### On the sum of iterations of the Euler function.

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Let F denote the finite field of q elements. O. Ahmadi and A. Menezes have recently considered the question about the possible number of elements with zero trace in polynomial bases of F over F. Here we show that the Weil bound implies that there is such a basis with n + O(log n) zero-trace elements.

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 ${}_{a,p}(X,Y)$.

For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u,v) ∈ [1,U] × [1,V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have gcd(u,v) =1, which are also known as primitive points. Such points have a nice geometric interpretation as points on the modular hyperbola which are "visible" from the origin.

For positive integers m and N, we estimate the rational exponential sums with denominator m over the reductions modulo m of elements of the set ℱ(N) = {s/r : r,s ∈ ℤ, gcd(r,s) = 1, N ≥ r > s ≥ 1} of Farey fractions of order N (only fractions s/r with gcd(r,m) = 1 are considered).

We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field ${\mathbb{F}}_{p}$ of $p$ elements, such that the discriminant $D\left(E\right)$ of the quadratic number field containing the endomorphism ring of $E$ over ${\mathbb{F}}_{p}$ is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).

We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product $(ab+1)(ac+1)(bc+1)$ for distinct positive integers $a$, $b$ and $c$. In particular, we show that, under some natural conditions on rational functions $F,G,H\in \u2102\left(X\right)$, the number of distinct zeros and poles of the shifted products $FH+1$ and $GH+1$ grows linearly with $degH$ if $degH\ge max\{degF,degG\}$. We also obtain a version of this result for rational functions over a finite field.

For an integer $n\ge 2$ we denote by $P\left(n\right)$ the largest prime factor of $n$. We obtain several upper bounds on the number of solutions of congruences of the form $n!+{2}^{n}-1\equiv 0\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}q)$ and use these bounds to show that $$\underset{n\to \infty}{lim\; sup}P(n!+{2}^{n}-1)/n\ge (2{\pi}^{2}+3)/18.$$

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