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So, I'm looking at the Voronoi Summation formula for the sum of divisor function, given by

$\displaystyle\sum_{n \leq x}'d(n) = x(\log x + 2\gamma -1) + \frac{1}{4} -\frac{x^{\frac{1}{2}}}{2 \pi} \displaystyle\sum_{n=1}^{\infty}\frac{d(n)}{n^{\frac{1}{2}}}(K_1(4\pi(nx)^\frac{1}{2}) + \frac{1}{2}\pi Y_1(4\pi (nx)^\frac{1}{2})$

where $d(n)$ is the divisor function, the tick means the final value of n is multiplied by 1/2 if n is a whole number, and $K_1$ and $Y_1$ are Bessel functions. You can find a reference to this and more details here: Voronoi Summation Formula as found in Ivic.

Suppose that I define higher indexed versions of the divisor function in the usual way, with

$d_k(n) = \displaystyle\sum_{j | n} d_{k-1}(j)$ and $d_1(n) = 1$

Are there Voronoi Summation formulas for expressing $\displaystyle\sum_{n \leq x}'d_k(n) $ ? Any references for such things? Or, alternatively, can anyone show me the steps leading to this identity? Outside of Ivic, I'm having a hard time tracking down good information on this.

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1 Answer 1

Another reference for Voronoi summation is Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, page 182 (and the discussion preceding that page). Voronoi used it to gives estimates for "the circle problem" and "the divisor problem," so any book that discusses those problems in detail might be another source for Voronoi's result (and generalizations, if there are any).

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This is a really handy reference, Gerry. I'm going to keep this question open in case anyone stumbles along who has answers to my specific case, but I appreciate this. –  Nathan McKenzie Nov 22 '11 at 18:16
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