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] elderly and timid single gentleman in Paris ... never drove down the Champs Elysees without expecting an accident, and commonly witnessing one; or found himself in the neighborhood of an official without calculating the chances of a bomb. So long as the rates of progress held good, these bombs would double in force and number every ten years.”
—Henry Brooks Adams (1838–1918)
“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)
“Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic.... It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.”
—Gottlob Frege (1848–1925)