Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.15714

On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers

Leonardo Carofiglio·Giacomo Cherubini·Alessandro Gambini
View original source ↗Read source document ↗Export metadata ↓

Abstract

We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan--Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime $p$. The conjectures state that: $(i)$ $J_p$ is always finite and of the order $O(p^2(\log\log p)^{2+\epsilon})$; $(ii)$ the set of primes for which $J_p$ is minimal (called harmonic primes) has density $e^{-1}$ among all primes; $(iii)$ no harmonic number is divisible by $p^4$. We prove $(i)$ and $(iii)$ for all $p\leq 16843$ with at most one exception, and enumerate harmonic primes up to~$50\cdot 10^5$, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of about $30$ and $50$, respectively.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Leonardo Carofiglio, Giacomo Cherubini, Alessandro Gambini. 2025-03-19. On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers. https://arxiv.org/abs/2503.15714

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields

For any primes $p \neq \ell$, $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_{p^2}$ have a typical structure called $\ell$-volcanoes, and the structure is the core of Sutherland's supersingularity testing algorithm for elliptic curves. In this paper, by exploiting the properties of $3$-isogenies between twisted Hessian curves, we show that when $p \equiv 2 \pmod{3}$ and $\ell = 3$, every ordinary twisted Hessian curve defined over $\mathbb{F}_p$ lies on the surface of the $3$-volcano. As an application, we give an improved version of Sutherland's supersingularity testing algorithm specialized to twisted Hessian curves defined over $\mathbb{F}_p$ with $p \equiv 2 \pmod{3}$. We also give a generalization of the known fact that any supersingular $j$-invariant is a cube in $\mathbb{F}_{p^2}$; we show that for any twisted Hessian curve $H(a,d)$ defined over $\mathbb{F}_{p^2}$, its $j$-invariant is not a cube in $\mathbb{F}_{p^2}$ if and only if $H(a,d)$ is ordinary and lies on the floor of a $3$-volcano.

math.NT↗
arXiv2026

Effective estimates for exponential sums with multiplicative coefficients

Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$ for every $x\ge1$. If $|\alpha-a/q|\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \] with effective implied constants. Montgomery and Vaughan proved this with second term $NR^{-1/2}(\log R)^{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $NR^{-1/2}\sqrt{\log R\log\log R}$. We remove the factor $\sqrt{\log R}$ from Bachman's second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.

math.NT↗
arXiv2026

Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers

In this paper, we will study a trivariate extension of the Cauchy numbers of both the first kind (also called Gregory coefficients) and the second kind (also called N\"orlund numbers) via the Laurent expansion of the reciprocal of any positive integer power (which is called the order) of multiple polylogarithms. In the case of logarithm, we will show by the WZ method that for each order $\ell>1$ some Gregory coefficient of order $\ell$ must vanish, in contrast to the fact that all classical Gregory coefficients are nonzero. We also prove in this higher order logarithm case that the sequence is eventually alternating for each fixed order, a property enjoyed by the classical Gregory coefficients. In the most general setting, we conjecture that these new sequences are all eventually positive, which is supported by strong numerical evidence. Finally, we confirm this conjecture in the special case of polylogarithms and double polylogarithms. As a by product, for each zeta value and double zeta value, we find an infinite family of identities expressing its reciprocal as a sum of a rational number and an improper integral.

math.NT↗