For positive integer n, let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma _0^{-}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>σ</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the difference between the number of odd divisors of n and the number of even divisors of n. Recently, Merca represented <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sigma _0^{-}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>σ</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a_m^{-}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>a</mi> <mi>m</mi> <mo>-</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a_m^{-}(n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>a</mi> <mi>m</mi> <mo>-</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the difference between the number of parts <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\equiv m \pmod {2m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≡</mo> <mi>m</mi> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the number of parts <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\equiv 0 \pmod {2m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in all the partitions of n. At the end of his paper, Merca posed two conjectures involving <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma _0^{-}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>σ</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$a_m^{-}(n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>a</mi> <mi>m</mi> <mo>-</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we confirm the two conjectures of Merca based on some transformation formulas of q-series and a result due to Pólya and Szegő.

Proofs of two conjectures of Merca on the number of divisors

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发表日期：

2025年11月21日

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Yue Qi,
Olivia X. M. Yao

摘要

For positive integer n, let $\sigma _0^{-}(n)$ $σ_{0}^{-} (n)$ denote the difference between the number of odd divisors of n and the number of even divisors of n. Recently, Merca represented $\sigma _0^{-}(n)$ $σ_{0}^{-} (n)$ in terms of $a_m^{-}(n)$ $a_{m}^{-} (n)$ , where $a_m^{-}(n) $ $a_{m}^{-} (n)$ is the difference between the number of parts $\equiv m \pmod {2m}$ $\equiv m (mod 2 m)$ and the number of parts $\equiv 0 \pmod {2m}$ $\equiv 0 (mod 2 m)$ in all the partitions of n. At the end of his paper, Merca posed two conjectures involving $\sigma _0^{-}(n)$ $σ_{0}^{-} (n)$ and $a_m^{-}(n) $ $a_{m}^{-} (n)$ . In this paper, we confirm the two conjectures of Merca based on some transformation formulas of q-series and a result due to Pólya and Szegő.