On Averages of Shifted Convolutions with Applications to $GL(2)$ and $GL(3)$ Fourier Coefficients

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This is a Quantitative Study study.

Jongchon Kim

doi:10.48550/arxiv.2509.24152

September 29, 2025Open Access

On Averages of Shifted Convolutions with Applications to $GL(2)$ and $GL(3)$ Fourier Coefficients

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Key Points

Non-trivial upper bounds are established for weighted averages involving shifted convolutions.
The study applies square-root cancellation over short intervals for GL(2) Fourier coefficients.
General multiplicative functions are explored through the lens of shifted sums.
The Hardy-Littlewood circle method is employed to tackle Fourier coefficients effectively.

Implication

This analysis reveals non-trivial upper bounds for shifted convolutions involving GL(2) and GL(3), suggesting new insights into Fourier coefficients.

Abstract

In this paper, we study the average of shifted sum for general multiplicative functions. As applications, we prove non-trivial upper bounds for weighted averages of shifted convolutions involving $GL(2)$ and $GL(3)$ Fourier coefficients without smoothing. We apply square-root cancellation on average over short intervals for $GL(2)$ Fourier coefficients with the standard Hardy-Littlewood circle method.

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Cite This Study

Jongchon Kim (2025) studied this question.

synapsesocial.com/papers/68f5fcce8d54a28a75cf1b7b https://doi.org/10.48550/arxiv.2509.24152

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On Averages of Shifted Convolutions with Applications to $GL(2)$ and $GL(3)$ Fourier Coefficients

Key Points

Implication

Abstract

Discussion

Cite This Study

On Averages of Shifted Convolutions with Applications to $GL(2)$ and $GL(3)$ Fourier Coefficients

Key Points

Implication

Abstract

Discussion

Cite This Study

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