For a positive integer $n$ we write $\phi \left(n\right)$ for the Euler function of $n$. In this note, we show that if $b>1$ is a fixed positive integer, then the equation $$\phi \left(x\frac{b^n-1}{b-1}\right)=y\frac{b^m-1}{b-1},\text{where}x,y\in \{1,...,b-1\},$$ has only finitely many positive integer solutions $(x,y,m,n)$.

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola ${}_{a,p}(X,Y)=(x,y):xy\equiv a\left(modp\right),1\le x\le X,1\le y\le Y$. We give asymptotic formulas for the average values ${\sum }_{\begin{array}{c}(\end{array}x,y){\in }_{a,p}(X,Y)x\ne y*}\phi \left(\right|x-y\left|\right)/|x-y|$ and ${\sum }_{\begin{array}{c}(\end{array}x,y){\in }_{a,p}(X,X)x\ne y*}\phi \left(\right|x-y\left|\right)$ with the Euler function φ(k) on the differences between the components of points of ${}_{a,p}(X,Y)$.

We introduce the real valued real analytic function κ(t) implicitly defined by $e^{2\pi i\kappa \left(t\right)}=-e^{-2i\vartheta \left(t\right)}\left({\zeta }^{\text{'}}(1/2-it)\right)/\left({\zeta }^{\text{'}}(1/2+it)\right)$ (κ(0) = -1/2). By studying the equation κ(t) = n (without making any unproved hypotheses), we show that (and how) this function is closely related to the (exact) position of the zeros of Riemann’s ζ(s) and ζ’(s). Assuming the Riemann hypothesis and the simplicity of the zeros of ζ(s), it follows that the ordinate of the zero 1/2 + iγₙ of ζ(s) is the unique solution to the equation κ(t) = n.

We consider an approximation to the popular conjecture about representations of integers as sums of four squares of prime numbers.

Let $K$ be a totally real cyclic number field of degree $n$ that is the product of two distinct primes and such that the class number of the $n$-th cyclotomic field equals 1. We derive certain necessary and sufficient conditions for the existence of a Minkowski unit for $K$.