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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:
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 ...
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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 ...
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A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
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 ...
A chemical file format is a type of data file which is used specifically for depicting molecular data. One of the most widely used is the chemical table file format, which is similar to Structure Data Format files. They are text files that represent multiple chemical structure records and associated data fields.