# A Formula for Primes in Arithmetic Progressions

A Formula for Primes in Arithmetic Progressions

Let and be integers with . If and are relatively prime, then the arithmetic progression where contains an infinite number of primes. The number of such primes that are less than or equal to is a function of usually denoted by (x).

q

a

q>0

q

a

qn+a

n=0,1,2,3,…

x

x

π

q,a

The graph of (x) is an irregular step function that jumps by 1 for every such that is prime.

π

q,a

x

x=qn+a

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

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