<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ f $</EquationSource> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ g $</EquationSource> <EquationSource Format="MATHML"><math> <mi>g</mi> </math></EquationSource> </InlineEquation> be two distinct normalized Hecke eigenforms of even integral weights for the full modular group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \textrm{SL}_2(\mathbb {Z}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Fix integers <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \ell , \mathfrak {u} \geqslant 3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>,</mo> <mi mathvariant="fraktur">u</mi> <mo>⩾</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We obtain asymptotic formulas for the sums of the coefficients of an <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ L $</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-function constructed by applying the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ \ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-fold to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ f $</EquationSource> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> </InlineEquation> and the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ \mathfrak {u} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">u</mi> </math></EquationSource> </InlineEquation>-fold to <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ g $</EquationSource> <EquationSource Format="MATHML"><math> <mi>g</mi> </math></EquationSource> </InlineEquation>, evaluated at integers <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ n \leqslant X + 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩽</mo> <mi>X</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ n \equiv 1 \ (\textrm{mod}\,{q}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≡</mo> <mn>1</mn> <mspace width="4pt" /> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="0.166667em" /> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we investigate shifted convolution sums of these coefficients, employing a kernel function that extends the analytic framework originally developed by Ivić and Tenenbaum.</p>

Shifted convolution sums and asymptotic analysis of the coefficients of an L-function associated with Hecke eigenforms

期刊论文
发表日期：

2026年1月3日

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Naveen K. Godara,
Anuj Jakhar

摘要

Let $ f $ $f$ and $ g $ $g$ be two distinct normalized Hecke eigenforms of even integral weights for the full modular group $ \textrm{SL}_2(\mathbb {Z}) $ ${SL}_{2} (Z)$ . Fix integers $ \ell , \mathfrak {u} \geqslant 3 $ $ℓ, u ⩾ 3$ . We obtain asymptotic formulas for the sums of the coefficients of an $ L $ $L$ -function constructed by applying the $ \ell $ $ℓ$ -fold to $ f $ $f$ and the $ \mathfrak {u} $ $u$ -fold to $ g $ $g$ , evaluated at integers $ n \leqslant X + 1 $ $n ⩽ X + 1$ satisfying $ n \equiv 1 \ (\textrm{mod}\,{q}) $ $n \equiv 1 (mod q)$ . Furthermore, we investigate shifted convolution sums of these coefficients, employing a kernel function that extends the analytic framework originally developed by Ivić and Tenenbaum.