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

Abstract

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