 If formula_3 is a positive integer and formula_4 is coprime to formula_3, we let formula_6 denote the number of primes less than or equal to formula_2 which are equal to formula_4 modulo formula_3.Dirichlet's theorem on primes in arithmetic progressions then tells us
that

where formula_11 is Euler's totient function.If we then define the error function

where the max is taken over all formula_4 coprime to formula_3, then the Elliott–Halberstam conjecture is the assertion that
for every formula_15 and formula_16 there exists a constant formula_17 such that

for all formula_19.