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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 ...
for any real number . Fix real numbers <, and let be a continuously differentiable function ... is Mertens function and () = = = +. This ...
It is an odd, composite, positive, real integer, composed of a prime (3) and a prime squared (11 2). 363 is a deficient number and a perfect totient number. 363 is a palindromic number in bases 3, 10, 11 and 32. 363 is a repdigit (BB) in base 32. The Mertens function returns 0. [1] Any subset of its digits is divisible by three.
A Meertens number is a sociable Meertens number with =, and a amicable Meertens number is a sociable Meertens number with =. The number of iterations i {\displaystyle i} needed for F b i ( n ) {\displaystyle F_{b}^{i}(n)} to reach a fixed point is the Meertens function's persistence of n {\displaystyle n} , and undefined if it never reaches a ...
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).
1255 = Mertens function zero, number of ways to write 23 as an orderless product of orderless sums, [109] number of partitions of 23 [202] 1256 = 1 × 2 × (5 2) 2 + 6, [203] Mertens function zero; 1257 = number of lattice points inside a circle of radius 20 [120] 1258 = 1 × 2 × (5 2) 2 + 8, [203] Mertens function zero; 1259 = highly ...