In mathematics, a Voronoi formula is an equality involving

Fourier coefficients A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...

s, with the coefficients twisted by additive characters on either side. It can be regarded as a

Poisson summation formula In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function (mathematics), function to values of the function's continuous Fourier transform. Consequently, the pe ...

for

non-abelian group In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ...

s. The Voronoi (summation) formula for GL(2) has long been a standard tool for studying analytic properties of automorphic forms and their ''L''-functions. There have been numerous results coming out the Voronoi formula on GL(2). The concept is named after

Georgy Voronoy Georgy Feodosevich Voronyi (; ; 28 April 1868 – 20 November 1908) was an Imperial Russian mathematician of Ukrainians, Ukrainian descent noted for defining the Voronoi diagram. Biography Voronyi was born in the village of Zhuravka, Pyriatyn, in ...

Classical application

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 In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is t ...

. The former estimates the size of ''d''(''n''), the number of positive divisors of an integer ''n''.

Dirichlet Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In Mathematical analysis, analysis, h ...

proved :

D(X)= \sum_^X d(n) - X \log X - (2\gamma -1)X=O(X^)

where

\gamma

Euler's constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limit of a sequence, limiting difference between the harmonic series (math ...

≈ 0.57721566. Gauss’ circle problem concerns the average size of :

r_2 (n) = \#\,

for which

Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

gave the estimate :

\Delta(X)=\sum_^X r_2 (n)-\pi X=O(X^).

Each problem has a geometric interpretation, with ''D''(''X'') counting lattice points in the region

\

, and

\Delta(X)

lattice points in the disc

\

. 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 In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

Modern formulation

Let ''ƒ'' be a Maass cusp form for the

modular group In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...

''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 :

\sum_n  a(n)e(an/c)\omega(n) = \sum_n a(n)e(-\bar a n/c)\Omega(n),

where

\bar

is a

multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...

of ''a'' modulo ''c'' and Ω is a certain integral

Hankel transform In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scalin ...

of ''ω''. (see )

References

*{{Citation , last1=Good , first1=Anton , title=Cusp forms and eigenfunctions of the Laplacian, year=1984 , journal=Mathematische Annalen , volume=255 , issue=4 , pages=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). Automorphic forms Analytic number theory