Mertens function

In number theory, the Mertens function is defined for all positive integers n as M ( n ) = ∑ k = 1 n μ ( k ) , {\displaystyle M(n)=\sum _{k=1}^{n}\mu (k),} where μ ( k ) {\displaystyle \mu (k)} is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive real numbers as follows: M ( x ) = M ( ⌊ x ⌋ ) . {\displaystyle M(x)=M(\lfloor x\rfloor ).} Less formally, M ( x ) {\displaystyle M(x)} is the count of square-free integers up to x that have an even number of prime factors, minus the count of those that have an odd number. The first 143 M(n) values are (sequence A002321 in the OEIS) The Mertens function slowly grows in positive and negative directions both on average and in peak value, oscillating in an apparently chaotic manner passing through zero when n has the values 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, ... (sequence A028442 in the OEIS). Because the Möbius function only takes the values −1, 0, and +1, the Mertens function moves slowly, and there is no x such that |M(x)| > x. H. Davenport demonstrated that, for any fixed h, ∑ n = 1 x μ ( n ) exp ⁡ ( i 2 π n θ ) = O ( x log h ⁡ x ) {\displaystyle \sum _{n=1}^{x}\mu (n)\exp(i2\pi n\theta )=O\left({\frac {x}{\log ^{h}x}}\right)} uniformly in θ {\displaystyle \theta } . This implies, for θ = 0 {\displaystyle \theta...