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arXiv · 2608.24878

Oscillation of partial sums of the M\"obius function and zeros of Riemann's zeta function

János Pintz
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Abstract

The oscillation of M(x), the partial sum of the M\"obius function has been in the focus of researchers in the theory of primes since the famous conjecture of Mertens in 1905 (formulated in a weaker form by Stieltjes in 1885 in a letter to Hermite). The average order of the modulus of M(x) in an interval of type [0,Y] is clearly in connection with the distribution of primes. The author proved at the beginning of 1980's that this average tends unconditionally to infinity with Y. The present work shows that (similarly to the average order of the error term of the Prime Number Theorem) this average agrees with great accuracy for large values of Y with the largest error term of the Riemann-von Mangoldt prime number formula for the value Y.

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János Pintz. 2026-08-25. Oscillation of partial sums of the M\"obius function and zeros of Riemann's zeta function. https://arxiv.org/abs/2608.24878

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