The convolution sum $\sum_{\begin{matrix} m = 1 \\ m \equiv a (mod 4) \end{matrix}}^{n - 1} σ (m) σ (n - m)$ is evaluated for $a \in {0, 1, 2, 3}$ and all $n \in ℕ$ . This completes the partial evaluation given in the paper of J. G. Huard, Z. M. Ou, B. K. Spearman, K. S. Williams.

Explicit decomposition of a rational prime in a cubic field.

Alaca, Şaban; Spearman, Blair K.; Williams, Kenneth S. — 2006

International Journal of Mathematics and Mathematical Sciences

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Representations by sextenary quadratic forms whose coefficients are 1, 2 and 4

Nineteen quaternary quadratic forms

On the two-dimensional theta functions of the Borweins

Seven octonary quadratic forms

Evaluation of the sums ∑ m = 1 m ≡ a ( mod 4 ) n - 1 σ ( m ) σ ( n - m )

Explicit decomposition of a rational prime in a cubic field.

Evaluation of the sums $\sum_{\begin{matrix} m = 1 \\ m \equiv a (mod 4) \end{matrix}}^{n - 1} σ (m) σ (n - m)$