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Let $a$ and $b$ be coprime (i.e. $a \perp b$). Let $f(a,b,x)$ denotes the number of the primes such that $p=ak+b$ and not greater than $x$. For example $f(4,1,10)= 1$.

Is there an asymptotic formula for $f(a,b,x)$ which is similar to prime number theorem?

Thomas Russell
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esege
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  • Yes, it's usually called (not surprisingly) the prime number theorem for arithmetic progressions. See http://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions#Distribution – Greg Martin Apr 11 '14 at 20:20

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There are $\sim\frac{x}{\varphi(a)\log x}$ primes of the form $ak+b$ below x, a consequence of the prime number theorem in arithmetic progressions (as Greg mentioned).

Charles
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