Formula to calculate the sum of an arithmetic function in analytic number theory
In
mathematics, and more particularly in
analytic number theory, Perron's formula is a formula due to
Oskar Perron to calculate the sum of an
arithmetic function, by means of an inverse
Mellin transform.
Statement
Let
be an
arithmetic function, and let
![{\displaystyle g(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2488dab1de3a3ebacbe0323ff23761827c794b7)
be the corresponding
Dirichlet series. Presume the Dirichlet series to be
uniformly convergent for
. Then Perron's formula is
![{\displaystyle A(x)={\sum _{n\leq x}}'a(n)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }g(z){\frac {x^{z}}{z}}\,dz.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f09327fca06ea05d537525261465ac8f711f0c8b)
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. The integral is not a convergent
Lebesgue integral; it is understood as the
Cauchy principal value. 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
![{\displaystyle g(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}=s\int _{1}^{\infty }A(x)x^{-(s+1)}dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1adab804b2ba120f6e7c24a98c18f79613c4cf7)
This is nothing but a
Laplace transform 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:
![{\displaystyle \zeta (s)=s\int _{1}^{\infty }{\frac {\lfloor x\rfloor }{x^{s+1}}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcbeb59908a99442df9f037aa930bff75f4c2277)
and a similar formula for
Dirichlet L-functions:
![{\displaystyle L(s,\chi )=s\int _{1}^{\infty }{\frac {A(x)}{x^{s+1}}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c04397a011a9be6c32d2535d62dbe23a459bf6)
where
![{\displaystyle A(x)=\sum _{n\leq x}\chi (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0249b5f1b8ce01774bfb0394a6cf9ed432280830)
and
is a
Dirichlet character. Other examples appear in the articles on the
Mertens function and the
von Mangoldt function.
Generalizations
Perron's formula is just a special case of the Mellin discrete convolution
![{\displaystyle \sum _{n=1}^{\infty }a(n)f(n/x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }F(s)G(s)x^{s}ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b32ff0ac72e20a8ca75e177b2df5e1f92eaf2217)
where
![{\displaystyle G(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab46694685fddd9aaf6314908e4be76a5be2acf)
and
![{\displaystyle F(s)=\int _{0}^{\infty }f(x)x^{s-1}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f56537a4f905d5c0717f1ba934e9f5f411da28)
the Mellin transform. The Perron formula is just the special case of the test function
for
the
Heaviside step function.
References