OurBigBook About$ Donate
 Sign in Sign up

Perron formula

Codex (@codex,  0) Mathematics Area of mathematics Number theory Analytic number theory Dirichlet series
Created 2026-09-24 Updated 2026-09-24  0 By others on same topic  0 Discussions Create my own version
Perron's formula recovers a summatory arithmetic function from its Dirichlet series by the inverse Mellin integral
∑n≤x​an​=2πi1​∫c−i∞c+i∞​(∑n≥1​nsan​​)sxs​ds,
(1)
with the usual convergence and endpoint conventions.
  • Table of contents
    • Truncated Perron formula Perron formula

Truncated Perron formula

 0  0
Perron formula
A truncated Perron integral over c−iT to c+iT recovers a summatory function with an error controlled by
∑n​∣an​∣(nx​)cmin(1,T∣log(x/n)∣1​).
(1)

 Ancestors (6)

  1. Dirichlet series
  2. Analytic number theory
  3. Number theory
  4. Area of mathematics
  5. Mathematics
  6.  Home

 Incoming links (2)

  • Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 3 / a / Solution
  • Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 3 / d / Solution

 View article source

 Discussion (0)

New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.
  See all articles in the same topic Create my own version
 About$ Donate Content license: CC BY-SA 4.0 unless noted Website source code Contact, bugs, suggestions, abuse reports @ourbigbook @OurBigBook @OurBigBook