arXiv subjects

Ben Eltschig

Publications and source records attributed to Ben Eltschig.

3 recordsLinked to original sources

arXiv2026

ABC implies that Ramanujan's tau function misses almost all primes

Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan's tau-function. Recently, Xiong showed that these prime values form a subset of the primes with density at most $2/11$. Assuming the $abc$ Conjecture, we prove the stronger upper bound \[ S(X):=\#\{\ell\le X:\ \ell\ \text{prime and } |\tau(n)|=\ell \text{ for some } n\ge 1\} = O(X^{13/22}), \] which implies that Ramanujan's tau-function misses a density 1 subset of the primes. We give a heuristic suggesting that $S(X)$ should nevertheless be infinite, with predicted order of magnitude \[ S(X)\asymp \frac{C X^{\frac{1}{11}}}{(\log X)^2}. \] The main engine in this note was formalized and produced automatically in Lean/Mathlib by AxiomProver from a natural-language statement of the problem.

math.NT↗

arXiv2026

Almost all primes are partially regular

For odd primes $p$, we let $K_p:=\mathbb{Q}(\zeta_p)$ be the $p$th cyclotomic field and let $\omega$ denote its Teichmuller character. For $\alpha>1/2$, we say that an odd prime $p$ is partially regular if the eigenspaces of the $p$-Sylow subgroup of $\operatorname{Cl}(K_p)$ under the Galois action vanish for all characters $\omega^{p-2k}$ with \[ 2\le 2k \le \frac{\sqrt{p}}{(\log p)^{\alpha}}. \] Equivalently, $p\nmid \operatorname{num}(B_{2k})$ throughout this range. We prove that a density-one subset of primes is partially regular in this sense. By Leopoldt reflection, this yields a partial Vandiver Theorem: for a density-one set of primes $p$, the even eigenspaces $A_p(\omega^{2k})$ vanish for all even $2k$ satisfying the inequality above. This result has consequences for Kubota-Leopoldt $p$-adic $L$-functions, congruences between cusp forms and Eisenstein series, and $p$-torsion in algebraic $K$-groups. The theorem proving partial regularity for almost all $p$ is fully formalized in Lean/Mathlib and was produced automatically by AxiomProver from a natural-language statement of the conjecture.

math.NT↗

arXiv2026

Dead ends in square-free digit walks

We study "dead ends" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite square-free walks occur with probability near $1$, corresponding to an asymptotic dead-end density of $\approx 5.218\times 10^{-5}$. We prove that the true asymptotic dead-end density satisfies \[ c_{\mathrm{dead}} \approx 1.317\times 10^{-9}, \] roughly a factor of $\sim 4\times 10^4$ smaller than the prediction. For every base $b\geq 2$, we prove that dead-end densities exist and are given by a closed-form expression (as a finite alternating sum of Euler products). The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the problem.

math.CO↗