Distribution of values of Hardy sums over Chebyshev polynomials

Jiankang Wang; Zhefeng Xu; Minmin Jia; Jiankang Wang; Zhefeng Xu; Minmin Jia

doi:10.3934/math.2024186

AIMS Mathematics

2024, Volume 9, Issue 2: 3788-3797. doi: 10.3934/math.2024186

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Research article

Distribution of values of Hardy sums over Chebyshev polynomials

Research Center for Number Theory and Its Applications, Northwest University, Xi'an 710127, China

Received: 24 November 2023 Revised: 31 December 2023 Accepted: 05 January 2024 Published: 10 January 2024
MSC : 11F20, 11B83

This paper mainly studied the distribution of values of Hardy sums involving Chebyshev polynomials. By using the method of analysis and the arithmetic properties of Hardy sums and Chebyshev polynomials of the first kind, we obtained a sharp asymptotic formula for the hybrid mean value of Hardy sums $ S_5(h, q) $ involving Chebyshev polynomials of the first kind. In addition, we also gave the value of Hardy sums $ S(h, q) $ and $ S_{3}(h, q) $ involving Chebyshev polynomials. Finally, we found the reciprocal formulas of $ S_{3}(h, q) $ and $ S_{4}(h, q) $ involving Chebyshev polynomials of the first kind.
- Hardy sums,
- Chebyshev polynomials,
- values
Citation: Jiankang Wang, Zhefeng Xu, Minmin Jia. Distribution of values of Hardy sums over Chebyshev polynomials[J]. AIMS Mathematics, 2024, 9(2): 3788-3797. doi: 10.3934/math.2024186

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Abstract

This paper mainly studied the distribution of values of Hardy sums involving Chebyshev polynomials. By using the method of analysis and the arithmetic properties of Hardy sums and Chebyshev polynomials of the first kind, we obtained a sharp asymptotic formula for the hybrid mean value of Hardy sums $ S_5(h, q) $ involving Chebyshev polynomials of the first kind. In addition, we also gave the value of Hardy sums $ S(h, q) $ and $ S_{3}(h, q) $ involving Chebyshev polynomials. Finally, we found the reciprocal formulas of $ S_{3}(h, q) $ and $ S_{4}(h, q) $ involving Chebyshev polynomials of the first kind.

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