mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, and more particularly in

analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...

, Perron's formula is a formula due to

Oskar Perron Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician. He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differentia ...

to calculate the sum of an

arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition th ...

, by means of an inverse

Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used ...

Statement

Let

\

be an

, and let :

g(s)=\sum_^ \frac

be the corresponding

Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in anal ...

. Presume the Dirichlet series to be uniformly convergent for

\Re(s)>\sigma

. Then Perron's formula is :

A(x) = ' a(n) =\frac\int_^ g(z)\frac \,dz.

Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when ''x'' is an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

. The integral is not a convergent

Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...

; it is understood as the

Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by ...

. The formula requires that ''c'' > 0, ''c'' > σ, and ''x'' > 0.

Proof

An easy sketch of the proof comes from taking Abel's sum formula :

g(s)=\sum_^ \frac=s\int_^  A(x)x^ dx.

This is nothing but a

Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...

under the variable change

x = e^t.

Inverting it one gets Perron's formula.

Examples

Because of its general relationship to Dirichlet series, the formula is commonly applied to many number-theoretic sums. Thus, for example, one has the famous integral representation for the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

: :

\zeta(s)=s\int_1^\infty \frac\,dx

and a similar formula for Dirichlet ''L''-functions: :

L(s,\chi)=s\int_1^\infty \frac\,dx

where :

A(x)=\sum_ \chi(n)

and

\chi(n)

is a

Dirichlet character In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi: \mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: # \chi(ab) = \ch ...

. Other examples appear in the articles on the

Mertens function In number theory, the Mertens function is defined for all positive integers ''n'' as : M(n) = \sum_^n \mu(k), where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive r ...

and the

von Mangoldt function In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mang ...

Generalizations

Perron's formula is just a special case of the formula :

\sum_^ a(n)f(n/x)= \frac \int_^F(s)G(s)x^ds

where :

G(s)= \sum_^ \frac

and :

F(s)= \int_^f(x)x^dx

the Mellin transform. The Perron formula is just the special case of the test function

f(1/x)=\theta (x-1),

for

\theta(x)

the

Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...

References

* Page 243 of * * {{cite book , last=Tenenbaum , first=Gérald , translator=C.B. Thomas , year=1995 , title=Introduction to analytic and probabilistic number theory , series=Cambridge Studies in Advanced Mathematics , volume=46 , publisher=

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, location=Cambridge , isbn=0-521-41261-7 , zbl=0831.11001 Theorems in analytic number theory Calculus Integral transforms Summability methods