Voronoi formula

To Voronoy and his contemporaries, the formula appeared tailor-made to evaluate certain finite sums. That seemed significant because several important questions in number theory involve finite sums of arithmetic quantities. In this connection, let us mention two classical examples, Dirichlet’s divisor problem and the Gauss’ circle problem. The former estimates the size of d(n), the number of positive divisors of an integer n. Dirichlet proved

${\displaystyle D(X)=\sum _{n=1}^{X}d(n)-X\log X-(2\gamma -1)X=O(X^{1/2})}$

where ${\displaystyle \gamma }$ is Euler’s constant ≈ 0.57721566. Gauss’ circle problem concerns the average size of

${\displaystyle r_{2}(n)=\#\{(x,y)\in \mathbb {Z} ^{2}\mid x^{2}+y^{2}=n\},}$

for which Gauss gave the estimate

${\displaystyle \Delta (X)=\sum _{n=1}^{X}r_{2}(n)-\pi X=O(X^{1/2}).}$

Each problem has a geometric interpretation, with D(X) counting lattice points in the region ${\displaystyle \{x,y>0,xy\leq X\}}$, and ${\displaystyle \Delta (X)}$ lattice points in the disc ${\displaystyle \{x^{2}+y^{2}\leq X\}}$. These two bounds are related, as we shall see, and come from fairly elementary considerations. In the series of papers Voronoy developed geometric and analytic methods to improve both Dirichlet’s and Gauss’ bound. Most importantly in retrospect, he generalized the formula by allowing weighted sums, at the expense of introducing more general integral operations on f than the Fourier transform.

Let ƒ be a Maass cusp form for the modular group PSL(2,Z) and a(n) its Fourier coefficients. Let a,c be integers with (a,c) = 1. Let ω be a well-behaved test function. The Voronoi formula for ƒ states

${\displaystyle \sum _{n}a(n)e(an/c)\omega (n)=\sum _{n}a(n)e(-{\bar {a}}n/c)\Omega (n),}$

where ${\displaystyle {\bar {a}}}$ is a multiplicative inverse of a modulo c and Ω is a certain integral Hankel transform of ω. (see Good (1984))

• Good, Anton (1984), "Cusp forms and eigenfunctions of the Laplacian", Mathematische Annalen, 255 (4): 523–548, doi:10.1007/bf01451932
• Miller, S. D., & Schmid, W. (2006). Automorphic distributions, L-functions, and Voronoi summation for GL(3). Annals of mathematics, 423–488.
• Voronoï, G. (1904). Sur une fonction transcendente et ses applications à la sommation de quelques séries. In Annales Scientifiques de l'École Normale Supérieure (Vol. 21, pp. 207–267).