In mathematics, and more particularly in analytic number theory, 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 differen ...

to calculate the sum of an

arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...

, 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 use ...

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 analyti ...

. Presume the Dirichlet series to be

uniformly convergent In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitra ...

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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

. The integral is not a convergent

Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...

; 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. Formulation Depending on the type of singularity in the integrand ...

. 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 its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...

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: :

\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: :1) \c ...

. 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 re ...

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 Man ...

Generalizations

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

\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 (1850–1925), the value of which is zero for negative arguments and one for positive argume ...

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 is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

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