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A note on Ramanujan's arithmetical function τ(n)

Published online by Cambridge University Press:  24 October 2008

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

The function τ (n), defined by the equation

is considered by Ramanujan* in his memoir “On certain arithmetical functions”

The associated Diriċhlet series

converges when σ = > σ0, for a sufficiently large positive σ0.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 25 , Issue 2 , April 1929 , pp. 121 - 129
Copyright
Copyright © Cambridge Philosophical Society 1929

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References

* Trans. Camb. Phil. Soc. 22 (1916), 159–184 (Collected papers, No. 18).Google Scholar

† Proc. Camb. Phil. Soc. 19 (1920), 117–124 (communicated 1917).Google Scholar

‡ Collected papers, 153–154.Google Scholar

§ Hardy, G. H., “Note on Ramanujan's arithmetical function τ(n)”, Proc. Camb. Phil. Soc. 23 (1927), 675–680.CrossRefGoogle Scholar

* Wigert, S., “Sur quelques functions arithmétiques”, Acta Math. 37 (1914), 113–140 (§ 4).CrossRefGoogle Scholar

† Comptes Rendus, 04 1914, 1012–4.Google Scholar

‡ Acta Math. 37 (1914), 193–238 (226).CrossRefGoogle Scholar

* Compare Hardy, G. H. and Ramanujan, S., “Asymptotic formulae in combinatory analysis”, Proc. London Math. Soc. (2), 17 (1918), 75–115, Lemma 4·31. (Ramanujan's Collected Papers, No. 36.)CrossRefGoogle Scholar

† Proc. Camb. Phil. Soc. 23 (1927), 675–680, Lemma 1.CrossRefGoogle Scholar

* Watson, G. N., Theory of Bessel functions (1922), 177 (8). It is at this stage that the proof follows the classical method of Wigert (loc. cit.).Google Scholar

* See, for example, Hobson, E. W., Theory of functions of a real variable, 2 (1926), 288–289.Google Scholar

† Whittaker, and Watson, , Modern analysis (1920), 246.Google Scholar

‡ Bromwich, , Theory of infinite series (1908), 453, Theorem B.Google Scholar

§ Since , it follows that (4·1) is absolutely and uniformly convergent if .Google Scholar

* Comptes Rendus, 178 (1924), 303.Google ScholarHobson, , loc. cit., 626.Google Scholar

† (P, q) denotes the greatest common factor of p and q.Google Scholar

‡ The change of order follows from Lemma 3 and the theorem quoted from Hobson, as in (3.61). If σ>7, Lemma 2 and the absolute convergence test are sufficient.7,+Lemma+2+and+the+absolute+convergence+test+are+sufficient.>Google Scholar