- Title
- On Dirichlet series for sums of squares
- Creator
- Borwein, Jonathan Michael; Choi, Kwok-Kwong Stephen
- Relation
- The Ramanujan Journal Vol. 7, Issue 1-3, p. 95-127
- Publisher Link
- http://dx.doi.org/10.1023/A:1026230709945
- Publisher
- Springer
- Resource Type
- journal article
- Date
- 2003
- Description
- Hardy and Wright (An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979) recorded elegant closed forms for the generating functions of the divisor functions σk(n) and σk/2(n) in the terms of Riemann Zeta function ζ(s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if fiand gi are completely multiplicative, then we have [formula could not be replicated] where Lf(s):= Σ∞/n=1 f(n)n-s is the Dirichlet series corresponding to f. Let rN(n) be the number of solutions of x²/1 +...+ x²/N = n and r₂,P(n) be the number of solutions of x² + Py² = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of ζ(s) and Dirichlet L-functions, for the generating functions of rN(n), r²/N(n),r2,P(n) and r2,P(n)² for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.
- Subject
- Dirichlet series; sums of squares; closed forms; binary quadratic forms; disjoint discriminants; L-functions
- Identifier
- http://hdl.handle.net/1959.13/940730
- Identifier
- uon:13081
- Identifier
- ISSN:1382-4090
- Language
- eng
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