A Formula for Primes in Arithmetic Progressions

start x axis at (from 0 to 400)

0

length of x axis (from 10 to 200)

50

pairs of zeros (from 0 to 100)

25

always show estimate using no L zeros

display

π

q,a

(x) values in a tooltip

primes in the progression

3n + 1

3n + 2

4n + 1

4n + 3

10n + 1

10n + 3

10n + 7

10n + 9

Let

q

and

a

be integers with

q>0

. If

q

and

a

are relatively prime, then the arithmetic progression

qn+a

where

n=0,1,2,3,…

contains an infinite number of primes. The number of such primes that are less than or equal to

x

is a function of

x

usually denoted by

π

q,a

(x)

.

The graph of

π

q,a

(x)

is an irregular step function that jumps by 1 for every

x

such that

x=qn+a

is prime.

This Demonstration illustrates the remarkable fact that we can replicate the jumps of this step function by using a sum that involves zeros of Dirichlet

L

-functions. This means that those zeros somehow carry information about which numbers in the progression are prime.