CONVOLUTION SUMS INVOLVING THE DIVISOR FUNCTION

Abstract

The series

\begin{alignat*}{2} L_{r,4}(q)\amp=\sum_{n=0}^\infty\sigma(4n+r)q^{4n+r},\amp\quad r\amp=0,1,2,3, \\ M_{r,4}(q)\amp=\sum_{n=0}^\infty\sigma_3(4n+r)q^{4n+r},\amp\quad r\amp=0,1,2,3, \\ N_{r,4}(q)\amp=\sum_{n=0}^\infty\sigma_5(4n+r)q^{4n+r},\amp\quad r\amp=0,1,2,3, \end{alignat*}

are evaluated and used to prove convolution formulae such as

$$ \sum_{m\le n}\sigma(4m-3)\sigma(4n-(4m-3))=4\sigma_3(n)-4\sigma_3(\tfrac12n). $$

AMS 2000 Mathematics subject classification: Primary 11A25