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
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:
“Faith in reason as a prime motor is no longer the criterion of the sound mind, any more than faith in the Bible is the criterion of righteous intention.”
—George Bernard Shaw (1856–1950)
“The world is so full of a number of things,
I’m sure we should all be as happy as kings.”
—Robert Louis Stevenson (1850–1894)
“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)