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The average value of an exponential function over the lattice points of a circle

Published online by Cambridge University Press:  24 October 2008

J. R. Wilton
J. R. Wilton
Affiliation:
Trinity College
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Extract

1. The subject-matter of this communication I believe to be new, but after Lemma 1 the method is classical; Lemma 1 is itself a particular case of a theorem which I have given elsewhere, and is a straightforward extension of a well-known result.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 24 , Issue 2 , April 1928 , pp. 171 - 179
Copyright
Copyright © Cambridge Philosophical Society 1928

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References

* See, for example, Landau, E., Vorlesungen über Zahlentheorie (1927), 2, 204–8.Google Scholar At the time of writing (August, 1927) this book, has not reached me; but I have made much use of the first part of Landau's elementary exposition, “Computo asintotico dei nodi di un reticulato entro un cerchio,” Rome Rendiconti (2), 3 (1926), 35–61.Google Scholar

† “The Lattice-points of an n-dimensional ellipsoid,” Journal London Math. Soc. 2 (1927), 227–33.Google Scholar

‡ References to the literature are given in my paper just cited.Google Scholar

§ Watson, G. N., Theory of Bessel Functions (1922), 77 (2).Google Scholar

‖ Theorems B, B′ and B″.Google Scholar

* Theorem A.Google Scholar

† Wilton, J. R., “Some applications of a transformation of series,” Proc. London Math. Soc. (2) 27 (1927), 81–104—a special case of equation (3·22).Google Scholar

* Watson, , loc. cit. 199.Google Scholar

* Acta Math. 11 (1888), 19–24Google Scholar; quoted by Whittaker, and Watson, , Modern Analysis (ed. 3), 280, Ex. 8.Google Scholar

* The definition of such a function is given by Hardy, , Orders of Infinity (1924), 17 (§ 3·2).Google Scholar

† Borel, E., Rend. di Palermo, 27 (1909), 247–271 (269).CrossRefGoogle ScholarSee also Bernstein, F., Math. Annalen, 71 (1912), 417–439 (430, Satz 4).CrossRefGoogle Scholar The “contradiction” spoken of by Bernstein in the first line of p. 431 rests upon a misunderstanding. See Borel, , Math. Annalen, 72 (1912), 578–584 (583)CrossRefGoogle Scholar, and Bernstein, ibid. 585–587.

The lemma is employed by Hardy, and Littlewood, , Acta Math. 37 (1914), 155–239 (215) in a way which suggested to me Theorem B″.CrossRefGoogle Scholar