Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\kappa _{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>κ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\kappa _{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>κ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> for the full modular group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma =\textrm{SL}\hspace{0.55542pt}(2,\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mtext>SL</mtext> <mspace width="0.55542pt" /> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda _{\textrm{sym}^{i}\!f\times \textrm{sym}^{j}g}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <msup> <mtext>sym</mtext> <mi>i</mi> </msup> <mspace width="-0.166667em" /> <mi>f</mi> <mo>×</mo> <msup> <mtext>sym</mtext> <mi>j</mi> </msup> <mi>g</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the n-th normalized coefficient of the Dirichlet expansion of the Rankin–Selberg L-function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L(\textrm{sym}^{i}\!f\hspace{0.55542pt}{\times }\hspace{1.111pt}\textrm{sym}^{j}g,s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <msup> <mtext>sym</mtext> <mi>i</mi> </msup> <mspace width="-0.166667em" /> <mi>f</mi> <mspace width="0.55542pt" /> <mo>×</mo> <mspace width="1.111pt" /> <msup> <mtext>sym</mtext> <mi>j</mi> </msup> <mi>g</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> associated to f and g. In this paper, we mainly investigate the averages of shifted convolution sums related to the general divisor problem of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda _{\textrm{sym}^{i}\!f\times \textrm{sym}^{j}g}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <msup> <mtext>sym</mtext> <mi>i</mi> </msup> <mspace width="-0.166667em" /> <mi>f</mi> <mo>×</mo> <msup> <mtext>sym</mtext> <mi>j</mi> </msup> <mi>g</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for any given positive integers <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$i,j\geqslant 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Similar results can also be obtained in the setting of the Hecke–Maass cusp forms on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. These results extend the previous results in this direction.

On averages of shifted convolutions associated to coefficients of Rankin–Selberg L-functions

期刊论文
发表日期：

2026年7月23日

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Guodong Hua

摘要

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights $\kappa _{1}$ $κ_{1}$ and $\kappa _{2}$ $κ_{2}$ for the full modular group $\Gamma =\textrm{SL}\hspace{0.55542pt}(2,\mathbb {Z})$ $Γ = SL (2, Z)$ , and let $\lambda _{\textrm{sym}^{i}\!f\times \textrm{sym}^{j}g}(n)$ $λ_{{sym}^{i} f \times {sym}^{j} g} (n)$ be the n-th normalized coefficient of the Dirichlet expansion of the Rankin–Selberg L-function $L(\textrm{sym}^{i}\!f\hspace{0.55542pt}{\times }\hspace{1.111pt}\textrm{sym}^{j}g,s)$ $L ({sym}^{i} f \times {sym}^{j} g, s)$ associated to f and g. In this paper, we mainly investigate the averages of shifted convolution sums related to the general divisor problem of $\lambda _{\textrm{sym}^{i}\!f\times \textrm{sym}^{j}g}(n)$ $λ_{{sym}^{i} f \times {sym}^{j} g} (n)$ , for any given positive integers $i,j\geqslant 1$ $i, j ⩾ 1$ . Similar results can also be obtained in the setting of the Hecke–Maass cusp forms on $\Gamma $ $Γ$ . These results extend the previous results in this direction.