Applications of the L‐functions ratios conjectures

By J. Brian Conrey, N. Snaith. 2005

153 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9ed93865f369b8dcbba0491294aab2e8a40e3efb

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Autocorrelation of ratios of L-functions (2007, methodology): “In particular, letting (5.17) Hζ,t(w1, . . . , wK+L; γ; δ) = ∏L ℓ=1 χ(s − wK+ℓ)1/2 ∏K k=1 χ(s + wk) 1/2 YUAζ(w1, . . . , wK;−wK+1, . . . ,−wK+L; γ; δ) the conjecture may be reformulated as ∫ T 0 ∏K k=1 ζ(s + αk) ∏K+L ℓ=K+1 ζ(1 − s − αℓ) ∏Q q=1 ζ(s + γq) ∏R r=1 ζ(1 − s + δr) dt = ∫ T 0 ∏K k=1 χ(s + α”
More than 41% of the zeros of the zeta function are on the critical line (2010, background): “e heuristics that can save a lot of time. In particular, the ratios conjecture [CFZ] can rather quickly allow one to express molliﬁed moments of L-functions as certain multiple contour integrals; see [CS] for a variety of examples of such calculations. Even then, it takes some signiﬁcant work to sim”
Non‐vanishing of the symmetric square L‐function at the central point (2008, background): “In fact, from a family’s symmetry type, using the Ratios Conjecture, one can make predictions about non-vanishing by the use of mollifiers (see [2]).”
Conjectures for the integral moments and ratios of L-functions over function fields (2014, background): “For more details about these applications see [17].”
On the zeros of Riemann's zeta-function on the critical line (2016, methodology): “However, Conrey and Snaith [8] have formulated specific conjectures for it by using the Ratios Conjectures.”