 is the optimization problem inf { f (0) : f ∈ A LP } . Thismeans that we can interpret our optimization problems for Dedekind zeta functions as in-terpolants between the corresponding problem for the zeta function from [10] and the Cohn-Elkies sphere packing bound in dimension 1 from [6].Since we can pack 1 ball of radius 1 / H ( x ) = ( − | x | for | x | ≤ , , which lies in A L P and satsifies H (0) = 1. Since Z n ( H ) = n + 1 /
3, this provides the proof forthe last part of Theorem 1.2.Since H is supported in [ − ,
1] and n Z n ( f ) is close to f (0) for large n , this raises thequestion whether for large n the optimization problem inf {Z n ( f ) : f ∈ A LP } also has a nearoptimal solution among the functions supported in [ − , f of the form p ( x ) e − πx in A LP with f (0) = 1 (since f (1) = 0 implies f ( n ) = ˆ f ( n ) = 0 for all integers n by complementaryslackness), the question arises whether we can use optimization over functions supported in[ − ,
1] to find a function f in A LP satisfying f (0) = 1 and 2 R f ( x ) x dx ≈ . − in the bound c n ≤ (1 + 10 − ) n + 0 . f hasnonnegative Fourier transform and supp( f ) ⊆ [ − , g with supp( g ) ⊆ [ − / , /
2] such that f ( x ) = g ∗ g ∗ ( x ) = Z ∞−∞ g ( y ) g ( y − x ) dy. As mentioned by Gallagher [17] the existence of such a g follows from the Paley-Wienertheorem and a theorem by Krein [2, p. 154]. The first theorem says that the nonnegativefunction ˆ f is analytic and hence can be approximated by nonnegative cosine polynomials,and the second theorem says that a nonnegative cosine polynomial supported on [ − ,
1] ofthe form f ( x ) = | g ( x ) | for some function g supported on [ − / , / d ≥
1, we model f via a positive semidefinite matrix X as follows:(3.11) f ( x ) = d X i,j =0 X i,j b i ∗ b ∗ j ( x ) , b i ( x ) = ( x i for | x | ≤ / , . Then f ( x ) ≤ | x | ≥ f is supported on [ − , X can be decomposed as P rank( X ) k =1 v k v T k we haveˆ f ( x ) = rank( X ) X k =1 d X i,j =1 ( v k ) i ( v k ) j ˆ b i ( x )ˆ b j ( x ) = rank( X ) X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i =1 ( v k ) i ˆ b i ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ . Direct computations showsˆ f (0) = d X i,j =0 i + 1)( j + 1)2 i + j A i,j and f ( x ) = d X i,j =0 X i,j Z T/ − T/ x y i ( y − x ) j dy, which shows f ( x ) is a polynomial whose coefficients are linear functions in the entries of X ,and thus ˆ f (0), f (0), and R f ( x ) x dx are linear functionals in the entries of X and thereforeinf {Z n ( f ) : X (cid:23) , ˆ f (0) = 1 } is a semidefinite program.By solving this problem for n = 1 and d = 1, we recover the bound c ≤ / n = 1 and d = 40, we recover the best possible bound c ≤ / − / cot(2 − / ) from [9] to within 70 decimals of accuracy. For n = 2 , , c ≤ . c ≤ . c ≤ . d = 40). As is to be expected, these bounds are not as good as the bounds computedin Theorem 1.2 through optimization over Schwartz functions.To answer the above questions we set n = 10 , which gives use the bound c ≤ +0 . . . . . This shows that although the Cohn-Elkies bound in dimension 1 has a sharpsolution f with supp( f ) ⊆ [ − , c n ≤ (1 + 10 − ) + 0 . − , f (0) = 1, then we just get the bound c n ≤ n + 1 / − ,
1] to removethe 10 − term in Theorem 1.2. Perhaps the optimal functions for the problems inf {Z n ( f ) : f ∈ A LP } are difficult to construct Schwartz funcions in the same way that the optimalfunctions for the Cohn-Elkies bound in dimension 8 and 24 are difficult to construct Schwartzfunctions [7, 31]. As for the optimization over unbounded functions, the code to compute theabove bounds and to verify the correctness of the bounds using interval arithmetic is includedin the arXiv version of this paper.As a final remark we note that since f ( x ) is a polynomial in x whose coefficients are linearin the entries of X , we can write the constraint f ( x ) ≤ | x | ≥ − T, T ] by replacing 1 / T /
AIR CORRELATION FOR DEDEKIND ZETA FUNCTIONS OF ABELIAN EXTENSIONS 15
References [1] Bezanson, J., A. Edelman, S. Karpinski and V.B. Shah, “Julia: A fresh approach to numerical comput-ing.”
SIAM Rev.
59 (2017): 65-98.[2] Achieser, N. I.. “Theory of Approximation”, New York, 1956.[3] Bauer, Peter J. “Zeros of Dirichlet L -series on the critical line.” Acta Arithmetica
Journal f¨ur die reine und angewandte Mathematik(Crelles Journal) arXiv preprint arXiv:1708.04122 (2017).[6] Cohn, Henry and Noam Elkies. “New upper bounds on sphere packings I.”
Annals of Mathematics
Ann. of Math. arXiv preprint arXiv:1603.04759 (2016).[9] Cheer, A. Y., and D. A. Goldston. “Simple zeros of the Riemann zeta-function.”
Proceedings of theAmerican Mathematical Society
Analytic Number Theory and Diophantine Problems . Birkh¨auser Boston, 1987. 87-114.[12] Conrey, Brian, Henryk Iwaniec, and Kannan Soundararajan. “Asymptotic large sieve.” arXiv preprintarXiv:1105.1176 (2011).[13] Conrey, J. Brian, Henryk Iwaniec, and Kannan Soundararajan. “Critical zeros of Dirichlet L -functions.” Journal f¨ur die reine und angewandte Mathematik (Crelles Journal)
Multiplicative Number Theory . Vol. 74. Springer Science & Business Media, 2013.[15] Farmer, David W., Steven M. Gonek, and Yoonbok Lee. “Pair correlation of the zeros of the derivativeof the Riemann ξ -function.” Journal of the London Mathematical Society
Proceedings of ISSAC ’17 (2017): 157-164.[17] Gallagher, P. X. “Pair correlation of zeros of the zeta function.”
Journal f¨ur die Reine und AngewandteMathematik.
362 (1985): 72-86.[18] Goldston, Daniel A. “Large Differences Between Consecutive Prime Numbers.” Thesis. University ofCalifornia, Berkeley. 1981.[19] Goldston, Daniel A., and Hugh L. Montgomery. “Pair correlation of zeros and primes in short intervals.”
Analytic number theory and Diophantine problems . Birkh¨auser Boston, 1987. 183-203.[20] Iwaniec, Henryk and Kowalski, Emmanuel.
Analytic Number Theory . Vol. 53. American MathematicalSoc., 2004.[21] Johansson, F. “Arb: efficient arbitrary-precision midpoint-radius interval arithmetic.”
IEEE Transac-tions on Computers
Bulletin of the AmericanMathematical Society ζ (1 / it ).” Communications inMathematical Physics
Handbuch der Lehre von der Verteilung der Primzahlen . Vol. 1. 2000.[25] Montgomery, Hugh L. “The pair correlation of zeros of the zeta function.”
Proc. Symp. Pure Math.
Vol.24. 1973.[26] Nakata, N. “A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefi-nite programming solver: SDPA-GMP,-QD and-DD.”
In IEEE International Symposium on Computer-Aided Control System Design , 232 (2010): 29-34.[27] Neukirch, J¨urgen.
Algebraic Number Theory . Vol. 322. Springer Science & Business Media, 2013.[28] Odlyzko, Andrew M. “On the distribution of spacings between zeros of the zeta function.”
Mathematicsof Computation [29] P´olya, George, G´abor Szeg¨o. “Problems and theorems in analysis. II”, Classics in Mathematics, Springer-Verlag, 1998; Translated from German by C. E. Billigheimer; Reprint of the 1976 English translation.[30] Rudnick, Ze´ev, and Peter Sarnak. “Zeros of principal L -functions and random matrix theory.” DukeMathematical Journal
Ann. of Math. L -functions.” The QuarterlyJournal of Mathematics (2016): 1-23.[33] Yildirim, C. Yal¸cin. “The pair correlation of zeros of Dirichlet L -functions and primes in arithmeticprogressions.” manuscripta mathematicamanuscripta mathematica