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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Partitions into Primes


Author: Yifan Yang
Journal: Trans. Amer. Math. Soc. 352 (2000), 2581-2600
MSC (2000): Primary 11P82; Secondary 11M26, 11N05
Published electronically: February 14, 2000
MathSciNet review: 1621714
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Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the asymptotic behavior of the partition function $p_{\Lambda} (n)$ defined by $\sum ^{\infty }_{n=0}p_{\Lambda} (n)x^{n} =\prod ^{\infty }_{m=1}(1-x^{m})^{-\Lambda (m)}$, where $\Lambda (n)$ denotes the von Mangoldt function. Improving a result of Richmond, we show that $\log p_{\Lambda} (n)=2\sqrt {\zeta (2)n}+O(\sqrt n\exp \{-c(\log n) (\log _{2} n)^{-2/3}(\log _{3} n)^{-1/3}\})$, where $c$ is a positive constant and $\log _{k}$ denotes the $k$ times iterated logarithm. We also show that the error term can be improved to $O(n^{1/4})$ if and only if the Riemann Hypothesis holds.


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Additional Information

Yifan Yang
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: yfyang@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02386-2
PII: S 0002-9947(00)02386-2
Received by editor(s): March 3, 1998
Published electronically: February 14, 2000
Article copyright: © Copyright 2000 American Mathematical Society