Mertens' Theorems are a set of three related results proved by Franz Mertens describing the asymptotic behavior of sums and products involving the prime numbers, including the growth rate of the sum of the reciprocals of the primes up to a given bound and the constant, now called the Mertens constant, that governs that growth. They refine earlier results on the density of the primes and stand as precursors to the later Prime Number Theorem.
Facts
StatementMertens' theorems are three related asymptotic results on the primes: the sum of the reciprocals of the primes up to x grows like log(log(x)) plus the Mertens constant, the product over primes up to x of (1 - 1/p) is asymptotic to e^(-gamma)/log(x), and a related bound on the density of primes up to x. Proved by Franz Mertens in 1874. 1 Classification
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Sources
1. Mertens' Theorems (Wikipedia)
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In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.
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