On Dirichlet Series for Sums of Squares

Ramanujan Journal

Volumn 7, Issue 1-3, 2003, Pages 95-127

  Borwein, Jonathan Michael ^a   Choi, Kwok Kwong Stephen ^a  

^a SIMON FRASER UNIVERSITY   (Canada)

Author keywords

Binary quadratic forms; Closed forms; Dirichlet series; Disjoint discriminants; L functions; Sums of squares

Indexed keywords

EID: 1642541641     PISSN: 13824090     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1026230709945     Document Type: Article

References (30)

1. G.E. Andrews, "The fifth and seventh order mock theta functions," Transactions of the AMS 293 (1986), 113-134.
2. P. Bateman, "On the representation of a number as the sum of three squares," Transactions of the AMS 71 (1951), 70-101.
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 & Sons, New York, 1987. Paperback, 1998.
4. L. Caditz, "A note on the multiplication formulas for the Bernoulli and Enter polynomials," Proceedings of the AMS 4 (1953), 184-188.
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. R.E. Crandall, "New representations for the Madelung constant, " Experimental Mathematics 8(4) (1999), 367-379.
7. R.E. Crandall, "Signal processing applications in additive number theory," (2001) preprint.
8. R. Crandall and S. Wagon, "Sums of squares: Computational aspects," (2001) preprint.
9. J.A. Ewell, "New representations of Ramanujan's tau function," Proc. Amer. Math. Soc. 128 (1999), 723-726.
10. M. Glasser and I. Zucker, "Lattice Sums," Theoretical Chemistry: Advances and Perspectives, 5 (1980), 67-139.
11. E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, 1985.
12. G.H. Hardy, Collected Papers, Oxford University Press, 1969, Vol. I.
13. G.H. Hardy, Ramanujan, Cambridge University Press, 1940. Revised Amer. Math. Soc., 1999.
14. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979.
15. L.K. Hua, Introduction to Number Theory, Springer-Verlag, 1982.
16. H. Jwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, Vol. 17, AMS, 1997.
17. A.A. Karatsuba, Basic Analytic Number Theory, Springer-Verlag, 1991.
18. 2," Ö sterreichisch-Ungarisch Slowakisches Koloquium Über Ober Zahlentheorie (Maria Trost, 1992), 63-67, Grazer Math. Ber., 318 Karl-Franzens-Univ. Graz, Graz, 1993.
19. E. Landau, Vorlesungen über Zahlentheorie, Leipzig, Hirzel, 1927.
20. E. Landau, Collected works, Vol. 4. (German) (Edited and with a preface in English by P.T. Bateman, L. Minsky, H.L. Montgomery, W. Schaal, I.J. Schoenberg, W. Schwarz and H. Wefelscheid. Thales-Verlag, Essen, 1986.)
21. M.R. Murty, Problems in Analytic Number Theory.
22. W. Nowak, "Zum Kreisproblem," österreich. Akad. Wiss. Math.-Natur. K1. Sitzungsber. II 194(410) (1985), 265-271.
23. H. Rademacher, Topics in Analytic Number Theory, Springer-Verlag, 1973.
24. S. Ramanujan, "Some formulae in the analytic theory of numbers," Messenger of Math. 45 (1916), 81-84.
25. R. Rankin, "Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions (I), (II), (III)," Proc. Cambridge Philos. Soc. 35, 36 (1939), (1940), 351-356, 357-372, 150-151.
26. M.M. Robertson and I.J. Zucker, "Exact values for some two-dimensional lattice sums," J. Phys. A: Math. Gen. 8 (1975), 874-881.
27. H.E. Rose, A Course in Number Theory, Oxford Science Publications, 2nd edn., 1994.
28. D. Shanks, "Calculation and applications of Epstein zeta functions", Math. Comp. 29 (1975), 271-287.
29. E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford Science Publications, 2nd edn., 1986.
30. G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1966.