Prime Number Theorem - Prime Number Theorem For Arithmetic Progressions

Prime Number Theorem For Arithmetic Progressions

Let denote the number of primes in the arithmetic progression a, a + n, a + 2n, a + 3n, … less than x. Dirichlet and Legendre conjectured, and Vallée-Poussin proved, that, if a and n are coprime, then

\pi_{n,a}(x) \sim \frac{1}{\phi(n)}\mathrm{Li}(x),

where φ(·) is the Euler's totient function. In other words, the primes are distributed evenly among the residue classes modulo n with gcd(a, n) = 1. This can be proved using similar methods used by Newman for his proof of the prime number theorem.

The Siegel–Walfisz theorem gives a good estimate for the distribution of primes in residue classes.

Read more about this topic:  Prime Number Theorem

Famous quotes containing the words prime, number, theorem and/or arithmetic:

    “I did not know I was in my prime until afterwards.”
    —Mason Cooley (b. 1927)

    “He is the richest man who knows how to draw a benefit from the labors of the greatest number of men, of men in distant countries, and in past times.”
    —Ralph Waldo Emerson (1803–1882)

    “To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
    —Albert Camus (1913–1960)

    “O! O! another stroke! that makes the third.
    He stabs me to the heart against my wish.
    If that be so, thy state of health is poor;
    But thine arithmetic is quite correct.”
    —A.E. (Alfred Edward)