<p>In his last letter to Hardy, Ramanujan introduced seventeen mock theta functions. Recently, Das established several congruences modulo small powers of 3 between the coefficients of two second order mock theta functions, namely, Ramanujan’s <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu _2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A_2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and further conjectured one congruence modulo 2187 and three congruences modulo 6561 between the coefficients of these mock theta functions. These conjectures were subsequently proved by the second author. In this paper, we establish two infinite families of congruences modulo high powers of 3 relating the coefficients of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mu _2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A_2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, one of which confirms a conjecture previously proposed by the second author (2025).</p>

Congruences Modulo Powers of 3 between the Coefficients of Mock Theta Functions

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发表日期：

2026年2月2日

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Yu Guan,
Dazhao Tang

摘要

In his last letter to Hardy, Ramanujan introduced seventeen mock theta functions. Recently, Das established several congruences modulo small powers of 3 between the coefficients of two second order mock theta functions, namely, Ramanujan’s $\mu _2(q)$ $μ_{2} (q)$ and $A_2(q)$ $A_{2} (q)$ , and further conjectured one congruence modulo 2187 and three congruences modulo 6561 between the coefficients of these mock theta functions. These conjectures were subsequently proved by the second author. In this paper, we establish two infinite families of congruences modulo high powers of 3 relating the coefficients of $\mu _2(q)$ $μ_{2} (q)$ and $A_2(q)$ $A_{2} (q)$ , one of which confirms a conjecture previously proposed by the second author (2025).