The partition function modulo 3 in arithmetic progressions

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• 作者：Geoffrey D. Smith ; Lynnelle Ye
• 关键词：Partitions ; Congruences ; Modular forms
• 刊名：The Ramanujan Journal
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：39
• 期：3
• 页码：603-608
• 全文大小：357 KB
• 参考文献：1.Ahlgren, S., Ono, K.: Congruences and conjectures for the partition function. In: \(q\) -Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000). Contemporary Mathematics, vol. 291 , pp. 1–10. American Mathematical Society, Providence (2001)<br>2.Bellaïche, J., Khare, C.: Hecke Algebras of Modular Forms Modulo \(p\) . http://​people.​brandeis.​edu/​jbellaic/​preprint/​Heckealgebra6.​pdf (preprint)<br>3.Boylan, M., Ono, K.: Parity of the partition function in arithmetic progressions. II. Bull. Lond. Math. Soc. 33(5), 558–564 (2001)MathSciNet CrossRef MATH <br>4.Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular Functions of One Variable, II (Proceedings of the International Summer School, University Antwerp, Antwerp, 1972). Lecture Notes in Mathematics, vol. 349, pp. 143–316. Springer, Berlin (1973)<br>5.Folsom, A., Kent, Z.A., Ono, K.: \(\ell \) -adic properties of the partition function. Adv. Math. 229(3), 1586–1609 (2012). Appendix A by Nick Ramsey<br>6.Nicolas, J.-L., Serre, J.-P.: Formes modulaires modulo 2: l’ordre de nilpotence des opérateurs de Hecke. C. R. Math. Acad. Sci. Paris 350(7–8), 343–348 (2012)MathSciNet CrossRef <br>7.Nicolas, J.-L., Serre, J.-P.: Formes modulaires modulo 2: structure de l’algèbre de Hecke. C. R. Math. Acad. Sci. Paris 350(9–10), 449–454 (2012)MathSciNet CrossRef <br>8.Ono, K.: Parity of the partition function in arithmetic progressions. J. Reine Angew. Math. 472, 1–15 (1996)MathSciNet MATH <br>9.Ono, K.: Distribution of the partition function modulo \(m\) . Ann. Math. (2), 151(1), 293–307 (2000)<br>10.Radu, C.-S.: A proof of Subbarao’s conjecture. J. Reine Angew. Math. 672, 161–175 (2012)MathSciNet MATH <br>11.Serre, J.-P.: Valeurs propres des opérateurs de Hecke modulo \(l\) . In: Journées Arithmétiques de Bordeaux (Conference University of Bordeaux, 1974), pp. 109–117. Astérisque, Nos. 24–25. Society Mathematics France, Paris (1975)<br>12.Subbarao, M.V.: Some remarks on the partition function. Am. Math. Mon. 73, 851–854 (1966)MathSciNet CrossRef MATH <br>
• 作者单位：Geoffrey D. Smith (1) <br> Lynnelle Ye (2) <br><br>1. Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT, 06511, USA <br> 2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA <br>
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics<br>Number Theory<br>Field Theory and Polynomials<br>Combinatorics<br>Fourier Analysis<br>Functions of a Complex Variable<br>
• 出版者：Springer U.S.
• ISSN：1572-9303

文摘

Let \(p(n)\) be the partition function. Ahlgren and Ono conjectured that every arithmetic progression contains infinitely many integers \(N\) for which \(p(N)\) is not congruent to \(0\;(\mathrm{mod}\;3)\). Radu proved this conjecture in 2010 using the work of Deligne and Rapoport. In this note, we give a simpler proof of Ahlgren and Ono’s conjecture in the special case where the modulus of the arithmetic progression is a power of \(3\) by applying a method of Boylan and Ono and using the work of Bellaïche and Khare generalizing Nicolas and Serre’s results on the local nilpotency of the Hecke algebra.