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:
“One wants in a Prime Minister a good many things, but not very great things. He should be clever but need not be a genius; he should be conscientious but by no means strait-laced; he should be cautious but never timid, bold but never venturesome; he should have a good digestion, genial manners, and, above all, a thick skin.”
—Anthony Trollope (1815–1882)
“Love has its name borrowed by a great number of dealings and affairs that are attributed to it—in which it has no greater part than the Doge in what is done at Venice.”
—François, Duc De La Rochefoucauld (1613–1680)
“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)
“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”
—Gottlob Frege (1848–1925)