In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors of Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)\in\mathbb{Z}[x]$, $a$ is a positive integer and $b$ an arbitrary positive integer.

In this paper, we give the order of magnitude for the summatory function of the number of divisors of the least common multiple of $p^i-1$ for $i=1,2,\ldots,k$ when $p\le x$ is prime.

In this paper we characterize off-diagonal Carleson embeddings for both Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper-half plane. We use these results to obtain embedding relations and pointwise multipliers between these spaces.

We consider the solutions to the inequality \[|p_1^c + \cdots + p_s^c - R| < R^{-\eta}\] (where $c > 1$, $c \not\in \mb N$ and $\eta$ is a small positive number; $R$ is large). We obtain new ranges of $c$ for which this has many solutions in primes $p_1, \ldots, p_s$, for $s = 2$ (and `almost all' $R$), $s=3$, 4 and~5. We also consider the solutions to the equation in integer parts \[[p_1^c] + \cdots + [p_s^c] = r\] where $r$ is large. Again $c> 1$, $c\not\in \mb N$. We obtain new ranges of $c$ for which this has many solutions in primes, for $s=3$ and 5.

We give a further elaboration of our previous works on abstract intersection theory for zeta-functions, emphasizing geometric aspects. We review and streamline Weil's proof of the Riemann hypothesis for curves over finite fields and then present it in cohomological terms. Then by using a spectral interpretation of the Riemann zeta-function we construct a model of our abstract intersection theory for the Riemann zeta-function as an analogue of the cohomological version of Weil's classical intersection theory.