This paper investigates the distribution of a composite arithmetic function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Z_{f,g}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> constructed from the Hecke eigenvalues of two distinct normalized primitive eigenforms f and g. We establish an asymptotic formula for the sum of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Z_{f,g}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in reduced residue classes <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n \equiv c \bmod q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≡</mo> <mi>c</mi> <mspace width="0.277778em" /> <mo>mod</mo> <mspace width="0.277778em" /> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(c,q)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, valid for a range of the modulus q determined by a subconvexity-based error exponent. As a corollary, we obtain a refined error term for the sum in the absence of the modulus restriction. Furthermore, we examine the correlation of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Z_{f,g}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with m-full kernel functions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\nu (n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> considered by Venkatasubbareddy and Sankaranarayanan [2024, J. Number Theory] through shifted convolution sums. Our results show that while the main term remains consistent across these problems, the power-saving exponents are sensitive to the arithmetic weights and the sparsity of the m-full numbers.

Sums of the coefficients of L-functions in residue classes and their applications

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2026年6月2日

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Rupam Barman,
Naveen K. Godara,
Anuj Jakhar

摘要

This paper investigates the distribution of a composite arithmetic function $Z_{f,g}(n)$ $Z_{f, g} (n)$ constructed from the Hecke eigenvalues of two distinct normalized primitive eigenforms f and g. We establish an asymptotic formula for the sum of $Z_{f,g}(n)$ $Z_{f, g} (n)$ in reduced residue classes $n \equiv c \bmod q$ $n \equiv c mod q$ with $(c,q)=1$ $(c, q) = 1$ , valid for a range of the modulus q determined by a subconvexity-based error exponent. As a corollary, we obtain a refined error term for the sum in the absence of the modulus restriction. Furthermore, we examine the correlation of $Z_{f,g}(n)$ $Z_{f, g} (n)$ with m-full kernel functions $\nu (n)$ $ν (n)$ considered by Venkatasubbareddy and Sankaranarayanan [2024, J. Number Theory] through shifted convolution sums. Our results show that while the main term remains consistent across these problems, the power-saving exponents are sensitive to the arithmetic weights and the sparsity of the m-full numbers.