Primes are uniformly distributed

Question

Is the following true?

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Let p, r, n are positive integers with p>1.

U(p, r, n) denotes the number of primes less than n that are equal to r (mod p).

For any prime p and pair of integers r1, r2 between 1 and p-1, we have:

The ratio U(p,r1,n) / U(p,r2,n) has limit 1 as n goes to infinity.

The limit is $1$ but one can also say interesting things about how often it is below or above $1$. See the references here: http://en.wikipedia.org/wiki/Chebyshev%27s_bias — Erick Wong, Jun 08 '13 at 01:12
possible duplicate of What is the distribution of primes modulo $n$? also answered here http://math.stackexchange.com/questions/68981/primes-sum-ratio — Zander, Jun 25 '13 at 22:34

André Nicolas · Answer 1 · 2013-06-08T00:09:41.613

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Yes, this is true. It is a strengthening of the Prime Number Theorem to primes in arithmetic progressions.

More is true. Let $m$ be a positive integer, not necessarily prime. Let $a$ and $a'$ be numbers relatively prime to $m$. Then the ratio of the number of primes $\le n$ of the form $mk+a$ to the number of primes $\le n$ of the form $mk+a'$ approaches $1$ as $n\to\infty$.

edited Jun 08 '13 at 00:09

answered Jun 08 '13 at 00:04

André Nicolas

514,336

See also http://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions . – Qiaochu Yuan Jun 08 '13 at 00:04
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Slightly more relevant: http://en.wikipedia.org/wiki/Prime_number_theorem_for_arithmetic_progressions#Prime_number_theorem_for_arithmetic_progressions – Erick Wong Jun 08 '13 at 01:10

Primes are uniformly distributed

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