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:

    “In time, after a dozen years of centering their lives around the games boys play with one another, the boys’ bodies change and that changes everything else. But the memories are not erased of that safest time in the lives of men, when their prime concern was playing games with guys who just wanted to be their friendly competitors. Life never again gets so simple.”
    —Frank Pittman (20th century)

    “Strange goings on! Jones did it slowly, deliberately, in the bathroom, with a knife, at midnight. What he did was butter a piece of toast. We are too familiar with the language of action to notice at first an anomaly: the ‘it’ of ‘Jones did it slowly, deliberately,...’ seems to refer to some entity, presumably an action, that is then characterized in a number of ways.”
    —Donald Davidson (b. 1917)

    “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)