Functiones et Approximatio Commentarii Mathematici

On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function

B. Ramakrishnan, Brundaban Sahu, and Anup Kumar Singh

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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.

Article information

Source
Funct. Approx. Comment. Math., Volume 58, Number 2 (2018), 233-244.

Dates
First available in Project Euclid: 2 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.facm/1512183764

Digital Object Identifier
doi:10.7169/facm/1695

Mathematical Reviews number (MathSciNet)
MR3816077

Zentralblatt MATH identifier
06924930

Subjects
Primary: 11E25: Sums of squares and representations by other particular quadratic forms 11F11: Holomorphic modular forms of integral weight
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables

Keywords
representation numbers of quadratic forms modular forms of one variable Ramanujan tau function

Citation

Ramakrishnan, B.; Sahu, Brundaban; Singh, Anup Kumar. On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function. Funct. Approx. Comment. Math. 58 (2018), no. 2, 233--244. doi:10.7169/facm/1695. https://projecteuclid.org/euclid.facm/1512183764


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