Applications of theL-functions ratios conjectures

Abstract

In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted L‐functions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of L‐functions, mollified moments of L‐functions and discrete averages over zeros of the Riemann zeta function. In particular, using the ratios conjectures we easily derive the answers to a number of notoriously difficult computations.

Autocorrelation of ratios of $L$-functions

“…Conrey and Snaith [10] have an alternative formulation for this conjecture with a subscript-free notation.…”

“…Since very little is known about mean values of ratios, we are not able to give any objective evidence for such a small error term. Also, we have not specified the allowable range for the shifts α, γ and δ. Conrey and Snaith [10] suggest that in the case of the zeta-function one should allow shifts with imaginary part T 1− .…”

“…The ratios conjectures contain much more information and can be used to make very precise conjectures about the distribution of zeros of L-functions. In addition to the examples we give in Section 7 of this paper, Conrey and Snaith [10] have given a large number of applications.…”

Riemann Zeros and Random Matrix Theory

“…Upon accepting Conjecture 4.1 we can now fully understand the two-point correlation function in Figure 3. By taking a logarithmic derivative of (4.17) with respect to the variables α and β, subsequently setting γ = α and δ = β and performing a double integration, integrated against a suitable test function around a contour that encloses the zeros with 0 < γ j ≤ T , we obtain a detailed expression for the two-point correlation function [28]. …”

“…Any random matrix calculation applied to the Riemann zeta function could equally be applied to such an L-function (see, for example, [62]). In addition, Katz and Sarnak [50,51] proposed that collections of Lfunctions gathered into natural families also show random matrix statistics, and much work has been done using random matrix theory to calculate moments and zero statistics in this case, too, leading to surprising new applications [12,13,16,26,27,28,32,45,46,47,52,55,56,66].…”