Perron's formula

Let $\backslash \{a(n)\backslash \}$ be an arithmetic function, and let

:$g(s)=\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{a(n)\}\{n^\{s\}\}$

be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for $\backslash Re(s)>\backslash sigma$. Then Perron's formula is

:$A(x)\; =\; \{\backslash sum\_\{n\backslash le\; x\}\}\text{'}\; a(n)\; =\backslash frac\{1\}\{2\backslash pi\; i\}\backslash int\_\{c-i\backslash infty\}^\{c+i\backslash infty\}\; g(z)\backslash frac\{x^\{z\}\}\{z\}\; \backslash ,dz.$

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. 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.

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

:$g(s)=\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{a(n)\}\{n^\{s\}\; \}=s\backslash int\_\{1\}^\{\backslash infty\}\; A(x)x^\{-(s+1)\; \}\; dx.$

This is nothing but a Laplace transform under the variable change $x\; =\; e^t.$ Inverting it one gets Perron's formula.

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:

:$\backslash zeta(s)=s\backslash int\_1^\backslash infty\; \backslash frac\{\backslash lfloor\; x\backslash rfloor\}\{x^\{s+1\}\}\backslash ,dx$

and a similar formula for Dirichlet L-functions:

:$L(s,\backslash chi)=s\backslash int\_1^\backslash infty\; \backslash frac\{A(x)\}\{x^\{s+1\}\}\backslash ,dx$

where

:$A(x)=\backslash sum\_\{n\backslash le\; x\}\; \backslash chi(n)$

and $\backslash chi(n)$ is a Dirichlet character. Other examples appear in the articles on the Mertens function and the von Mangoldt function.

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

:$\backslash sum\_\{n=1\}^\{\backslash infty\}\; a(n)f(n/x)=\; \backslash frac\{1\}\{2\backslash pi\; i\}\; \backslash int\_\{c-i\backslash infty\}^\{c+i\backslash infty\}F(s)G(s)x^\{s\}ds$

where

:$G(s)=\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{a(n)\}\{n^\{s\}\}$

and

:$F(s)=\; \backslash int\_\{0\}^\{\backslash infty\}f(x)x^\{s-1\}dx$

the Mellin transform. The Perron formula is just the special case of the test function $f(1/x)=\backslash theta\; (x-1),$ for $\backslash theta(x)$ the Heaviside step function.

• Page 243 of {{Apostol IANT}}
• {{mathworld|urlname=PerronsFormula|title=Perron's formula}}
• {{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 |location=Cambridge | isbn=0-521-41261-7 | zbl=0831.11001 }}

Category:Theorems in analytic number theory

Category:Calculus

Category:Integral transforms

Category:Summability methods