We give a new heuristic for all of the main terms in the integral moments of various families of primitive $L$-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical $L$-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

https://web.williams.edu/Mathematics/sjmiller/public_html/ntandrmt/handouts/conrey/ConreyFarmerKeatingRubinsteinSnaith_IntegralMomentsLFns.pdf

Integral moments of L-functions

Montgomery’s pair correlation conjecture predicts the asymptotic behavior of the function N(T, β) defined to be the number of pairs γ and γ′ of ordinates of nontrivial zeros of the Riemann zetafunction satisfying 0 0, using Montgomery’s formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval [−β, β] in a way to minimize the L1 ( R, { 1− ( sinπx πx )2} dx ) -error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work of P. X. Gallagher [18] in 1985, where the case β ∈ 1 2 N was considered using non-extremal majorants and minorants.

/pdf/hilbert-spaces-and-the-pair-correlationof-zeros-of-the-4eg27p0nqg.pdf

Hilbert spaces and the pair correlationof zeros of the Riemann zeta-function

In this paper s = σ + it will denote a complex variable, where σ and t are real, and T will denote a large parameter. The relations between the zeros of a function and the zeros of its derivatives have been the object of much study. The case of the Riemann zeta-function ζ(s) presents many puzzles beginning with the Riemann hypothesis (RH). Speiser [11] showed that RH is equivalent to ζ ′(s) having no zeros in 0 < σ < 2 . From Riemann’s original work (proofs for some parts of which were provided later by other mathematicians), it is well-known that the nontrivial zeros of ζ(s), to be denoted by ρ = β + iγ, are to be found only in the

On Small Distances Between Ordinates of Zeros of ζ(s) and ζ′(s)

Pair correlation estimates for the zeros of the zeta function via semidefinite programming

http://www.fuchs-braun.com/media/86c03deef7daa654ffff800bfffffff0.pdf

Prime pairs and the zeta function

To study the distribution of pairs of zeros of the Riemann zeta-function, Montgomery introduced the function F(\alpha) = F_T(\alpha) = \left({T\over 2\pi}\log T\right)^{-1} \sum_{0<\gamma,\gamma ' \le T} T^{i\alpha(\gamma -\gamma ')}w(\gamma-\gamma '), where and denote the imaginary parts of zeros of the Riemann zeta-function, and .
 Assuming the Riemann Hypothesis, Montgomery proved an asymptotic formula for when , and made the conjecture
 that for any bounded .
 In this paper we use an approximation for the prime
 indicator function together with a new mean value
 theorem for long Dirichlet polynomials and tails of
 Dirichlet series to prove that, assuming the Generalized
 Riemann Hypothesis for all Dirichlet -functions, then
 for any we have uniformly for and all .1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.

On the Pair Correlation of Zeros of the Riemann Zeta-Function

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On the Pair Correlation of Zeros of the Riemann Zeta-Function