Applications of the<i>L</i>-functions ratios conjectures

2007/01/22 by J. B. Conrey, J. Brian Conrey, N. C. Snaith · 9 citations
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Advanced Algebra and Geometry

paper · doi:10.1112/plms/pdl021

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.

Citations

• Integral moments of L-functions
• Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189

Cited by

• Sums of three cubes over a function field
• Ratios conjecture of cubic L-functions of prime moduli
• Low-lying zeros of a family of quadratic Hecke L-functions via ratios conjecture
• The mollified fourth moment of Dirichlet L-functions
• Integer moments of the derivatives of the Riemann zeta function
• Low-lying zeros of Hilbert modular L-functions weighted by powers of central L-values
• The Ratios Conjecture and upper bounds for negative moments of\n L-functions over function fields
• Critical zeros of the Riemann zeta-function
• The Riemann Hypothesis: Past, Present and a Letter Through Time

Related

• The Ratios conjecture for real Dirichlet characters and multiple Dirichlet series
• The Ratios Conjecture and upper bounds for negative moments of\n L-functions over function fields
• Conjectures for the Integral Moments and Ratios of L-functions in Even characteristic
• Autocorrelation of ratios of L-functions
• Modeling Convolutions of L-Functions