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 2 k (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
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 21 + … + x 2 N = 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 2 N (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.
Research supported by NSERC and by the Canada Research Chair Programme.
In memory of Robert A. Rankin
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References
G.E. Andrews, “The fifth and seventh order mock theta functions,” Transactions of the AMS 293 (1986), 113–134.
P. Bateman, “On the representation of a number as the sum of three squares,” Transactions of the AMS 71 (1951), ‘70–101.
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.
L. Carlitz, “A note on the multiplication formulas for the Bernoulli and Euler polynomials,” Proceedings of the AMS 4 (1953), 184–188.
R.D. Connors and J.P. Keating, “Degeneracy moments for the square billiard,” J. Phys. G: Nucl. Part. Phys. 25 (1999), 555–562.
R.E. Crandall, “New representations for the Madelung constant,” Experimental Mathematics 8 (4) (1999), 367–379.
R.E. Crandall, “Signal processing applications in additive number theory,” (2001) preprint.
R. Crandall and S. Wagon, “Sums of squares: Computational aspects,” (2001) preprint.
J.A. Ewell, “New representations of Ramanujan’s tau function,” Proc. Amer. Math. Soc. 128 (1999), 723–726.
M. Glasser and I. Zucker, “Lattice Sums,” Theoretical Chemistry: Advances and Perspectives, 5 (1980), 67–139.
E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, 1985.
G.H. Hardy, Collected Papers,Oxford University Press, 1969, Vol. I.
G.H. Hardy, Ramanujan, Cambridge University Press, 1940. Revised Amer. Math. Soc., 1999.
G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979.
L.K. Hua, Introduction to Number Theory, Springer-Verlag, 1982.
H. Jwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, Vol. 17, AMS, 1997.
A.A. Karatsuba, Basic Analytic Number Theory, Springer-Verlag, 1991.
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.
E. Landau, Vorlesungen über Zahlentheorie, Leipzig, Hirzel, 1927.
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.)
M.R. Marty, Problems in Analytic Number Theory.
W. Nowak, “Zum Kreisproblem,” österreich. Akad. Wiss. Math.-Natur. K1. Sitzungsber. II 194 (4–10) (1985), 265–271.
H. Rademacher, Topics in Analytic Number Theory, Springer-Verlag, 1973.
S. Ramanujan, “Some formulae in the analytic theory of numbers:’ Messenger of Math. 45 (1916), 81–84.
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.
M.M. Robertson and I.J. Zucker, “Exact values for some two-dimensional lattice sums,” Z Phys. A: Math. Gen. 8 (1975), 874–881.
H.E. Rose, A Course in Number Theory, Oxford Science Publications, 2nd edn., 1994.
D. Shanks, “Calculation and applications of Epstein zeta functions”, Math. Comp. 29 (1975), 271–287.
E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford Science Publications, 2nd edn., 1986.
G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1966.
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Borwein, J.M., Choi, KK.S. (2003). On Dirichlet Series for Sums of Squares. In: Berndt, B., Ono, K. (eds) Number Theory and Modular Forms. Developments in Mathematics, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6044-6_9
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