In this paper, the generating functions of Garvan’s so-called k-ranks are used, to define a family of mock Eisenstein series. The k-rank moments are then expressed as partition traces of these functions. We explore the modular properties of this new family, give recursive formulas for them involving divisor like sums, and prove that their Fourier coefficients are integral. Furthermore, we show that these functions lie in an algebra that is generated only by derivatives up to a finite order but is nevertheless closed under differentiation. In the process, we also answer a question raised by Bringmann, Pandey and van Ittersum by showing that the divisor like sum <Equation ID="Equ38"> <EquationSource Format="TEX">$\begin{aligned} \left( 1-2^{\ell -1} \right) \frac{B_\ell }{2\ell }+ \sum _{2n-1 \ge bm \ge b} (2n-bm)^{\ell -1} q^{mn} - \sum _{m-1 \ge 2bn \ge 2b} (m-2bn)^{\ell -1} q^{mn}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfenced close=")" open="("> <mn>1</mn> <mo>-</mo> <msup> <mn>2</mn> <mrow> <mi>ℓ</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mfenced> <mfrac> <msub> <mi>B</mi> <mi>ℓ</mi> </msub> <mrow> <mn>2</mn> <mi>ℓ</mi> </mrow> </mfrac> <mo>+</mo> <munder> <mo>∑</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>≥</mo> <mi>b</mi> <mi>m</mi> <mo>≥</mo> <mi>b</mi> </mrow> </munder> <msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mi>b</mi> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ℓ</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow> <mi mathvariant="italic">mn</mi> </mrow> </msup> <mo>-</mo> <munder> <mo>∑</mo> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>≥</mo> <mn>2</mn> <mi>b</mi> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mi>b</mi> </mrow> </munder> <msup> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> <mi>b</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ℓ</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow> <mi mathvariant="italic">mn</mi> </mrow> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>has a quasi-completion, when <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$b\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> is odd.

Mock modular forms from the k-rank moments

期刊论文
发表日期：

2026年3月26日

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Kilian Rausch

摘要

In this paper, the generating functions of Garvan’s so-called k-ranks are used, to define a family of mock Eisenstein series. The k-rank moments are then expressed as partition traces of these functions. We explore the modular properties of this new family, give recursive formulas for them involving divisor like sums, and prove that their Fourier coefficients are integral. Furthermore, we show that these functions lie in an algebra that is generated only by derivatives up to a finite order but is nevertheless closed under differentiation. In the process, we also answer a question raised by Bringmann, Pandey and van Ittersum by showing that the divisor like sum $\begin{aligned} \left( 1-2^{\ell -1} \right) \frac{B_\ell }{2\ell }+ \sum _{2n-1 \ge bm \ge b} (2n-bm)^{\ell -1} q^{mn} - \sum _{m-1 \ge 2bn \ge 2b} (m-2bn)^{\ell -1} q^{mn}, \end{aligned}$ $\begin{matrix} (1 - 2^{ℓ - 1}) \frac{B_{ℓ}}{2 ℓ} + \sum_{2 n - 1 \geq b m \geq b} {(2 n - b m)}^{ℓ - 1} q^{mn} - \sum_{m - 1 \geq 2 b n \geq 2 b} {(m - 2 b n)}^{ℓ - 1} q^{mn}, \end{matrix}$ has a quasi-completion, when $b\ge 3$ $b \geq 3$ is odd.