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Log-concavity of the partition function

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Abstract

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 \(p(n)\).

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Notes

  1. Conjecture 4.3 was proved by Chen et al. [8].

  2. See http://tinyurl.com/kkc6fwf.

  3. Personal communication.

  4. See e.g., Speyer’s calculation in http://tinyurl.com/nyq2zrn.

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Acknowledgments

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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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA , 90095, USA

    Stephen DeSalvo & Igor Pak

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  1. Stephen DeSalvo
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  2. Igor Pak
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Correspondence to Stephen DeSalvo.

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The second author was partially supported by the NSF.

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DeSalvo, S., Pak, I. Log-concavity of the partition function. Ramanujan J 38, 61–73 (2015). https://doi.org/10.1007/s11139-014-9599-y

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