Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.2771

Approximations to Euler's constant

Kh. Hessami Pilehrood·T. Hessami Pilehrood
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study a problem of finding good approximations to Euler's constant $γ=\lim_{n\to\infty}S_n,$ where $S_n=\sum_{k=1}^n\frac{1}{n}-\log(n+1),$ by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence $S_n$ can be significantly improved if $S_n$ is replaced by linear combinations of $S_n$ with integer coefficients. In this paper, considering more general linear transformations of the sequence $S_n$ we establish new accelerating convergence formulae for $γ.$ Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kh. Hessami Pilehrood, T. Hessami Pilehrood. 2007-08-21. Approximations to Euler's constant. https://arxiv.org/abs/0708.2771

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

KEEP EXPLORING

Related papers

arXiv2026

Hecke Subalgebras and Local Newforms for the Metaplectic Double Cover of $\mathrm{SL}_2(\mathbb{Q}_p)$

Let $p$ be an odd prime, and let $\overline{\mathrm{SL}_2(\mathbb{Z}_p)}$ and $\overline{K_0(p^n)}$ denote the inverse images of $\mathrm{SL}_2(\mathbb{Z}_p)$ and the congruence subgroup $K_0(p^n)$ in the metaplectic double cover $\widetilde{\mathrm{SL}}_2(\mathbb{Q}_p)$. For $n\geq2$, we study the subalgebra of the genuine Hecke algebra $H(\widetilde{\mathrm{SL}}_2(\mathbb{Q}_p)//\overline{K_0(p^n)},η)$ consisting of functions supported in $\overline{\mathrm{SL}_2(\mathbb{Z}_p)}$, where $η$ is the genuine extension of either the trivial or the nontrivial quadratic character modulo $p$. We give an explicit basis and a presentation by generators and relations, and prove that this subalgebra is commutative of dimension $2n$. We give the multiplicity-free decomposition of $\operatorname{Ind}_{\overline{K_0(p^n)}}^{\overline{\mathrm{SL}_2(\mathbb{Z}_p)}}η$, determining the dimensions of its irreducible constituents and their associated Hecke eigenvalues. This Hecke subalgebra acts on the $(\overline{K_0(p^n)},η)$-fixed spaces in irreducible admissible genuine representations of $\widetilde{\mathrm{SL}}_2(\mathbb{Q}_p)$. We compute its action explicitly on newvectors of prescribed $η$-type in principal series, Steinberg, even Weil, and supercuspidal representations. For supercuspidal representations, we construct the vectors by compact induction and use Ishimoto's conductor and dimension formulas. Finally, we relate our operator $\mathcal{W}_{n-1}$ to Ishimoto's local realization of Ueda's twisting operator.

math.NT↗
arXiv2026

Ekedahl-Oort strata meeting the supersingular locus

We give an explicit combinatorial criterion for an Ekedahl-Oort (EO) stratum in the moduli stack of principally polarized abelian varieties of dimension $g$ in characteristic $p > 2$ to meet the supersingular locus. The criterion is obtained by translating a non-emptiness criterion for basic affine Deligne-Lusztig varieties, into an explicit condition on the elementary sequences indexing the EO strata. Furthermore, we give explicit bounds for the number of EO strata of $p$-rank zero which are disjoint from the supersingular locus. As a result, we prove that asymptotically almost every EO stratum of $p$-rank zero meets the supersingular locus. As an application, we also give a lower bound for the number of EO strata meeting the supersingular locus on unitary PEL Shimura varieties of signature $(a,b)$.

math.NT↗
arXiv2026

A parametric Catalan-type congruence for generalized central trinomial coefficients

Let $p>3$ be a prime, and let $b,c$ be integers with $p\nmid b+2c$. We obtain an explicit congruence modulo $p^2$ for the sum of $\binom{2k}{k}T_{2k}(b,c^2)/(4^k(k+1)(b+2c)^{2k})$ from $k=0$ to $(p-1)/2$, where $T_n(b,c)$ denotes the generalized central trinomial coefficient. This gives a parametric formula for the sum considered in a question of Wang and Cui. The proof uses a constant term representation and a logarithmic generating function to reduce the sum to two coefficients of an algebraic generating function. A differential identity and a coefficientwise polynomial congruence then yield the desired evaluation. The formula also applies when $p$ divides $b$ or $b-2c$.

math.NT↗