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On counterexamples to the Mertens conjecture

Seungki Kim, Phong Q. Nguyen

TL;DR

This paper refines the upper bound on the smallest counterexample to the Mertens conjecture by applying state-of-the-art lattice point enumeration to an optimized, unbalanced lattice that encodes the Diophantine approximation underlying the Mertens problem. By replacing extensive LLL-based trials with pruned lattice enumeration, and by carefully tuning lattice parameters, the authors obtain $\mathfrak x<\exp(1.96\times 10^{19})$, a dramatic improvement over prior bounds and well below certain growth conjectures. The work clarifies practical implementation details (BKZ/Lattice reduction, pruning strategies, and high-precision zeta-zero data) and demonstrates robust sanity checks and potential extensions to related questions about $M(x)$ and the zeta-zero distribution. It also outlines several directions for future research, including further reductions of $\mathfrak x$, insights into the growth of $q(x)$, and implications for linear relations among zeta zeros, highlighting the continued relevance of lattice techniques in analytic number theory.

Abstract

We use state-of-art lattice algorithms to improve the upper bound on the lowest counterexample to the Mertens conjecture to $\approx \exp(1.96 \times 10^{19})$, which is significantly below the conjectured value of $\approx \exp(5.15 \times 10^{23})$ by Kotnik and van de Lune [KvdL04].

On counterexamples to the Mertens conjecture

TL;DR

This paper refines the upper bound on the smallest counterexample to the Mertens conjecture by applying state-of-the-art lattice point enumeration to an optimized, unbalanced lattice that encodes the Diophantine approximation underlying the Mertens problem. By replacing extensive LLL-based trials with pruned lattice enumeration, and by carefully tuning lattice parameters, the authors obtain , a dramatic improvement over prior bounds and well below certain growth conjectures. The work clarifies practical implementation details (BKZ/Lattice reduction, pruning strategies, and high-precision zeta-zero data) and demonstrates robust sanity checks and potential extensions to related questions about and the zeta-zero distribution. It also outlines several directions for future research, including further reductions of , insights into the growth of , and implications for linear relations among zeta zeros, highlighting the continued relevance of lattice techniques in analytic number theory.

Abstract

We use state-of-art lattice algorithms to improve the upper bound on the lowest counterexample to the Mertens conjecture to , which is significantly below the conjectured value of by Kotnik and van de Lune [KvdL04].

Paper Structure

This paper contains 14 sections, 1 theorem, 34 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A review of the previous works
  3. The original argument of OR85
  4. Results on the smallest counterexample
  5. Our experiment
  6. Lattice point enumeration
  7. Choice of lattice
  8. Constraints on the parameters
  9. Implementation details
  10. Discussions
  11. Further research topics
  12. More improvement on the lowest counterexample
  13. On the growth order of $q(x)$
  14. Linear relations among the zeroes of $\zeta(s)$

Key Result

Theorem 2.1

Let If there exists $y \in [e^7, e^{50000}]$ with $|h_P(y)| > 1+e^{-40}$, then $\mathfrak x < \exp(y + \sqrt{y})$.

Figures (3)

  • Figure 1: Profile of a 1-round BKZ-84 reduced basis of the Mertens lattice for $N=120$, $\nu = 130$, $\nu_y = 100$, $\nu_t = 15$, and $\alpha^* = \alpha\exp(-1.5 \cdot 10^{-6}\gamma^2)$.
  • Figure 2: Correlations between $h_{StR}(y)$ and partial sums for $N=120$.
  • Figure 3: Correlations between $\|\mathbf{u}-\mathbf{t}\|^2$ and $h_{StR}(y)$ for $N=120$, where $\mathbf{u}$ is the lattice point corresponding to $y$.

Theorems & Definitions (1)

  • Theorem 2.1: Pintz Pintz