Iterated Dirichlet series and the inverse of Ramanujan’s sum

Kenneth R. Johnson
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 9, 2003, Number 3, Pages 65–72
Full paper (PDF, 2246 Kb)

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Authors and affiliations

Kenneth R. Johnson
North Dakota State University
Fargo, ND

Abstract

The theory of Dirichlet series having number theoretic functions of a single variable as coefficients has a rich history. In this paper we present a parallel theory for iterated Dirichlet series with number theoretic functions of two variables as coefficients and find the Dirichlet product inverse of Ramanujan’s sum. The results presented here are easily accessible to an Advanced Calculus or undergraduate Number Theory course.

AMS Classification

• 11A25

References

1. D. R. Anderson and T. M. Apostol, The Evaluation of Ramanujan’s Sum and Generalizations, Duke Journal, 20 (1953), 211-216.
2. Tom M. Apostol, Introduction to Analytic Number Theory, New York-Heidelberg-Berlin: Springer-Verlag(1976).
3. S. Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers, Cambridge Philosophical Transactions, 22 (1918), 259-276.