PreprintCC BY 4.0arXiv
Oscillation of partial sums of the Möbius function and zeros of Riemann's zeta function
Abstract
The oscillation of M(x), the partial sum of the Möbius 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.
Related
- More than 8/9 zeros of the Riemann zeta function are simple or on the critical line
- Explicit formula for the discrete Laplace transform of the Möbius function, related special functions, and a criterion for the Riemann hypothesis
- One-level density of zeros of the Riemann zeta function weighted by its special values
- Half-sized gaps between zeros of the Riemann zeta function
- Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals
BibTeX
@misc{pintz2026oscillation,
title = {{Oscillation of partial sums of the Möbius function and zeros of Riemann's zeta function}},
author = {János Pintz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24878},
eprint = {2608.24878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24878},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Oscillation of partial sums of the Möbius function and zeros of Riemann's zeta function
AU - János Pintz
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.24878
UR - https://arxiv.org/abs/2608.24878
AB - The oscillation of M(x), the partial sum of the Möbius 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.
ER -