# On Dirichlet Series for Sums of Squares

• Jonathan Michael Borwein
• Kwok-Kwong Stephen Choi
Part of the Developments in Mathematics book series (DEVM, volume 10)

## Abstract

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 σ k (n) and σ k 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 f i and g i are completely multiplicative, then we have
$$\sum\limits_{n = 1}^\infty {\frac{{({f_{1*{g_1}}})\cdot({f_{2*{g_2}}})(n)}}{{{n^s}}}} = \frac{{{L_{{f_1}{f_2}}}(s){L_{{g_1}{g_2}}}(s){L_{{f_1}{g_2}}}(s){L_{{g_1}{f_2}}}(s)}}{{{L_{{f_1}{f_2}{g_1}{g_2}}}(2s)}}$$
where $${L_f}(s): = \sum\nolimits_{n = 1}^\infty {f(n){n^{ - s}}}$$ is the Dirichlet series corresponding to f. Let r N (n) be the number of solutions of x 1 2 + … + x N 2 = n and r 2, p (n) be the number of solutions of x 2 + Py 2 = 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 r N (n), r N 2 (n), r 2, p (n) and r 2, 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.

## Key words

Dirichlet series sums of squares closed forms binary quadratic forms disjoint discriminants L-functions

## References

1. 1.
G.E. Andrews, “The fifth and seventh order mock theta functions,” Transactions of the AMS 293 (1986), 113–134.
2. 2.
P. Bateman, “On the representation of a number as the sum of three squares,” Transactions of the AMS 71 (1951), ‘70–101.Google Scholar
3. 3.
J.M. Borwein and P.B. Borwein, Pi and the AGM. A study in analytic number theory and computational complexity, CMS, Monographs and Advanced Texts, 4. John Wiley and Sons, New York, 1987. Paperback, 1998.Google Scholar
4. 4.
L. Carlitz, “A note on the multiplication formulas for the Bernoulli and Euler polynomials,” Proceedings of the AMS 4 (1953), 184–188.
5. 5.
R.D. Connors and J.P. Keating, “Degeneracy moments for the square billiard,” J. Phys. G: Nucl. Part. Phys. 25 (1999), 555–562.
6. 6.
R.E. Crandall, “New representations for the Madelung constant,” Experimental Mathematics 8 (4) (1999), 367–379.
7. 7.
R.E. Crandall, “Signal processing applications in additive number theory,” (2001) preprint.Google Scholar
8. 8.
R. Crandall and S. Wagon, “Sums of squares: Computational aspects,” (2001) preprint.Google Scholar
9. 9.
J.A. Ewell, “New representations of Ramanujan’s tau function,” Proc. Amer. Math. Soc. 128 (1999), 723–726.
10. 10.
M. Glasser and I. Zucker, “Lattice Sums,” Theoretical Chemistry: Advances and Perspectives, 5 (1980), 67–139.Google Scholar
11. 11.
E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, 1985.Google Scholar
12. 12.
G.H. Hardy, Collected Papers,Oxford University Press, 1969, Vol. I.Google Scholar
13. 13.
G.H. Hardy, Ramanujan, Cambridge University Press, 1940. Revised Amer. Math. Soc., 1999.Google Scholar
14. 14.
G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979.Google Scholar
15. 15.
L.K. Hua, Introduction to Number Theory, Springer-Verlag, 1982.Google Scholar
16. 16.
H. Jwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, Vol. 17, AMS, 1997.Google Scholar
17. 17.
A.A. Karatsuba, Basic Analytic Number Theory, Springer-Verlag, 1991.Google Scholar
18. 18.
M. Kühleitner, “On a question of A. Schinzel concerning the sum E„X(r(n))2,” Österreichisch-UngarischSlowakisches Koloquium Über Zahlentheorie (Maria Trost, 1992 ), 63–67, Grazer Math. Ber., 318 KarlFranzens-Univ. Graz, Graz, 1993.Google Scholar
19. 19.
E. Landau, Vorlesungen über Zahlentheorie, Leipzig, Hirzel, 1927.
20. 20.
E. Landau, Collected works,Vol. 4. (German) (Edited and with a preface in English by P.T. Bateman, L. Mirsky, H.L. Montgomery, W. Schaal, I.J. Schoenberg, W. Schwarz and H. Wefelscheid. Thales-Verlag, Essen, 1986.)Google Scholar
21. 21.
M.R. Marty, Problems in Analytic Number Theory.Google Scholar
22. 22.
W. Nowak, “Zum Kreisproblem,” österreich. Akad. Wiss. Math.-Natur. K1. Sitzungsber. II 194 (4–10) (1985), 265–271.
23. 23.
24. 24.
S. Ramanujan, “Some formulae in the analytic theory of numbers:’ Messenger of Math. 45 (1916), 81–84.Google Scholar
25. R. Rankin, “Contributions to the theory of Ramanujan’s function r(n) and similar arithmetical functions (I), (II), (III),” Proc. Cambridge Philos. Soc. 35, 36 (1939) (1940), 351–356,357–372,150–151.Google Scholar
26. 26.
M.M. Robertson and I.J. Zucker, “Exact values for some two-dimensional lattice sums,” Z Phys. A: Math. Gen. 8 (1975), 874–881.
27. 27.
H.E. Rose, A Course in Number Theory, Oxford Science Publications, 2nd edn., 1994.Google Scholar
28. 28.
D. Shanks, “Calculation and applications of Epstein zeta functions”, Math. Comp. 29 (1975), 271–287.
29. 29.
E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford Science Publications, 2nd edn., 1986.Google Scholar
30. 30.
G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1966.Google Scholar