An Asymptotic Formula for Estimating the Mean of A Class of Indices

Wanchun Pu

doi:10.54097/g1cz3c37

Authors

Wanchun Pu

DOI:

https://doi.org/10.54097/g1cz3c37

Keywords:

Asymptotic formula, Riemann Zeta function, The principle of index pairs, Vinogradov method.

Abstract

Let , , with , . This article uses the exponential pair principle and Vinogradov method to first obtain some bound exponential sums of the type where , and then derives the asymptotic formula of . The asymptotic formula of is an important tool for studying the mean estimation of the exponential divisor functions and divisor functions. By using the asymptotic formula of , we can derive more accurate mean estimates of the divisor function and exponential divisor function. Therefore, with the help of the asymptotic formula of , we can have a deeper understanding of the divisor function and exponential divisor function, and then provide solutions for more number theory problems.

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References

[1] Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Mathematica Hungarica, vol. 77, no. 1/2, pp. 159–175, 1997.

[2] F. Zhao & J. Wu, On a sum involving the sum-of-divisors function, J. Math., 2021.

[3] W. Zhai , On a sum involving the Euler function, Journal of Number Theory, vol. 211, pp. 199-219, 2020.

[4] J. Ma, J. Wu & F. Zhao, On a generalization of Bordellès-Dai-Heyman-Pan-Shparlinski’s conjecture, J. Number Theory, 2022, 236: 334-348.

[5] A. Smati and J. Wu , On The Exponential Divisor Function, Pubi. Inst. Math. Beograd to appear. 61 (75), 1997, 21-32.

[6] J. Wu, On a sum involving the Euler totient function, Inda Math., 2019, 30: 536-541.

[7] A. Karatsuba, Estimates for trigonometric sums by Vinogradov’s method, and some applications, Proc. St. Inst. Math., 1971, 112: 251-265.

[8] Y.-F. S. Pétermann, About a theorem of Paolo Codecá’s and estimates for arithmetical convolutions, J. Number Theory, 30(1988),71-85; Addendum ibid.,36(1990) 322-327.

[9] S. W. Graham and G. Kolesnik, Van der Corput’s Method of Exponential Sums, Cambridge University Press, Cambridge, UK, 1991.