Functiones et Approximatio Commentarii Mathematici

The Moments of the Riemann Zeta-Function Part I: The fourth moment off the critical line

Aleksandar Ivić and Yoichi Motohashi

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Abstract

In this paper, the first part of a larger work, we prove the spectral decomposition of $$ \int_{-\infty}^\infty|\zeta(\sigma+it)|^4g(t) dt \qquad(\tfrac{1}{2} < \sigma < 1 {\rm {fixed}}), $$ where $g(t)$ is a suitable weight function of fast decay. This is used to obtain estimates and omega results for the function \begin{align*} E_2(T,\sigma) &: =\int_0^T|\zeta(\sigma+it)|^4 dt - {\zeta^4(2\sigma)\over\zeta(4\sigma)}T -{\frac{T}{3-4\sigma}}{\left(\frac{T}{2\pi} \right)}^{2-4\sigma}{\zeta^4(2-2\sigma)\over\zeta(4-4\sigma)}\cr & \quad- T^{2-2\sigma}(a_0(\sigma) + a_1(\sigma)\log T + a_2(\sigma)\log^2T), \end{align*} the error term in the asymptotic formula for the fourth moment of $|\zeta(\sigma+it)|$.

Article information

Source
Funct. Approx. Comment. Math., Volume 35 (2006), 133-181.

Dates
First available in Project Euclid: 16 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229442621

Digital Object Identifier
doi:10.7169/facm/1229442621

Mathematical Reviews number (MathSciNet)
MR2271611

Zentralblatt MATH identifier
1196.11116

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11F72: Spectral theory; Selberg trace formula 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations

Keywords
Fourth moment of the Riemann zeta-function spectral decomposition Hecke series hypergeometric function omega results

Citation

Ivić, Aleksandar; Motohashi, Yoichi. The Moments of the Riemann Zeta-Function Part I: The fourth moment off the critical line. Funct. Approx. Comment. Math. 35 (2006), 133--181. doi:10.7169/facm/1229442621. https://projecteuclid.org/euclid.facm/1229442621


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