Let λf (n) denote the normalized Fourier coefficients of a primitive holomorphic cusp form f(z) of even integral weight k for the full modular group. In this paper, we investigate the error terms of the summatory function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\sum }_{n\le x}{\lambda }_{f}^{2}\left({n}^{j}\right),j=\text{3,4},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msubsup> <mi>λ</mi> <mrow> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mfenced close=")" open="("> <msup> <mrow> <mi>n</mi> </mrow> <mi>j</mi> </msup> </mfenced> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mtext>3,4</mtext> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and establish the following Ω-results:<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sum_{n\le x}{\lambda }_{f}^{2}\left({n}^{3}\right)={c}_{3}x+\Omega \left({x}^{15/32}\right), \sum_{n\le x}{\lambda }_{f}^{2}\left({n}^{4}\right)={c}_{4}x+\Omega \left({x}^{12/25}\right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msubsup> <mi>λ</mi> <mrow> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mfenced close=")" open="("> <msup> <mrow> <mi>n</mi> </mrow> <mn>3</mn> </msup> </mfenced> <mo>=</mo> <msub> <mi>c</mi> <mn>3</mn> </msub> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Ω</mi> <mfenced close=")" open="("> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>15</mn> <mo stretchy="false">/</mo> <mn>32</mn> </mrow> </msup> </mfenced> <mo>,</mo> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msubsup> <mi>λ</mi> <mrow> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mfenced close=")" open="("> <msup> <mrow> <mi>n</mi> </mrow> <mn>4</mn> </msup> </mfenced> <mo>=</mo> <msub> <mi>c</mi> <mn>4</mn> </msub> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Ω</mi> <mfenced close=")" open="("> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>12</mn> <mo stretchy="false">/</mo> <mn>25</mn> </mrow> </msup> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation>where c3 and c4 are suitable constants.

Omega theorems for the coefficients of automorphic L-functions

期刊论文
发表日期：

2026年1月22日

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Dan Wang

摘要

Let λ_f (n) denote the normalized Fourier coefficients of a primitive holomorphic cusp form f(z) of even integral weight k for the full modular group. In this paper, we investigate the error terms of the summatory function ${\sum }_{n\le x}{\lambda }_{f}^{2}\left({n}^{j}\right),j=\text{3,4},$ $\sum_{n \leq x} λ_{f}^{2} (n^{j}), j = 3,4,$ and establish the following Ω-results:

$\sum_{n\le x}{\lambda }_{f}^{2}\left({n}^{3}\right)={c}_{3}x+\Omega \left({x}^{15/32}\right), \sum_{n\le x}{\lambda }_{f}^{2}\left({n}^{4}\right)={c}_{4}x+\Omega \left({x}^{12/25}\right),$ $\sum_{n \leq x} λ_{f}^{2} (n^{3}) = c_{3} x + Ω (x^{15 / 32}), \sum_{n \leq x} λ_{f}^{2} (n^{4}) = c_{4} x + Ω (x^{12 / 25}),$

where c₃ and c₄ are suitable constants.