The mean value of L(12,χ) in the hyperelliptic ensemble

By Julio Andrade, Jon P Keating. 2012

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Abstract We obtain an asymptotic formula for the first moment of quadratic Dirichlet L-functions over the rational function field at the central point s = 1 2 . Specifically, we compute the expected value of L ( 1 2 , χ ) for an ensemble of hyperelliptic curves of genus g over a fixed finite field as g → ∞ . Our approach relies on the use of the analogue of the approximate functional equation for such L-functions. The results presented here are the function field analogues of those obtained previously by Jutila in the number-field setting and are consistent with recent general conjectures for the moments of L-functions motivated by Random Matrix Theory.

Subfields: Algebra and Number Theory, Artificial Intelligence, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2306ecf5e294de252b63b953d78dd3281a603a0b

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In Algebra and Number Theory

In Artificial Intelligence

In Mathematical Physics

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Conjectures for the integral moments and ratios of L-functions over function fields (2014, methodology): “1− 1 |P|1+zi+zj × 1 2 Yk j=1 1− 1 |P|12+zj !−1 + Yk j=1 1+ 1 |P|21+zj !−1 + 1 |P| × 1+ 1 |P| −1 (3.6) . Remark 3. In the case when k= 1, this conjecture coincides with a theorem in [AK]. See section 5.1. Remark 4. Note that (3.3) is the function ﬁeld analogue of the formula (1.5.11) in [CFKR”