Partition values and central critical values of certain modular 𝐿-functions

Webb, John

doi:10.1090/S0002-9939-09-10188-0

Partition values and central critical values of certain modular $L$-functions
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Proc. Amer. Math. Soc. 138 (2010), 1263-1272 Request permission

Abstract:

Let $p(n)$ denote the number of partitions of a positive number $n$, let $\ell \in \{5,7,11\}$ and let $\delta _\ell$ be the least non-negative residue of $24^{-1}$ modulo $\ell$. In this paper we prove congruences modulo $\ell$ between $\frac {p(\ell n+ \delta _\ell )}{\ell }$ and ratios of central critical values of $L$-functions associated to twists of certain integer weight newforms. In 1999, Guo and Ono proved analogous results for $13\leq \ell \leq 31$.

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