AN IMPROVED UPPER BOUND FOR THE SMALLEST COUNTEREXAMPLE TO THE MERTENS CONJECTURE
Abstract
Let x be the smallest x > 1 with |M (x)| ≥ √
x, where M (x) = ∑︁ n≤x µ(n). We prove that
x < exp(3 484 971 332 733 178 124) < exp(3.485 × 10 18 ), improving the bound x < exp(1.96 × 10 19 ) of Kim and Nguyen by a factor of about 5.6 in the exponent. The proof follows the effective method of Pintz, as refined by Saouter and te Riele. The criterion gives a counterexample near e u , or earlier, when the absolute value of a Gaussian-weighted finite zero sum exceeds 1 by more than an explicit error. In the local case, the sum is compared with a positive Gaussian average of M (x)/ √ x, with both contour and smoothing errors controlled. The smoothing parameter k in the weight e -kγ 2 controls a trade-off: smaller k gives more weight to higher zeros but requires a higher cutoff to keep the error small. All previous records used k ≥ 3 • 10 -9 . We re-derive the error terms for general parameters and work at k = 1.5 • 10 -10 , at a point found by lattice-based search. This requires the sum over all 2 325 572 zeros of height below 1.3 • 10 6 , which we evaluate in rigorous ball arithmetic. We also recompute, from certified zeros, the bound on 1/ζ required on the relevant height interval.
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BibTeX
@misc{bi2026improved,
title = {{AN IMPROVED UPPER BOUND FOR THE SMALLEST COUNTEREXAMPLE TO THE MERTENS CONJECTURE}},
author = {Zhenyu Bi and Enze Hou},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05838582v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - AN IMPROVED UPPER BOUND FOR THE SMALLEST COUNTEREXAMPLE TO THE MERTENS CONJECTURE
AU - Zhenyu Bi
AU - Enze Hou
PY - 2026
DA - 2026/10/26
PB - HAL
UR - https://hal.science/hal-05838582v1
AB - Let x be the smallest x > 1 with |M (x)| ≥ √ x, where M (x) = ∑︁ n≤x µ(n). We prove that x < exp(3 484 971 332 733 178 124) < exp(3.485 × 10 18 ), improving the bound x < exp(1.96 × 10 19 ) of Kim and Nguyen by a factor of about 5.6 in the exponent. The proof follows the effective method of Pintz, as refined by Saouter and te Riele. The criterion gives a counterexample near e u , or earlier, when the absolute value of a Gaussian-weighted finite zero sum exceeds 1 by more than an explicit error. In the local case, the sum is compared with a positive Gaussian average of M (x)/ √ x, with both contour and smoothing errors controlled. The smoothing parameter k in the weight e -kγ 2 controls a trade-off: smaller k gives more weight to higher zeros but requires a higher cutoff to keep the error small. All previous records used k ≥ 3 • 10 -9 . We re-derive the error terms for general parameters and work at k = 1.5 • 10 -10 , at a point found by lattice-based search. This requires the sum over all 2 325 572 zeros of height below 1.3 • 10 6 , which we evaluate in rigorous ball arithmetic. We also recompute, from certified zeros, the bound on 1/ζ required on the relevant height interval.
ER -