Improving the error term in the mean value of L(\tfrac12,χ) in the hyperelliptic ensemble

2015/05/12 by Alexandra Florea, Florea, Alexandra
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1505.03094

arxiv created 2015/05/12 · arxiv updated 2015/05/13

Abstract

Andrade and Keating computed the mean value of quadratic Dirichlet L--functions at the critical point, in the hyperelliptic ensemble over a fixed finite field \mathbbF_q. Summing L(1/2,χ_D) over monic, square-free polynomials D of degree 2g+1, the main term is of size |D| log_q |D| (where |D|=q^2g+1) and Andrade and Keating bound the error term by |D|^{\frac 34+ (log_q(2))/(2)}. For simplicity, we assume that q is prime with q ≡ 1 \pmod 4. We prove that there is an extra term of size |D|^1/3 log_q|D| in the asymptotic formula and bound the error term by |D|^1/4+ε.

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