We show that the sequence of mantissas of the primorial numbers Pₙ, defined as the product of the first n prime numbers, is distributed following Benford's law. This is done by proving that the values of the first Chebyshev function at prime numbers are uniformly distributed modulo 1. We provide a convergence rate estimate. We also briefly treat some other sequences defined in the same way as Pₙ.

Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that ${\sum }_{n\le x}d²\left(n\right)=xP\left(logx\right)+E\left(x\right)$, where P(y) is a cubic polynomial in y and $E\left(x\right)=O\left(x^{3/5+\epsilon }\right)$, with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $E\left(x\right)=O\left(x^{1/2+\epsilon }\right)$. In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E\left(x\right)=O\left(x^{1/2}\left(logx\right)⁵loglogx\right)$. In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove $E\left(x\right)=O\left(x^{1/2}\left(logx\right)⁵\right)$.

Let k ≥ 1 denote any positive rational integer. We give formulae for the sums $S_{odd}(k,f)={\sum }_{\chi (-1)=-1}\left|L(k,\chi )\right|²$ (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums $S_{even}(k,f)={\sum }_{\chi (-1)=+1}\left|L(k,\chi )\right|²$ (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.

Let $\alpha$ be an algebraic integer of degree $d$ with conjugates ${\alpha }_1=\alpha ,{\alpha }_2,\cdots ,{\alpha }_d$. In the paper we give a lower bound for the mean value$$M_p\left(\alpha \right)=\sqrt[p]{\frac{1}{d}{\sum }_{i=1}^d|log|{\alpha }_i{\left|\right|}^p}$$when $\alpha$ is not a root of unity and $p\>1$.

Each of the Diophantine equations $A^4\pm n{B}^3=C^2$ has an infinite number of integral solutions $(A,B,C)$ for any positive integer $n$. In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when $A$, $B$ and $C$ are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions $(A,B,C)$ of the Diophantine equation $a{A}^3+c{B}^3=C^2$ for any co-prime integer pair $(a,c)$.