# Applications of the L‐functions ratios conjectures

@article{Conrey2005ApplicationsOT, title={Applications of the L‐functions ratios conjectures}, author={J. Brian Conrey and Nina C. Snaith}, journal={Proceedings of the London Mathematical Society}, year={2005}, volume={94} }

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…

## 117 Citations

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- Economics
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In this paper, we apply the ratio conjecture of $L$-functions to derive the lower order terms of the $1$-level density of the low-lying zeros of a family quadratic Hecke $L$-functions in the Gaussian…

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Abstract We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb{Q}$. We are primarily interested in the family of all twists…

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We test the predictions of (a weakened version of) the L-functions Ratios Conjecture for the family of cuspidal newforms of weight k and level N , with either k fixed and N → ∞ through the primes or…

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We prove special cases of the Ratios Conjecture for the family of quadratic Dirichlet L–functions over function fields. More specifically, we study the average of L(1/2 + α, χD)/L(1/2 + β, χD), when…

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Abstract. Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted Lfunctions. These ratios conjectures are very powerful and can be…

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