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<p>It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (J. Reine Angew. Math. 179:97–128, 1938) and Atkin and O’Brien (Trans. Am. Math. Soc. 126:442–459, 1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for the analogous rank parity function is <i>f</i>(<i>q</i>), the first example of a mock theta-function that Ramanujan mentioned in his last letter to Hardy. Recently we proved congruences modulo powers of 5 for the rank parity function, and here we extend these congruences for powers of 7. We also show how these congruences imply congruences modulo powers of 5 and 7 for the coefficients of the related third order mock theta-function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega (q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, using Atkin-Lehner involutions and transformation results of Zwegers. Finally we prove a family of congruences modulo powers of 7 for the crank parity function.</p>

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Congruences modulo powers of 5 and 7 for the crank and rank parity functions and related mock theta-functions

  • 期刊论文

  • 发表日期：

    2026年5月18日

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  • Dandan Chen,
  • Rong Chen,
  • Frank Garvan

摘要

It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (J. Reine Angew. Math. 179:97–128, 1938) and Atkin and O’Brien (Trans. Am. Math. Soc. 126:442–459, 1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for the analogous rank parity function is f(q), the first example of a mock theta-function that Ramanujan mentioned in his last letter to Hardy. Recently we proved congruences modulo powers of 5 for the rank parity function, and here we extend these congruences for powers of 7. We also show how these congruences imply congruences modulo powers of 5 and 7 for the coefficients of the related third order mock theta-function \(\omega (q)\) ω ( q ) , using Atkin-Lehner involutions and transformation results of Zwegers. Finally we prove a family of congruences modulo powers of 7 for the crank parity function.