1. Introduction. In a recent article [6], the positive definite ternary quadratic forms that can possibly represent all odd positive integers were found. There are only twenty-three such forms (up to equivalence). Of these, the first nineteen were proven to represent all odd numbers. The next four are listed as "candidates". The aim of the present paper is to show that one of the candidate forms h = x² + 3y² + 11z² + xy + 7yz does represent all odd (positive) integers, and that it is regular in...

We describe the extension of the multiplication on a not-necessarily-discrete topological monoid to its flow compactification. We offer two applications. The first is a nondiscrete version of Hindman’s Theorem, and the second is a characterization of the projective minimal and elementary flows in terms of idempotents of the flow compactification of the monoid.

We study Mellin transforms $\widehat{N}\left(s\right)={\int }_{1-}^{\infty }{x}^{-s}dN\left(x\right)$ for which $N\left(x\right)-x$ is periodic with period $1$ in order to investigate ‘flows’ of such functions to Riemann’s $\zeta \left(s\right)$ and the possibility of proving the Riemann Hypothesis with such an approach. We show that, excepting the trivial case where $N\left(x\right)=x$, the supremum of the real parts of the zeros of any such function is at least $\frac{1}{2}$.We investigate a particular flow of such functions ${\left\{\widehat{N_{\lambda }}\right\}}_{\lambda \ge 1}$ which converges locally uniformly to $\zeta \left(s\right)$ as $\lambda \to 1$, and show that they exhibit features similar to $\zeta \left(s\right)$. For example, $\widehat{N_{\lambda }}\left(s\right)$...

On établit les majorations $\left|M\left(x\right)\right|\le \frac{0.002969x}{{(logx)}^{1/2}}$, valable pour $x\ge 142194,\left|M\left(x\right)\right|\le \frac{0.6437752x}{logx}$ qui est la meilleure majoration possible en $\frac{x}{logx}$ valable pour tout $x\>1(\left|M\left(5\right)\right|=2=\frac{0.6437752\times 5}{log5})$, et d’autres analogues. On montre enfin comment trouver des majorations effectives $\left|M\left(x\right)\right|\>\frac{c_{k}x{(loglogx)}^{2k}}{{(logx)}^k}$ pour tout $k$.

Nous exprimons certaines séries d’Epstein normalisées en $s=2$ comme combinaisons linéaires de dilogarithmes de Bloch-Wigner en des nombres algébriques des corps $\mathbb{Q}\left(\sqrt{\Delta }\right)$ pour les discriminants $\Delta$ associés à la forme quadratique.

L’objet de cet article est d’obtenir une formule pour la fonction zêta des hauteurs classique à partir de la fonction zêta des hauteurs multiple de La Bretèche, et d’utiliser cette formule pour prolonger de manière méromorphe la fonction zêta des hauteurs. En particulier, il est montré que celle-ci peut être prolongée au demi-plan $\{s\in ℂ:\Re es\>\frac{3}{4}\}$ et que la frontière naturelle de son domaine naturel de méromorphie est $\{s\in ℂ:\Re es=\frac{3}{4}\}$.