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where the infinite product extends over all prime numbers p. [2] The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is valid for all complex s.
An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, chiefly that the distribution of prime numbers is intimately connected with the zeros of the analytically ...
The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Adolf Piltz in 1884. [1] Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers. The formal statement of the hypothesis follows.
In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the asymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of for intervals near a number ).
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]
For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that x / ln(x) is a good approximation to π(x), in the sense that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1:
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. [1] [2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π 0 (x), which is equal to π(x) − 1 ⁄ 2 if x is exactly a prime number, and equal to π(x) otherwise.