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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.
37 is the sixth floor of imaginary parts of non-trivial zeroes in the Riemann zeta function. [18] It is in equivalence with the sum of ceilings of the first two such zeroes, 15 and 22. [19] The secretary problem is also known as the 37% rule by %.
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 .
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 Mertens function can be expressed as a sum over Farey fractions as = + where is the Farey sequence of order n. This formula is used in the proof of the Franel–Landau theorem . [ 24 ]
In particular, the matrix is not invertible precisely when the Mertens function is zero (or is close to changing signs). As a corollary of the disproof [ 1 ] of the Mertens conjecture , it follows that the Mertens function changes sign, and is therefore zero, infinitely many times, so the Redheffer matrix A n {\displaystyle A_{n}} is singular ...
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 ...
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.