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Further aspects of the evaluation of σ(m. n≠0, 0) (am2 + bnm + cn2)−s

Published online by Cambridge University Press:  24 October 2008

I. J. Zucker  and
M. M. Robertson
I. J. Zucker
Affiliation:
Department of Physics, University of Surrey, Guildford
M. M. Robertson
Affiliation:
Department of Mathematics, University of Surrey, Guildford
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Extract

In some previous publications, Zucker and Robertson [13, 14, 15] for certain cases evaluated exactly the double sum

with a, b, c integers. In (1·1) the summation is over all integer values of m and n, both positive and negative, but excluding the case where m and n are simultaneously zero. The term ‘exact’ used here is in the sense introduced by Glasser [5], and means that S may be expressed as a linear sum of products of pairs of Dirichlet L-series. S is then said to be solvable. Whether S may be solved or not depends on the properties of the related binary quadratic form (am2 + bmn + cn2) = (a, b, c). The cases considered in [15] were when a > 0 and d = b2 − 4ac < 0, i.e. (a, b, c) was positive definite. When this is so, Glasser [5] conjectured that S was solvable if and only if (a, b, c) had one reduced form per genus, i.e. the reduced forms of (a, b, c) were disjoint. Zucker and Robertson [15] were in fact able to solve S for all (a, b, c) for which d < 0 and whose reduced forms were disjoint. A complete list of solutions may be found in [6].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 1 , January 1984 , pp. 5 - 13
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1] Baker, A. and Schinzel, A.. On the least integers represented by the genera of binary quadratic forms. Acta Arith. 18 (1971), 137–144.CrossRefGoogle Scholar
[2] Berndt, B. C.. A double summation. SIAM Rev. 16 (1974), 555–557.CrossRefGoogle Scholar
[3] Chowla, S.. An extension of Heilbrom's class number theorem. Quart. J. Math. Oxford Ser. 5 (1934), 304–307.CrossRefGoogle Scholar
[4] Durell, C. V. and Robson, A.. Advanced Trigonometry, ch. ix and xii (G. Bell, 1967).Google Scholar
[5] Glasser, M. L.. The evaluation of lattice sums. II. Number theoretic approach. J. Math. Phys. 14 (1973), 701–703.CrossRefGoogle Scholar
[6] Glasser, M. L. and Zucker, I. J.. Lattice sums. Theoret. Chem. Adv. & Perspectives 5 (1980), 67–139.CrossRefGoogle Scholar
[7] Grosswald, E.. Negative discriminants of binary quadratic forms with one class in each genus. Acta Arith. 8 (1963), 295–306.CrossRefGoogle Scholar
[8] Lorenz, L.. Bidrag til tallenes theori. Tidsskrift for Math. 1 (1871), 97–114.Google Scholar
[9] Selberg, A. and Chowla, S.. On Epstein's zeta-function. J. reine angew. Math. 227 (1967), 86–110.Google Scholar
[10] Siegel, C. L.. Advanced Analytic Number Theory, vol. 62 (Tata Institute, 1980).Google Scholar
[11] Whittaker, E. T. and Watson, G. N.. Modern Analysis, 4th edn., ch. xiii (C.U.P., 1927).Google Scholar
[12] Zucker, I. J.. The evaluation in terms of Γ-functions of the periods of elliptic curves admitting complex multiplication. Math. Proc. Cambridge Philos. Soc. 82 (1977), 111–118.CrossRefGoogle Scholar
[13] Zucker, I. J. and Robertson, M. M.. Exact values for some two dimensional lattice sums. J. Phys. A 8 (1975), 874–881.Google Scholar
[14] Zucker, I. J. and Robertson, M. M.. Some properties of Dirichlet L-series. J. Phys. A 9 (1976), 1207–1214.CrossRefGoogle Scholar
[15] Zucker, I. J. and Robertson, M. M.. A systematic approach to the evaluation of Σ(m n≠0,0) (am2+bmn+cn2)−s. J. Phys. A 9 (1976), 1215–1224.Google Scholar