Let $H\left(x\right)={\sum }_{n\le x}\frac{\phi \left(n\right)}{n}-\frac{6}{{\pi }^2}x$. Motivated by a conjecture of Erdös, Lau developed a new method and proved that $\#\{n\le T:H(n\left)H\right(n+1)\<0\}\gg T.$ We consider arithmetical functions $f\left(n\right)={\sum }_{d\mid n}\frac{b_d}{d}$ whose summation can be expressed as ${\sum }_{n\le x}f\left(n\right)=\alpha x+P(log\left(x\right))+E\left(x\right)$, where $P\left(x\right)$ is a polynomial, $E\left(x\right)=-{\sum }_{n\le y\left(x\right)}\frac{b_n}{n}\psi \left(\frac{x}{n}\right)+o\left(1\right)$ and $\psi \left(x\right)=x-\lfloor x\rfloor -1/2$. We generalize Lau’s method and prove results about the number of sign changes for these error terms.

We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field ${𝔽}_p$ of $p$ elements, such that the discriminant $D\left(E\right)$ of the quadratic number field containing the endomorphism ring of $E$ over ${𝔽}_p$ 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).

This paper studies integer solutions to the $abc$ equation $A+B+C=0$ in which none of $A,B,C$ have a large prime factor. We set $H(A,B,C)=max\left(\right|A|,|B|,|C\left|\right)$, and consider primitive solutions ($\gcd (A,B,C)=1$) having no prime factor larger than ${(logH(A,B,C))}^{\kappa }$, for a given finite $\kappa$. We show that the $abc$ Conjecture implies that for any fixed $\kappa \<1$ the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed $\kappa \>8$ the $abc$ equation has infinitely many primitive solutions....

We solve a problem of Bombieri, stated in connection with the «prime number theorem» for function fields.