In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 Diric ...
, 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 differential ...
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 used i ...
.
Statement
Let
be 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 ...
, and let
:
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 arbitrarily s ...
for
. Then Perron's formula is
:
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 Lebe ...
; 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
:
This is nothing but a
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
under the variable change
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 \operatorname(s) > ...
:
:
and a similar formula for
Dirichlet ''L''-functions:
:
where
:
and
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) \chi ...
. 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 Mangold ...
.
Generalizations
Perron's formula is just a special case of the Mellin discrete convolution
:
where
:
and
:
the Mellin transform. The Perron formula is just the special case of the test function
for
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
A university press is an academic publishing hou ...
, location=Cambridge , isbn=0-521-41261-7 , zbl=0831.11001
Theorems in analytic number theory
Calculus
Integral transforms
Summability methods