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June 2018 On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function
B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh
Funct. Approx. Comment. Math. 58(2): 233-244 (June 2018). DOI: 10.7169/facm/1695

Abstract

In this paper, we find the number of representations of the quadratic form $x_1^2+ x_1x_2 + x_2^2 + x_3^2+ x_3x_4 + x_4^2 + \ldots + x_{2k-1}^2 + x_{2k-1}x_{2k} + x_{2k}^2,$ for $k=7,9,11,12,14$ using the theory of modular forms. By comparing our formulas with the formulas obtained by G.A. Lomadze, we obtain the Fourier coefficients of certain newforms of level $3$ and weights $7,9,11$ in terms of certain finite sums involving the solutions of similar quadratic forms of lower variables. In the case of $24$ variables, comparison of these formulas gives rise to a new formula for the Ramanujan tau function.

Citation

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B. Ramakrishnan. Brundaban Sahu. Anup Kumar Singh. "On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function." Funct. Approx. Comment. Math. 58 (2) 233 - 244, June 2018. https://doi.org/10.7169/facm/1695

Information

Published: June 2018
First available in Project Euclid: 2 December 2017

zbMATH: 06924930
MathSciNet: MR3816077
Digital Object Identifier: 10.7169/facm/1695

Subjects:
Primary: 11E25, 11F11
Secondary: 11E20

Rights: Copyright © 2018 Adam Mickiewicz University

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Vol.58 • No. 2 • June 2018
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