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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 ...
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 ...
The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for
The Möbius function () is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. [i] [ii] [2] It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula.
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 ...
The relative activity of a species i, denoted a i, is defined [4] [5] as: = where μ i is the (molar) chemical potential of the species i under the conditions of interest, μ o i is the (molar) chemical potential of that species under some defined set of standard conditions, R is the gas constant, T is the thermodynamic temperature and e is the exponential constant.