Using recent explicit asymptotic zero--free region and computations of zeros of the Riemann zeta function, obtained by Mossinghoff & Trudgian and Gourdon, respectively, we give an improvement for estimates of some functions related to distribution of primes, such as prime counting function, intervals containing at least one prime, Chebyshev's $\psi$ and $\vartheta$ functions.

If $0 < \gamma_1 \le \gamma_2 \le \gamma_3 \le \ldots$ denote ordinates of complex zeros of the Riemann zeta-function $\zeta(s)$, then several results involving the maximal order of $\gamma_{n+1}-\gamma_n$ and the sum $$ \sum_{0<\gamma_n\le T}{(\gamma_{n+1}-\gamma_n)}^k \qquad(k>0) $$ are proved.

In this paper, we study the diophantine equation $$y^{p} = f(x_{1}, x_{2}, ..., x_{r}),$$ where $f(x_{1}, x_{2}, ..., x_{r})$ is a real polynomial in variables $x_{1}, x_{2}, ..., x_{r}$ in $R$, a group of real numbers under the usual addition $+$, having the least element property.

Let $\rho$ denote the non-trivial zeros of the Riemann zeta function. We study the relative value distribution of $L(\rho+\sigma,\chi_{1})$ and $L(\rho+\sigma,\chi_{2})$, where $\sigma\in[0,1/2)$ is fixed and $\chi_{1}$, $\chi_{2}$ are two fixed Dirichlet characters to distinct prime moduli. For $\sigma>0$ we prove that a positive proportion of these pairs of values are linearly independent over $\mathbb{R}$, which implies that the arguments of the values are different. For $\sigma=0$ we show that, up to height $T$, the values are different for $cT$ of the Riemann zeros for some positive constant $c$.

A technique for generating new families of non-congruent numbers by appending a tail of primes to extend known families of non-congruent numbers is presented. These new non-congruent numbers are comprised of arbitrarily many prime factors belonging to two or three odd congruence classes modulo 8.

In the present paper, we substantially generalize one of the results obtained in our earlier paper [RM]. We present a solution of a problem of Waring type: if $F(x_1, \dots ,x_N)$ is a~symmetric form of odd degree $n\ge 9$ in $N=16\cdot 2^{n-9}$ variables, then for any $q\in \mathbb{Q}$, $q\neq 0$, the equation $F(x_i)=q$ has rational parametric solutions, that depend on $n-8$ parameters.

We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ of all algorithms that use $N$ information evaluations from the class $\Lambda$ in the $d$-dimensional case. The two classes $\Lambda$ considered in this paper are the class $\Lambda^{{\rm all}}$ consisting of all linear functionals and the class $\Lambda^{{\rm std}}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.

Schur proved the infinitude of primes in arithmetic progressions of the form $\equiv l\mod m$, such that $l^{2}\equiv1\mod m$, with non-analytic methods by ideas inspired from the famous proof Euclid gave for the infinitude of primes. Ram Murty showed that Schur's method has its limits given by the assumption Schur made. We will discuss analogous for the primes in the ring $\mathbb{F}_{q}[T]$.

We study integral solutions $n$ of the equation $A(n+k)=A(n)$, where $A$ is a certain multiplicative function related to Jordan's totient function.

We study elliptically parametrized families of elliptic curves with a point of order $13$ that do not arise from rational parametrizations. We also show that no elliptic curve over $\mathbb{Q}(\zeta_{13})^+$ can possess a rational point of order $13$.

We discuss a question of Datskovsky [1] about the minimal number of solutions to Schur-type equation $x_1+\ldots+x_{2n-1}=x_{2n}$ in a cyclic group $\mathbb{Z}_N$. We provide lower and upper bounds for this quantity.