Uniform asymptotics for the full moment conjecture of the Riemann zeta function

Details Uniform asymptotics for the full moment conjecture of the Riemann zeta function Uniform asymptotics for the full moment conjecture of the Riemann zeta function

2011-06-21

arxiv.org/abs/1106.4352v3

Mathematics

Number Theory

53 pages, 1 figure, 2 tables

Scientific paper

Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured formulas for the full asymptotics of the moments of $L$-functions. In the case of the Riemann zeta function, their conjecture states that the $2k$-th absolute moment of zeta on the critical line is asymptotically given by a certain $2k$-fold residue integral. This residue integral can be expressed as a polynomial of degree $k^2$, whose coefficients are given in exact form by elaborate and complicated formulas. In this article, uniform asymptotics for roughly the first $k$ coefficients of the moment polynomial are derived. Numerical data to support our asymptotic formula are presented. An application to bounding the maximal size of the zeta function is considered.

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