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The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $\begin{matrix} \sum_{\binom{(l, m) \in ℕ_{0}^{2}}{α l + β m = n}} σ (l) σ (m), \end{matrix}$ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by...

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

Ayşe Alaca, Şaban Alaca, Kenneth S. Williams (2009)

Czechoslovak Mathematical Journal

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.

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Eichler Commutation Relations for Theta Series of Nebentypus.

Ein Verschwindungssatz für automorphe Formen zur Siegelschen Modulgruppe.

Eine Bemerkung zur Summation von Eisenstein-Reihen

Eine Charakterisierung der Einstein-Reihe zur Siegelschen Modulgruppe.

Eisenstein cohomology of arithmetic groups. The case GL2.

Eisenstein series and fields of definition

Eisensteinkohomolgie für Gruppen vom Typ GU (2, 1).

Eisensteinkohomologie und Dedekindsummen für GL2 über imaginär-quadratischen Körpern.

Eisensteinreihen vom Gewicht n + l zur Siegelschen Modulgruppe n-ten Grades.

Elementare Berechnung der Multiplizitäten n-dimensionaler Spitzen.

Elliptische Fixpunkte und Drehfaktoren zur Modulgruppe in Quaternionenschiefkörpern über reellquadratischen Zahlkörpern.

Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

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

Examples of Eigenvalues of Hecke Operators on Siegel Cusp Forms of Degree Two.

Explicit evaluations of the Rogers-Ramanujan continued fraction.

Explicit formulas for the Fourier coefficients.

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