Related papers: Various observations on angles proceeding in geome…

E661 in the Enestrom index. This was originally published as "Variae considerationes circa series hypergeometricas" (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma…

This is a translation into English from the original Latin of Leonhard Euler's Exercitatio analytica, Nova Acta Academiae Scientarum Imperialis Petropolitinae 8 (1794), 69-72; E664 in the Enestrom index. In it Euler uses the infinite…

Translation from the Latin of Euler's "Demonstratio theorematis circa ordinem in summis divisorum observatum" (1760). E244 in the Enestroem index. In his previous paper E243, Euler stated the pentagonal number theorem and assuming it proved…

In the present paper we study the asymptotic behavior of trigonometric products of the form $\prod_{k=1}^N 2 \sin(\pi x_k)$ for $N \to \infty$, where the numbers $\omega=(x_k)_{k=1}^N$ are evenly distributed in the unit interval $[0,1]$.…

I present and analyze a quadratically convergent algorithm for computing the infinite product \prod_{n=1}^\infty (1 - tx^n) for arbitrary complex t and x satisfying |x| < 1, based on the identity \prod_{n=1}^\infty (1 - tx^n) =…

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12,…

Translation from the Latin of Euler's "Observatio de summis divisorum" (1752). E243 in the Enestroem index. The pentagonal number theorem is that $\prod_{n=1}^\infty (1-x^n)=\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This paper assumes…

E158 in the Enestrom index. Translation of the Latin original "Observationes analyticae variae de combinationibus" (1741). This paper introduces the problem of partitions, or partitio numerorum (the partition of integers). In the first part…

Translated from the Latin original "Evolutio producti infiniti $(1-x)(1-xx)(1-x^3)(1x^4)(1-x^5)(1-x^6)$ etc. in seriem simplicem" (1775). E541 in the Enestroem index. In this paper Euler is revisiting his proof of the pentagonal number…

We derive the infinite product of the tangent function expressed in terms of trigonometric expressions such as Eulers Sinc function and Vietes formula, along with their generalizations. All the results presented in this work are novel.

This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part…

In this study, we derive the infinite product representation of the $\operatorname{sinc}(\mathrm{z})$ function by expressing it in a trigonometric form, evoking similarities to Morrie's Law and Euler's Product formula, along with their…

Finite Euler product is known to be one of the classical zeta functions in number theory. In [1], [2] and [3], we have introduced some multivariable zeta functions and studied their definable probability distributions on R^d. They include…

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's…

It is well known that $\zeta(2)=\pi^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula:…

In this paper, we show that Riemann hypothesis (concerning zeros of the zeta function in the critical strip) is equivalent to the analytic continuation of Euler products obtained by restricting the Euler zeta product to suitable subsets…

This paper is a promenade around the product $$ P_n(x) := \sin x \sin 2x \sin 4x ... \sin 2^n x. $$

We focus on a well-known convergence phenomenon, the fact that the $\zeta$ zeros are the universal singularities of certain Euler products.

Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin…

This is a translation from the Latin of Euler's "Problema algebraicum de inveniendis quatuor numeris ex datis totidem productis uniuscuiusque horum numerorum in summas trium reliquorum", Opera Postuma 1 (1862), 282-287, reprinted in…