## Partitions into Primes

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- by Yifan Yang PDF
- Trans. Amer. Math. Soc.
**352**(2000), 2581-2600 Request permission

## 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.## References

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## Additional Information

**Yifan Yang**- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- MR Author ID: 633505
- Email: yfyang@math.uiuc.edu
- Received by editor(s): March 3, 1998
- Published electronically: February 14, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 2581-2600 - MSC (2000): Primary 11P82; Secondary 11M26, 11N05
- DOI: https://doi.org/10.1090/S0002-9947-00-02386-2
- MathSciNet review: 1621714