Abstract
The partition function p(n) has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the “circle method” to estimate the size of p(n), which was later perfected by Rademacher who obtained an exact formula. Recently, Chan and Wang considered the fractional partition functions, defined for \(\alpha \in {\mathbb {Q}}\) by \(\sum _{n = 0}^\infty p_{\alpha }(n)x^n := \prod _{k=1}^\infty (1-x^k)^{-\alpha }\). In this paper we use the Rademacher circle method to find an exact formula for \(p_\alpha (n)\) and study its implications, including log-concavity and the higher-order generalizations (i.e., the Turán inequalities) that \(p_\alpha (n)\) satisfies.
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Notes
All computations in this section were done with Wolfram Mathematica.
References
Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics, 2nd edn. Springer, New York (1990)
Bevilacqua, E., Chandran, K., Choi, Y.: Ramanujan Congruences for Fractional Partition Functions. Unpublished (2019)
Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass Forms and Mock Modular Forms: Theory and Applications, vol. 64. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (2017)
Chan, H.H., Wang, L.: Fractional powers of the generating function for the partition function. Acta Arithmetica 187, 59–80 (2019)
Chen, W., Jia, D., Wang, L.: Higher order Turán inequalities for the partition function. Trans. Am. Math. Soc. (2018)
Craven, T., Csordas, G.: Jensen polynomials and the Turán and Laguerre inequalities. Pac. J. Math. 136(2), 241–260 (1989)
DeSalvo, S., Pak, I.: Log-concavity of the partition function. Ramanujan J. 38(1), 61–73 (2015)
Griffin, M., Ono, K., Rolen, L., Zagier, D.: Jensen polynomials for the Riemann zeta function and other sequences. Proc. Natl. Acad. Sci. USA 116(23), 11103–11110 (2019)
Hardy, G.H., Ramanujan, S.: Asymptotic formulæ in combinatory analysis. Proc. Lond. Math. Soc. 2(1), 75–115 (1918)
Helfgott, H.A.: The ternary Goldbach conjecture is true. arXiv e-prints, page arXiv:1312.7748 (2013)
Keith, W.J.: Restricted \(k\)-color partitions. Ramanujan J. 40(1), 71–92 (2016)
Larson, H., Wagner, I.: Hyperbolicity of the partition Jensen polynomials. Res. Number Theory 5(2), 19 (2019)
Nicolas, J.-L.: Sur les entiers \(n\) pour lesquels il y a beaucoup de groupes abéliens d’ordre \(n\). Ann. Insti. Fourier 28(4), 1–16 (1978)
Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., B.V. Saunders, (eds.) NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.23 of 2019-06-15.
Paris, R.: An Inequality for the Bessel function \({J}_{\nu }(\nu x)\). SIAM J. Math. Anal. 15(1), 203–205 (1984)
Rademacher, H.: On the Partition Function \(p(n)\). Proc. Lond. Math. Soc. s2–43(1), 241–254 (1938)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)
Acknowledgements
The authors would like to thank Ken Ono, Larry Rolen, and Ian Wagner for suggesting the problem and their guidance. The research was supported by the generosity of the Asa Griggs Candler Fund, the National Security Agency under grant H98230-19-1-0013, and the National Science Foundation under Grants 1557960 and 1849959.
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Iskander, J., Jain, V. & Talvola, V. Exact formulae for the fractional partition functions. Res. number theory 6, 20 (2020). https://doi.org/10.1007/s40993-020-00195-0
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DOI: https://doi.org/10.1007/s40993-020-00195-0