Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T05:14:35.484Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-vanishing of the partition function modulo odd primes l

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Scott Ahlgren
Scott Ahlgren
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania, 16802-6401, U.S.A.
Get access

Abstract

Let p(n) be the usual partition function. Let l be an odd prime, and let r (mod t) be any arithmetic progression. If there exists an integer n ≡ r (mod t) such that p(n) ≢ 0 (mod l), then, for large X,

MSC classification

Secondary: 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Mathematika , Volume 46 , Issue 1 , June 1999 , pp. 185 - 192
Copyright
Copyright © University College London 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A.Ahlgren, S.. Distribution of parity of the partition function in arithmetic progressions. Indag. Math., 10 (1999), 173–181.CrossRefGoogle Scholar
E.Erdős, P.. Über die kleinste quadratfreíe Zahl einer arithmetischen Reihe. Monatsh. Math., 64 (1960), 314–316.CrossRefGoogle Scholar
N.-R.-SNicolas, J. L, Ruzsa, I. Z. and Sárkőzy, A. (with an appendix by J.-P. Serre) On the parity of additive representation functions. J. Number Theory, 73 (1998), 292–317.CrossRefGoogle Scholar
O.1Ono, K.. Parity of the partition function in arithmetic progressions. J. reine angew. Math., 472 (1996), 1–15.Google Scholar
O.2Ono, K.. The partition function in arthmetic progressions. Math. Ann., 312 (1998), 251–260.Google Scholar
O.-SOno, K. and Skinner, C.. Non-vanishing of quadratic twists of modular L-functions, Invent. Math. 134 (1998), 651–660.Google Scholar
St.Sturm, J.. On the congruence of modular forms. Springer Lecture Notes, 1240 (1984).Google Scholar