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Note on Ramanujan's arithmetical function τ (n)

Published online by Cambridge University Press:  24 October 2008

G. H. Hardy
G. H. Hardy
Affiliation:
Trinity College
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In his remarkable memoir ‘On certain arithmetical functions’* Ramanujan considers, among other functions of much interest, the. function τ(n) defined by

This function is important in the theory of the representation of a number as a sum of 24 squares. In fact

where r24 (n) is the number of representations;

where σs(n) is the sum of the 8th powers of the divisors of n, and the sum of those of its odd divisors; and

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 23 , Issue 6 , April 1927 , pp. 675 - 680
Copyright
Copyright © Cambridge Philosophical Society 1927

References

* Ramanujan, S., ‘On certain arithmetical functions’, Trans. Camb. Phil. Soc., 22 (1916), 159–184.Google Scholar

† Ramanujan, l.c., 179, 184. A function with argument ½n is zero when n is odd.

‡ Mordell, L. J., ‘On Mr Ramanujan“s empirical expansions of modular functions’, Proc. Camb. Phil. Soc., 19 (1920), 117–124 (a paper communicated in 1917).Google Scholar

* Ramanujan, l.c., 168, 171.

† That is to say τ(n)≠o(n5). See Ramanujan, l.c., 171.

‡ Ramanujan, l.c., 175.

* Hardy, G. H. and Littlewood, J. E., ‘Some problems of Diophantine approximation: II: The trigonometrical series associated with the elliptic theta-functions’, Acta Math., 37 (1914), 193–238.CrossRefGoogle Scholar

† 226 et seq.

* The function −6 log (1/λ) − 2πλ becomes negatively infinite when λ → 0 or λ → ∞, and has a maximum when πλ = 3. Here λ > ½.

* The maximnm of u αe −αy occurs for u = α/y, outside the range of integration because c > α.

† Fundamenta nova § 66.