Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T08:32:14.541Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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