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Franz Mertens (20 March 1840 – 5 March 1927) (also known as Franciszek Mertens) was a Polish mathematician. He was born in Schroda in the Grand Duchy of Posen, Kingdom of Prussia (now Środa Wielkopolska, Poland) and died in Vienna, Austria. The Mertens function M(x) is the sum function for the Möbius function, in the theory of arithmetic ...
In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens. [ 1 ] In the following, let p ≤ n {\displaystyle p\leq n} mean all primes not exceeding n .
Mertens function to n = 10 000 Mertens function to n = 10 000 000. In number theory, the Mertens function is defined for all positive integers n as = = (), where () is the Möbius function. The function is named in honour of Franz Mertens.
In mathematics, the Mertens conjecture is the statement that the Mertens function is bounded by . Although now disproven, it had been shown to imply the Riemann hypothesis . It was conjectured by Thomas Joannes Stieltjes , in an 1885 letter to Charles Hermite (reprinted in Stieltjes ( 1905 )), and again in print by Franz Mertens ( 1897 ), and ...
It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by Franz Mertens in 1874. [3] Thus Euler obtained a correct result by questionable means. Erdős's proof by upper and lower estimates
The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as the Mertens constant, Kronecker's constant (after Leopold Kronecker), Hadamard–de la Vallée-Poussin constant (after Jacques Hadamard and Charles Jean de la Vallée-Poussin), or the prime reciprocal constant, is a mathematical constant in number ...
Enjoy a classic game of Hearts and watch out for the Queen of Spades!
Mertens conjecture; Pólya conjecture, 1919 (1958) Ragsdale conjecture; Schoenflies conjecture (disproved 1910) [21] Tait's conjecture; Von Neumann conjecture; Weyl–Berry conjecture; Williamson conjecture; In mathematics, ideas are supposedly not accepted as fact until they have been rigorously proved. However, there have been some ideas that ...