Motivated by Newman's phenomenon for the Thue-Morse sequence $(-1)^{s(n)}$, where $s(n)$ is the sum of the binary digits of $n$, we investigate a similar problem for prime numbers. More specifically, for an integer $k\ge 2$, we explore the signs of $ S_k(x)=\sum_{p \le x} (-1)^{s_k(p)}$, where $s_k(n)$ is the sum of the last $k$ binary digits of $n$, and $p$ runs over the primes. We prove that $S_k(x)$ changes signs for infinitely many integers $x$, assuming that all Dirichlet $L$-functions attached to primitive characters modulo $2^k$ do not vanish on $(0,1)$. Our result is unconditional for $k\leq 18$. Furthermore, under stronger assumptions on the zeros of Dirichlet $L$-functions, we show that for $k\geq 4$, the sets $\{x> 2: S_k(x)>0\}$ and $\{x> 2: S_k(x)<0\}$ both have logarithmic density $1/2$.

We determine the convolution sum $\sum_{al+bm=n} \sigma(l) \sigma(m)$ for $(a,b)=(1,48), (3, 16), (1,54), (2,27)$ for all positive integers $n$. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of $n$ by the octonary quadratic forms $k(x_1^2 + x_1x_2 + x_2^2 +x_3^2 + x_3x_4 + x_4^2) + l(x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2)$ for $(k,l)=(1,16), (1,18), (2,9)$. A modular form approach is used.

Our aim in this paper is to show that the set $G_j(A)\cup \{0\}$, where $G_j(A)$ is the set of all J-invertible elements of a topological Jordan algebra $A$ without J-g.t.d.z, is isomorphic to $\mathbb{C}$ in the complex case. In the real case it is a $p$-normed quadratic J-division subalgebra of $A$. This result extends a well-known result of W. Żelazko ([11]) in the associative case to Jordan algebras.

We prove tight estimates for averages of the twisted Hooley $\Delta$-function over arbitrary number fields.

Let $\mathbb{F}_q$ be the finite field with $q=p^r$ elements, where $p$ is a prime. For a monic $m \in \mathbb{F}_q[T]$, let $h_m$ be the class number of the $m$th cyclotomic function field. The goal of this paper is to determine the $p$-divisibility of $h_m$ when $q = p$ $(r \ge 2)$ and $\deg m=2$.

We discuss the capitulation kernel associated to a degree $n$ covering using Čech cohomology and the Kummer sequence. The main result is a five-term exact sequence that relates the capitulation kernel to the Čech cohomology of the $n$-th roots of unity and a certain subquotient of the group of units modulo $n$-th powers.

If $X$ is a closed subscheme of torus $\mathbb{G}_{m,K}^g$ on a number field $K$, we define $Z_X$ as the union of all translated positive dimensional subtori contained in $X$. Vojta proved that the set of integral points on $X\setminus Z_X$ is finite. In this article, we consider a family of closed subschemes $V\to P$ of $\mathbb{G}_{m,K}^g$. We prove a semi-effective bound for the height of the integer points in $V_p\setminus Z_{V_p}$. We prove this result in the more general context of the Mordell-Lang plus Bogomolov problem and we obtain a corollary on the unit equation.

We introduce an Eisenstein series associated to a loxodromic element of cofinite Kleinian groups, namely the loxodromic Eisenstein series, and study its fundamental properties. It is the analogue of the hyperbolic Eisenstein series for Fuchsian groups of the first kind. We prove the convergence and the differential equation associated to the Laplace-Beltrami operator. We also prove the precise spectral expansion associated to the Laplace-Beltrami operator. Furthermore, we derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.