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Some arithmetical identities for Ramanujan's and divisor functions

Published online by Cambridge University Press:  17 April 2009

D.B. Lahiri
D.B. Lahiri
Affiliation:
Indian Statistical Institute, Calcutta, India.
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Abstract

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A new linear expression in σ(ν), ν = l, 2, …, n, which vanishes identically is established. A linear expression in σ(ν)'s has been found for σ3(n). A similar expression in σ3(ν)'s has been proved for σ7(n) also, Ramanujan's τ(n) = P24(n-1) is given in three different ways as linear expressions in σ2k+1(n) and σK(ν)'s with k = 1, 3, 5 respectively. Again, the coefficient p48(n-2) is expressed as a linear expression in σ11(v)'S and σ5(ν)'s. In establishing these results advantage is taken of the general theorem, also established, that the coefficients of the square of a power series whose coefficients satisfy a certain functional equation are expressible as linear functions of the latter coefficients.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 3 , December 1969 , pp. 307 - 314
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Hardy, G.H., Ramanujan (Cambridge University Press, Cambridge, 1940).Google Scholar
[2]Hardy, G.H. and Wright, E.M., An introduction to the theory of numbers (Clarendon Press, Oxford, 1938).Google Scholar
[3]Hurwitz, Adolf, Mathematische Werke, Band II, (Birkhaüser, Basel, 1933).Google Scholar
[4]Lahiri, D.B., “On Ramanujan's function τ(n) and the divisor function σk (n) – I”, Bull. Calcutta Math. Soc. 38 (1946), 193–206.Google Scholar
[5]Lahiri, D.B., “On Ramanujan's function τ(n) and the divisor function σk(n) – II”, Bull. Calcutta Math. Soc. 39 (1947), 33–52.Google Scholar
[6]Lahiri, D.B., “Identities connecting the partition, divisor and Ramanujan's functions”, Proc. Nat. Inst. Sci. India, 34A (1968). 96–103.Google Scholar
[7]Niven, I. and Zuckerman, H.S., An introduction to the theory of numbers, (John Wiley, New York, 1960).Google Scholar
[8]Sandham, H.F., “A square as the sum of 7 squares”, Quart. J. Math., Oxford Ser. (2), 4 (1953), 230–236.CrossRefGoogle Scholar
[9]Sandham, H.F., “A square as the sum of 9, 11 and 13 squares”, J. London Math. Soc. 29 (1954), 31–38.CrossRefGoogle Scholar