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Dirichlet series whose partial sums of coefficients have regular variation

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Abstract

We study pairs of Dirichlet series\(A(s) = \sum {_{n = 1}^\infty a(n)n^{ - s} } \) and\(P(s) = \sum {_{n = 2}^\infty p(n)n^{ - s} } \) in whicha(n) counts the number of objects of “size”n of some class of objects which is closed under formation of direct products and extraction of irreducible factors, andp(n) counts the number of objects of “size”n which are irreducible in this class. We prove Dirichlet series analogues of certain results about power series and use these results to prove some conjectures of Burris concerning first-order 0–1 laws.

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  1. Department of Mathematics, University of Michigan, East Hall 525 East University Ave., 48109-1109, Ann Arbor, MI, USA

    Jason P. Bell

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  1. Jason P. Bell
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Correspondence to Jason P. Bell.

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Bell, J.P. Dirichlet series whose partial sums of coefficients have regular variation. Isr. J. Math. 144, 343–365 (2004). https://doi.org/10.1007/BF02916717

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  • DOI: https://doi.org/10.1007/BF02916717

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