Elliott–Halberstam conjecture

In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who stated the conjecture in 1968.

Stating the conjecture requires some notation. Let ${\displaystyle \pi (x)}$, the prime counting function, denote the number of primes less than or equal to x. If q is a positive integer and a is coprime to q, we let ${\displaystyle \pi (x;q,a)}$ denote the number of primes less than or equal to x which are equal to a modulo q. Dirichlet's theorem on primes in arithmetic progressions then tells us that

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