Applications of the L-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.

References in corpus (4)

• On the averages of characteristic polynomials from classical groups
• Real zeros of quadratic Dirichlet L-functions
• Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)
• Random matrix theory and discrete moments of the Riemann zeta function

Cited by in corpus (11)

• Lower-Order Terms of the 1-Level Density of Families of Elliptic Curves
• On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class
• Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros
• A method for calculating spectral statistics based on random-matrix universality with an application to the three-point correlations of the Riemann zeros
• Low-lying zeros in families of holomorphic cusp forms: the weight aspect
• Modeling families of L-functions
• A representation theory approach to integral moments of L-functions over function fields
• On the nonvanishing of elliptic curve L-functions at the central point
• Non-vanishing of Dirichlet L-functions at the central point
• Lower order terms of the second moment of $S(t)$
• Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble