×

Asymptotic estimates for a class of summatory functions. (English) Zbl 0917.11045

The authors prove asymptotic formulae with [rather] sharp remainder terms for arithmetical functions \( n \mapsto a(n) \), representable as a Dirichlet convolution \( a = 1 \ast v \). Under suitable assumptions there is an asymptotic formula \[ \sum_{n \leq x} a(n) = L \cdot x + x^{1 - \gamma \over r} \cdot \sum^R_0 {\mathcal B}_m \cdot (\log x)^{\alpha - m - \rho} - \sum_{n \leq y} v(n) \cdot \psi\left( {x \over n} \right) + \text{ error term.} \] Moreover, formulae for \( \sum_{n \leq x} n^\beta \cdot a(n) \), and for the mean \( {1 \over x} \cdot \sum_{n \leq x} {\mathcal E}(n) \) of the error term are given.
The main result is applicable, for example, if the arithmetical function \( a \) is defined by the generating Dirichlet series \[ \sum_1^\infty {a(n) \over n^s} = \zeta(s) \cdot \xi(rs+\gamma) \cdot f(rs+\gamma), \] with the abbreviation \(\xi(s) = \zeta^{\alpha_1}(s) \cdot L^{\alpha_2} (s, \chi_2) \cdots L^{\alpha_n} (s, \chi_n)\), where \( r > 0 \) is an integer, \( 1 - r < \gamma \leq 1 \), \( \alpha_i \in {\mathbb C} \), and the \( \chi \) are non-principal characters \(\text{mod }m\), and \( f(rs+\gamma) = \sum {b_n \over n^s} \) is a Dirichlet series with abscissa of absolute convergence \( < {1 - \gamma \over r} \).

MSC:

11N37 Asymptotic results on arithmetic functions
11M41 Other Dirichlet series and zeta functions
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Balakrishnan, U.; Pétermann, Y.-F. S., The Dirichlet series of \(ζs ζ^α (sfs\), Acta Arithmetica, 75, 39-69 (1996) · Zbl 0846.11054
[2] U. Balakrishnan, Y.-F. S. Pétermann, Asymptotic estimates for a class of summatory functions, sequel; U. Balakrishnan, Y.-F. S. Pétermann, Asymptotic estimates for a class of summatory functions, sequel · Zbl 0999.11057
[3] Berndt, B. C., A new method in arithmetical functions and contour integration, Canad. Math. Bull, 16, 381-387 (1973) · Zbl 0276.10004
[4] Borevich, Z. I.; Shafarevich, I. R., Number Theory (1966), Academic Press: Academic Press New York/London
[5] Gegenbauer, L., Über einige zahlentheoretische Functionen, Sitzungber. Math.-Naturw. Cl. Akad. Wiss. Wien, 89-2, 37-79 (1884) · JFM 16.0149.01
[6] Gegenbauer, L., Zahlentheoretische Sätze, Denkschr. Math.-Naturw. Cl. Akad. Wiss. Wien, 57, 497-530 (1890)
[7] Ishibashi, M.; Kanemitsu, S., Some Asymptotic Formulae of Ramanujan. Some Asymptotic Formulae of Ramanujan, Springer Lect. Notes Math., 1434 (1990), p. 149-167 · Zbl 0716.11044
[8] Liu, Jianya, On an error term of Chowla I, J. Number Theory, 64, 20-35 (1997) · Zbl 0871.11062
[9] McCarthy, P. J., Generating functions of some products of arithmetical functions, Publ. Math. Debrecen, 38, 83-91 (1991) · Zbl 0722.11005
[10] Y.-F. S. Pétermann, On an estimate of Walfisz and Saltykov for an error term related to the Euler function; Y.-F. S. Pétermann, On an estimate of Walfisz and Saltykov for an error term related to the Euler function · Zbl 0917.11047
[11] Ramanujan, S., Some formulae in the theory of numbers, Mess. Math., XLV, 81-84 (1916)
[12] Sita Ramaiah, V.; Suryanarayana, D., On a method of Eckford Cohen, Bolletino U.M.I. Ser. 6A, 1, 1235-1251 (1982) · Zbl 0504.10023
[13] Tenenbaum, G., Introduction à la théorie analytique et probabiliste des nombres (1990), Institut Élie Cartan 13 · Zbl 0788.11001
[14] Walfisz, A., Weylsche Exponentialsummen in der neueren Zalhentheorie (1963), VEB Deutscher Verlag der Wissenschaften: VEB Deutscher Verlag der Wissenschaften Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.