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  2. Mertens function - Wikipedia

    en.wikipedia.org/wiki/Mertens_function

    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.

  3. Mertens conjecture - Wikipedia

    en.wikipedia.org/wiki/Mertens_conjecture

    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 ...

  4. Franz Mertens - Wikipedia

    en.wikipedia.org/wiki/Franz_Mertens

    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 ...

  5. Mertens' theorems - Wikipedia

    en.wikipedia.org/wiki/Mertens'_theorems

    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 ...

  6. Meissel–Mertens constant - Wikipedia

    en.wikipedia.org/wiki/Meissel–Mertens_constant

    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, Hadamard-de la Vallée-Poussin constant, or the prime reciprocal constant, is a ...

  7. 150 (number) - Wikipedia

    en.wikipedia.org/wiki/150_(number)

    150 is the sum of eight consecutive primes (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31). Given 150, the Mertens function returns 0. [1]150 is conjectured to be the only minimal difference greater than 1 of any increasing arithmetic progression of n primes (in this case, n = 7) that is not a primorial (a product of the first m primes).

  8. 114 (number) - Wikipedia

    en.wikipedia.org/wiki/114_(number)

    At 114, the Mertens function sets a new low of -6, a record that stands until 197. 114 is the smallest positive integer* which has yet to be represented as a 3 + b 3 + c 3, where a, b, and c are integers. It is conjectured that 114 can be represented this way. (*Excluding integers of the form 9k ± 4, for which solutions are known not to exist ...

  9. 159 (number) - Wikipedia

    en.wikipedia.org/wiki/159_(number)

    159 is: . the sum of 3 consecutive prime numbers: 47 + 53 + 59.; a Woodall number. [1]equal to the sum of the squares of the digits of its own square in base 15. [2]Only 5 numbers (greater than 1) have this property in base 15, none in base 10.