Experimental Mathematics

Conjectures and Experiments Concerning the Moments of $L (1/2, \chi_d)$

Matthew W. Alderson and Michael O. Rubinstein

Full-text: Open access

Abstract

We report on some extensive computations and experiments concerning the moments of quadratic Dirichlet $L$-functions at the critical point. We computed the values of $L (1/2, \chi_d)$ for $−5 × 10^{10} \lt d \lt 1.3 × 10^{10}$ in order to numerically test conjectures concerning the moments $\sum_{|d|\lt X} L (1/2, \chi_d)^k$. Specifically, we tested the full asymptotics for the moments conjectured by Conrey, Farmer, Keating, Rubinstein, and Snaith, as well as the conjectures of Diaconu, Goldfeld, Hoffstein, and Zhang concerning additional lower-order terms in the moments. We also describe the algorithms used for this large-scale computation.

Article information

Source
Experiment. Math., Volume 21, Issue 3 (2012), 307-328.

Dates
First available in Project Euclid: 13 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.em/1347541280

Mathematical Reviews number (MathSciNet)
MR2988582

Zentralblatt MATH identifier
1318.11104

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$

Keywords
Moments of $L (1/2, \chi_d)$

Citation

Alderson, Matthew W.; Rubinstein, Michael O. Conjectures and Experiments Concerning the Moments of $L (1/2, \chi_d)$. Experiment. Math. 21 (2012), no. 3, 307--328. https://projecteuclid.org/euclid.em/1347541280


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