A summation formula involving $\sigma (N)$

Abstract:

The ${L^2}$ theories are known of the summation formula involving ${\sigma _k}(n)$, the sum of the kth power of divisors of n, as coefficients, for all k except $k = 1$. In this paper, techniques are used to overcome the extra convergence difficulty of the case $k = 1$, to establish a symmetric formula connecting the sums of the form $\sum {\sigma _1}(n){n^{ - 1/2}}f(n)$ and $\sum {{\sigma _1}} (n){n^{ - 1/2}}g(n)$, where $f(x)$ and $g(x)$ are Hankel transforms of each other.