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The Convolution Sum Σ m<n/16 σ(m)σ(n – 16m)

  • Ayşe Alaca (a1), Şaban Alaca (a1) and Kenneth S. Williams (a1)

Abstract

The convolution sum $\sum{_{m<n/16}\,\sigma (m)\sigma (n\,}-16m)$ is evaluated for all $n\,\in \,\mathbb{N}$ . This evaluation is used to determine the number of representations of $n$ by the quadratic form $x_{1}^{2}\,+\,x_{2}^{2}\,+\,x_{3}^{2}\,+\,x_{4}^{2}\,+\,4x_{5}^{2}\,+\,4x_{6}^{2}\,+\,4x_{7}^{2}\,+\,4x_{8}^{2}$ .

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References

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[1] Berndt, B. C., Ramanujan's Notebooks. Part II. Springer-Verlag, New York, 1989.
[2] Berndt, B. C., Ramanujan's Notebooks. Part III. Springer-Verlag, New York, 1991.
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[7] Huard, J. G., Ou, Z. M., Spearman, B. K., and Williams, K. S., Elementary evaluation of certain convolution sums involving divisor functions. In: Number Theory for the Millennium, II. A K Peters, Natick, MA, 2002, pp. 229274.
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The Convolution Sum Σ m<n/16 σ(m)σ(n – 16m)

  • Ayşe Alaca (a1), Şaban Alaca (a1) and Kenneth S. Williams (a1)

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