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’s prime is elusive. You little girls, when you grow up, must be on the alert to recognize your prime at whatever time of your life it may occur. You must then live it to the full.”
—Muriel Spark (b. 1918)
“The growing good of the world is partly dependent on unhistoric acts; and that things are not so ill with you and me as they might have been, is half owing to the number who lived faithfully a hidden life, and rest in unvisited tombs.”
—George Eliot [Mary Ann (or Marian)
“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)
“Under the dominion of an idea, which possesses the minds of multitudes, as civil freedom, or the religious sentiment, the power of persons are no longer subjects of calculation. A nation of men unanimously bent on freedom, or conquest, can easily confound the arithmetic of statists, and achieve extravagant actions, out of all proportion to their means; as, the Greeks, the Saracens, the Swiss, the Americans, and the French have done.”
—Ralph Waldo Emerson (1803–1882)