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Asymptotic Estimates for a Class of Summatory Functions

✍ Scribed by U Balakrishnan; Y.-F.S Pétermann

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
402 KB
Volume
70
Category
Article
ISSN
0022-314X

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✦ Synopsis

We establish explicit expressions for both P and E in n x a(n)=P(x)+E(x)= principal term''+error term'', when the (complex) arithmetical function a has a generating function of the form (s) Z(s), where is the Riemann zeta function, and where Z has a representation as a Dirichlet series having an abscissa of absolute convergence smaller than 1 (and satisfying some other conditions). We obtain O-estimates (and in some cases 0-estimates) on E. We also obtain asymptotic expressions for n x n ; a(n) when the real number ; is not too small, and for

x 1 E(t) dt and n x E(n). This can be applied to a number of arithmetical functions a which have been studied in the literature with various methods. In most cases what we obtain improves on, or extends, the existing results.

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