It is proved that under some suitable conditions, the degree two functions in the Selberg class have infinitely many zeros on the critical line.

We prove new asymptotic formulas for some two-dimensional Farey-Brocot nets.

We prove that the Riemann Hypothesis holds if and only if $$I=\int_1^{+\infty}\bigl\{\Pi(x)-\Li(x)\bigr\}^2x^{-2}\,dx<+\infty$$ with $I=J$, where $J$ is some definite, computable real number ($1.266<J<1.273$). This provides us with a numerical test for the Riemann Hypothesis. The main interest of our test lies in the fact that it can also supply a \emph{goal}. Namely, having computed $J(a):=\int_1^a \bigl\{\Pi(x)-\Li(x)\bigr\}^2x^{-2}\,dx< J$ for a number of values of $a=a_n$, we can estimate a value $a$ for which, within our precision, we will have $J(a)\approx J$.

Let $k$ be a fixed non-zero integer, and let $x$ and $y$ be integers such that $$y^2=x^3+k.$$ We show that $$\log \max\{|x|,|y|\}<\min_{(c,d)\in S} \{c|k|(\log |k|)^d\},$$ where $$S=\{(10^{181},4), (10^{23},5), (10^{19},6)\}.$$

Let $F$ be a finite field with $q$ elements and characteristic $3.$ A sum $$M = M_{1}^4+\ldots+ M_{s}^4$$ of fourth powers of polynomials $M_1,\dots, M_{s}$ is a strict one if $ 4\deg M_i < 4 + \deg M$ for each $i= 1,\ldots, s.$ Our main results are: Let $P\in F[T]$ of degree $\geq 329.$ If $q>81$ is congruent to $1$ (mod. $4$), then $P$ is the strict sum of $9$ fourth powers; if $q=81$ or if $q>3$ is congruent to $3$ (mod $4$), then $P$ is the strict sum of $10$ fourth powers. If $q=3,$ every $P\in F[T]$ which is a sum of fourth powers is a strict sum of $12$ fourth powers, if $q=9,$ every $P\in F[T]$ which is a sum of fourth powers and whose degree is not divisible by $4$ is a strict sum of $8$ fourth powers; every $P\in F[T]$ which is a sum of fourth powers, whose degree is divisible by $4$ and whose leading coefficient is a fourth power is a strict sum of $7$ fourth powers.

We define a sequence of squarefree positive integers which arise naturally in the context of the solvability of a family of simultaneous Pell equations. It is proved that, apart from an explicitly given finite subset, each integer in this sequence has at least eight prime factors.

In this paper, we relax the condition in Theorem 2 of [1] to include a larger class of $N$. We then prove similar results on congruence subgroups other than $\Gamma_0(N)$.

There are no nontrivial integer solutions of $x^2+y^2+z^4=p^2$ for primes $p \equiv 7 \pmod 8$, even though there are no congruence obstructions.

Karin Halupczok [4] pointed out that we have stated an estimate in [3] that does not follow as easily as claimed. Although we are unable to obtain the claimed estimate, we prove a good enough estimate to (mostly) recover the theorems claimed in [3].