The Ramanujan Journal

, Volume 38, Issue 1, pp 61–73 | Cite as

Log-concavity of the partition function



We prove that the partition function \(p(n)\) is log-concave for all \(n>25\). We then extend the results to resolve two related conjectures by Chen and one by Sun. The proofs are based on Lehmer’s estimates on the remainders of the Hardy–Ramanujan and the Rademacher series for \(p(n)\).


Integer partition Partition function Log-concave sequence Asymptotic analysis Error estimates 

Mathematics Subject Classification

05A17 11N37 65G99 



The authors would like to thank William Chen, Christian Krattenthaler, Karl Mahlburg, Bruce Rothschild, Bruce Sagan for helpful remarks and suggestions. The authors are grateful to Richard Arratia for introducing us to the problem, to David Moews for bringing the work of Lehmer to our attention, and to Janine Janoski for informing us about the status of [14].


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

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