Abstract
In this article, we use elementary methods and the estimate for character sums to study the mean value properties of a new arithmetical function, and obtain a sharp asymptotic formula for it.
References
Andrea, S.: Least primitive root and simultaneous power non-residues. J. Number Theory 204, 246–263 (2019)
Anwar, M., Pappalardi, F.: On simultaneous primitive roots. Acta. Arith. 180, 35–43 (2017)
Apostol, T.M.: Introduction to analytic number theory. Springer-Verlag, New York (1976)
Banks, W.D., Shparlinski, I.E.: Sums with convolutions of Dirichlet characters. Manuscripta Math. 133, 105–144 (2010)
Berndt, B.C.: Character analogues of the poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7(4), 413–445 (1975)
Cohen, S.D., Trudgian, T.: On the least square-free primitive root modulo \(p\). J. Number Theory 170, 10–16 (2017)
Cohen, S.D., Trudgian, T.: Lehmer numbers and primitive roots modulo a prime. J. Number Theory 203, 68–79 (2019)
Hua, L.K.: Introduction to number theory. Science Press, Beijing (1979)
Ivić, A.: The Riemann zeta-function, the theory of the Riemann zeta-function with applications. John Wiley & Sons Inc, London (1985)
Munsch, M., Trudgian, T.: Square-full primitive roots. Int. J. Number Theory 14, 1013–1021 (2018)
Narkiewicz, W.: Classical problems in number theory. Polish Scientifc Publishers, WARSZAWA (1986)
Richert, H.E.: Über die anzahl abelscher gruppen gegebener Ordnung. II. Math. Z. 58(1), 71–84 (1953)
Srichan, T.: On the distribution of square-full and cube-full primitive roots. Period. Math. Hungar 80, 103–107 (2020)
Srichan, T.: A bound of sums with convolutions of Dirichlet characters. Notes Number Theory Discrete Math. 26, 70–74 (2020)
Sun, Q.: On primitive roots in a finite field. J. Sichuan Univ. Nat. Sci. Ed. 25, 133–139 (1988)
Tian, T., Qi, W.: Primitive normal element and its inverse in finite fields. Acta Math. Sinica. 49, 657–668 (2006)
Wang, J.P.: On Golomb’s conjecture. Sci. China (Ser A) 9, 927–935 (1987)
Wang, W.Q., Zhang, W.P.: A mean value related to primitive roots and Golomb’s conjectures. Abstract and Applied analysis , 908273 (2014)
Wang, T.T., Wang, X.N.: On the Golomb’s conjecture and Lehmer’s numbers. Open Math. 15, 1003–1009 (2017)
Zhang, W.P.: On a problem related to Golomb’s conjectures. J Syst Sci Complex 16, 13–18 (2003)
Zhang, W.P., Li, H.L.: Elementary number theory. Shaanxi Normal University Press, Xi’an (2013)
Zhang, W.P., Wang, T.T.: The primitive roots and a problem related to the Golomb conjecture. AIMS Math. 5, 3899–3905 (2020)
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This work is supported by the N. S. F. (11771351) of People’s Republic of China.
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Zhang, W., Srichan, T. & Lv, X. A new arithmetical function and its mean value properties. Bol. Soc. Mat. Mex. 27, 83 (2021). https://doi.org/10.1007/s40590-021-00390-8
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DOI: https://doi.org/10.1007/s40590-021-00390-8