Related papers: Various observations on angles proceeding in geome…

The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{\mu^2(d)}{\varphi(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right).…

Euler defines a function f(x) somehow as an infinite product and a generalization of [x], where [x] ist, what we now call following Legendre the Gamma-Funktion. He gets some recursive relationships for f(x), by applying some very nice…

One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n \alpha)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $\alpha$. Another…

We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…

The absolute zeta function for a scheme $X$ of finite type over $\mathbb{Z}$ satisfying a certain condition is defined as the limit as $p\to 1$ of the congruent zeta function for $X\otimes\mathbb{F}_p$. In 2016, after calculating absolute…

Noam Elkies and Everett Howe independently noticed a certain elegant product formula for the multiple integral \int_R \prod_{1 \le i < j \le k} (x_j-x_i) dx_1 \cdots dx_k, where the region $R$ is the set of $k$-tuples satisfying $0 < x_1 <…

Let $(a;q)_{\infty}$ be the $q$-Pochhammer symbol and $\mathrm{li}_2(x)$ be the dilogarithm function. Let $\prod_{\alpha,\beta,\gamma}$ be a finite product with every triple $(\alpha,\beta,\gamma)\in(\mathbb{R}_{>0})^3$ and…

The Riemann Hypothesis has been of central interest to mathematicians for a long time and many unsuccessful attempts have been made to either prove or disprove it. Since the Riemann zeta function is defined as a sum of the infinite number…

The generalized Euler number E_{n|k} counts the number of permutations of {1,2,...,n} which have a descent in position m if and only if m is divisible by k. The classical Euler numbers are the special case when k=2. In this paper, we study…

We classify singularities of Dirichlet series having Euler products which are rational functions for p and p^{-s} for p a prime number and give examples of natural boundaries from zeta functions of groups and height zeta functions.

This paper discusses some infinite trigonometric products which are characteristic functions of simple random walks on the real line; in fact, these define "random Riemann-$\zeta$ functions," a notion which is explained. The concept of…

Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the Birch and Swinnerton-Dyer conjecture asserts that $\prod \limits _{p \leq x} \frac{N_p}{p}…

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function $\sigma_{\alpha}(n):=\sum_{d|n}d^{\alpha}$. We obtain an exact identity relating the Dirichlet series…

This work is the final version of my master thesis. Many, but not all of its key results are already available as a preprint with Chiara Esposito and Stefan Waldmann on arxiv.org under the title "Convergence of the Gutt Star Product", which…

This is a translation of Euler's Latin paper "De fractionibus continuis observationes" into English. In this paper Euler describes his theory of continued fractions. He teaches, how to transform series into continued fractions, solves the…

The Euler graph has vertices labelled (n,k) for n=0,1,2,... and k=0,1,...,n, with k+1 edges from (n,k) to (n+1,k) and n-k+1 edges from (n,k) to (n+1,k+1). The number of paths from (0,0) to (n,k) is the Eulerian number A(n,k), the number of…

In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and…

Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given…

This appendix to the beautiful paper of Ihara puts it in the context of infinite global fields of our papers. We study the behaviour of Euler--Kronecker constant $\gamma\_{K}$ when the discriminant (respectively, the genus) tends to…

We study the convergence of a discrete Luenberger observer for the barotropic Euler equations in one dimension, for measurements of the velocity only. We use a mixed finite element method in space and implicit Euler integration in time. We…