The first counterexample to the Mertens conjecture is below exp(3.485·10¹⁸)

Let M(x) be the Mertens function, the sum of the Möbius function μ(n) over n ≤ x. Odlyzko and te Riele (1985) disproved the Mertens conjecture, |M(x)| < √x for all x > 1, without exhibiting a counterexample; |M(n)| < √n has been verified for 1 < n ≤ 10¹⁶. Let x₁ be the least integer n ≥ 1 with |M(n)| > √n. In this preprint I prove, unconditionally, that x₁ < exp(3.485·10¹⁸); in fact x₁ < 10^(1.51351·10¹⁸). The previous best bound I am aware of is exp(1.96·10¹⁹), by Kim and Nguyen (2025), stated for the least real x > 1 with |M(x)| ≥ √x (my bound also holds for that quantity); the exponent is smaller by a factor of about 5.6. This is an upper bound: it does not locate the first counterexample. If, in addition, an explicit hypothesis on the zeros of ζ holds (H5 in the paper: it follows from the Riemann hypothesis and can fail only for a zero with |γ| > 3·10¹²), then there is a counterexample x with ln x in an explicit window of width about 1.78·10⁴, and it satisfies M(x) > +√x. I do not claim that this window contains x₁. The proof is computer-assisted and introduces no new theory: it applies the criterion of Pintz and of Saouter–te Riele with a Gaussian kernel, as Kim and Nguyen do, with other parameters. It uses a sum over the 16 622 627 zeros of ζ with 0 < γ ≤ 7 999 035.172, evaluated in ball arithmetic (Arb), and Table 1 of Saouter–te Riele recomputed in ball arithmetic. Verification: the deposit contains the data and programs. The zeros and the sum can be recomputed rigorously with public software (Arb, via python-flint), and the inequality of Table 1 can be checked independently from Platt's public list of zeros (rigorous given the accuracy and completeness of that list). The analytic part of the proof can be checked by hand, following Appendix A. The file SHA256SUMS lists the SHA-256 checksum of every file and carries an OpenTimestamps timestamp (file SHA256SUMS.ots). The text of the paper is licensed under CC BY 4.0 and the programs under the MIT License. Status: preprint, not yet peer reviewed. The English version is the main text; a Spanish translation is included (the English version prevails).