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<p>In&#xa0;this paper, we investigate the fourth moments of two-term exponential sums weighted by Dirichlet characters.Using properties of quadratic residues and the third-order character <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \psi $</EquationSource> </InlineEquation> modulo a&#xa0;prime <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ p $</EquationSource> </InlineEquation>, we first present an&#xa0;explicit identity for the fourth moment of cubic exponential sums weighted by the Legendre symbol when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ (3,p-1)=3 $</EquationSource> </InlineEquation>.Subsequently, by counting solutions to certain congruence equations, we derive some results on the fourth mixed moments associated with quartic exponential sums weighted by an&#xa0;arbitrary Dirichlet character <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \chi\bmod p $</EquationSource> </InlineEquation>.</p>

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On the Fourth Moments of Exponential Sums Weighted by Dirichlet Characters

  • 期刊论文

  • 发表日期：

    2026年7月30日

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  • S. Cao,
  • T. Wang

摘要

In this paper, we investigate the fourth moments of two-term exponential sums weighted by Dirichlet characters.Using properties of quadratic residues and the third-order character $ \psi $ modulo a prime $ p $ , we first present an explicit identity for the fourth moment of cubic exponential sums weighted by the Legendre symbol when $ (3,p-1)=3 $ .Subsequently, by counting solutions to certain congruence equations, we derive some results on the fourth mixed moments associated with quartic exponential sums weighted by an arbitrary Dirichlet character $ \chi\bmod p $ .