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3 results

Generalized divisor functions in arithmetic progressions: II
Part of
- Multiplicative number theory
- Zeta and $L$-functions: analytic theory
D. T. Nguyen
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 1 / February 2024

Published online by Cambridge University Press:

16 November 2023, pp. 28-56
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We obtain a new bound on the second moment of modified shifted convolutions of the generalized threefold divisor function and show that, for applications, the modified version is sufficient.

Averages of shifted convolutions of d3(n)
Part of
S. Baier, T. D. Browning, G. Marasingha, L. Zhao
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 55 / Issue 3 / October 2012

Published online by Cambridge University Press:

12 April 2012, pp. 551-576
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We investigate the first and second moments of shifted convolutions of the generalized divisor function d3(n).

The Convolution Sum Σm<n/16σ(m)σ(n – 16m)
Ayşe Alaca, Şaban Alaca, Kenneth S. Williams
Journal:

Canadian Mathematical Bulletin / Volume 51 / Issue 1 / 01 March 2008

Published online by Cambridge University Press:

20 November 2018, pp. 3-14

Print publication:

01 March 2008
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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 $x_{1}^{2}\,+\,x_{2}^{2}\,+\,x_{3}^{2}\,+\,x_{4}^{2}\,+\,4x_{5}^{2}\,+\,4x_{6}^{2}\,+\,4x_{7}^{2}\,+\,4x_{8}^{2}$.