We give the best known error term of the $B$-process of van der Corput's method and we extend the class of exponential sums that can be estimated by using exponent pairs coming from the iteration of the $A$ and $B$ processes.

For $\zeta$ a transcendental real number, we consider the classical Diophantine exponents $w_{n}(\zeta)$ and $\widehat{w}_{n}(\zeta)$. They measure how small $\vert P(\zeta)\vert$ can be for an integer polynomial $P$ of degree at most $n$ and naive height bounded by $X$, for arbitrarily large and all large $X$, respectively. The discrepancy between the exponents $w_{n}(\zeta)$ and $\widehat{w}_{n}(\zeta)$ has attracted interest recently. Studying parametric geometry of numbers, W. Schmidt and L. Summerer were the first to refine the trivial inequality $w_{n}(\zeta)\geq \widehat{w}_{n}(\zeta)$. Y. Bugeaud and the author found another estimation provided that the condition $w_{n}(\zeta)>w_{n-1}(\zeta)$ holds. In this paper we establish an unconditional version of the latter result, which can be regarded as a proper extension. Unfortunately, the new contribution involves an additional exponent and is of interest only in certain cases.

Let $\phi$ denote the Euler totient function, defined by $id*\mu$ where $\mu$ is the M\"{o}bius function. We shall consider the $k$-th Riesz mean of the arithmetical function $n/\phi(n)$ for any positive integer $k \geq 2$ on the assumption of the Riemann Hypothesis. Our result is a refinement of Theorem 2 in A. Sankaranarayanan and S.K. Singh [6]. We also improve it upon the assumption of the Gonek-Hejhal Hypothesis.

Let $E$ be the elliptic curve $y^2=x^3+(i-2)x^2+x$ over the imaginary quadratic field $\mathbb{Q}(i)$. In this paper, we investigate the supersingular primes of $E$. We introduce the curve $C$ of genus two over $\mathbb{Q}$ covering a quotient of $E$ and for any prime number $p$, we state a condition (over $\mathbb{F}_p$) about the reduction of the jacobian variety of $C$ modulo $p$ which is equivalent to the existence of a supersingular prime of $E$ lying over $p$ (Theorem 5.10).

Let $m_1$ and $m_2$ be distinct square-free integers. We show that there exist infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{m_1D})$ and $\mathbb{Q}(\sqrt{m_2D})$ whose class numbers are both divisible by $3$ under the splitting conditions of prime numbers. This improves results of T. Komatsu and the author.

For any finitely generated subgroup $\Gamma$ of $\mathbb{Q}^*$, Pappalardi and the first--named author [1] found a formula to compute the density of the primes $\ell$ for which the reduction modulo $\ell$ of $\Gamma$ contains a primitive root modulo $\ell$. They conjectured a characterization of optimal groups, free or torsion, i.e. subgroups with maximal density. In this paper we prove their conjecture and give a similar characterization for optimal positive groups.

Motivated by a recent result of Farhi we show that for each $n\equiv \pm 1\pmod{6}$ the title Diophantine equation has at least two solutions in integers. As a consequence, we get that each (even) perfect number is a sum of three cubes of integers. Moreover, we present some computational results concerning the considered equation and state some questions and conjectures.

Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In recent years, it has emerged that a sharp asymptotic formula for the shifted convolution sum of the triple divisor function would be useful in evaluating the sixth moment of the Riemann zeta function. In this article, we study $D_{k,\ell}(x,h) = \sum_{n \le x} \tau_k(n) \tau_{\ell}(n+h)$ where $\tau_k$ and $\tau_{\ell}$ are the $k$-th and $\ell$-th divisor functions. The main result is a lower bound of the correct order of magnitude for $D_{k,\ell}(x,h)$, uniform in $h$. In addition, the conjectural asymptotic formula for $D_{k,\ell}(x,h)$ is studied. Using an argument of Ivi\'{c} [25], [26] and Conrey-Gonek [8] the leading term in the conjectural asymptotic formula is simplified. In addition, a probabilistic method is presented which gives the same leading term. Finally, we show that these two methods give the same answer as in a recent probabilistic argument of Terry Tao [41].