Abstract
Generalizing two results of Rieger [8] and Selberg [10] we give asymptotic formulas for sums of type
where χ is a suitable multiplicative function, f1,…, f r are “small” additive, prime-independent arithmetical functions and k, l are coprime. The proofs are based on an analytic method which consists of considering the Dirichlet series generated by \( \chi(n)z_{1}^{f_{1}(n)}\cdot... \cdot z_{r}^{f_{r}(n)},z_{1}\dots z_{r} \) complex.
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References
J.-M. De Koninck and A. Ivic. Topics in Arithmetical Functions. North-Holland Mathematics Studies 43. North Holland Publisher Company, Amsterdam-New York-Oxford, 1980.
H. Delange. Sur des formules de Atle Selberg. Acta Arithmetica, 19, 105–146, 1971.
J. Grah. Comportement moyen du cardinal de certains ensembles de facteurs premiers. Mh. ath., 118, 91–109, 1994.
B. Greuel. Über Summen zahlentheoretischer Funktionen. Arch. Math., 67, 383–394, 1996.
A. Ivic. The Riemann Zeta-Function. John Wiley & Sons, New York-Chichester-Brisbane— oronto-Singapore, 1985.
A. Ivic. Sums of products of certain arithmetical functions. Publ. Inst. Math., Nouv. Ser. 41(55), 31–41, 1987.
H. Nakaya. On the generalized divisor problem in arithmetic progressions. Sci. Rep. Kanazawa niv., 37 (1), 23–47, 1992.
G. J. Rieger. Zum Teilerproblem von Atle Selberg. Math. Nachr., 30, 181–192, 1965.
W. Schwarz und J. Spilker. Arithmetical Functions. London Mathematical Society, Lecture ote Series 184, 1994.
A. Selberg. Note on a paper by L.G. Sathe. J. Indian Math. Soc, 18, 83–87, 1954.
G. Tenenbaum. Introduction à la théorie analytique et probaliste des nombres. Inst. Elie Cartan, ancy 13, 1990.
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Greuel, B. Two results on arithmetical functions. Results. Math. 32, 80–86 (1997). https://doi.org/10.1007/BF03322527
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DOI: https://doi.org/10.1007/BF03322527