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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. This definition can be extended to positive real numbers as follows:
Mertens' proof does not appeal to any unproved hypothesis (in 1874), and only to elementary real analysis. It comes 22 years before the first proof of the prime number theorem which, by contrast, relies on a careful analysis of the behavior of the Riemann zeta function as a function of a complex variable. Mertens' proof is in that respect ...
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 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 ...
For Mertens' result on convergence of Cauchy products of series, see Cauchy product Topics referred to by the same term This disambiguation page lists articles associated with the title Mertens' theorem .
In the limit, the sum of the reciprocals of the primes < n and the function ln(ln n) are separated by a constant, the Meissel–Mertens constant (labelled M above). 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 ...
Chemical thermodynamics involves not only laboratory measurements of various thermodynamic properties, but also the application of mathematical methods to the study of chemical questions and the spontaneity of processes. The structure of chemical thermodynamics is based on the first two laws of thermodynamics. Starting from the first and second ...
Such functions occur in the theory of the zeta function controlling the behavior of its zeros; for example the function S(T) above has average size around (log log T) 1/2. As S ( T ) jumps by at least 2 at any counterexample to the Riemann hypothesis, one might expect any counterexamples to the Riemann hypothesis to start appearing only when S ...