Multiplicative inverses in short intervals

We give an alternative proof of a recent result by T.D. Browning and A. Haynes (arXiv:1204.6374v1) on multiplicative inverses in sequences of intervals and improve this result under additional conditions on the spacing of these intervals.Comment: 7 page

The large sieve with sparse sets of moduli

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then apply our result to the case when S consists of sqares. In this case we obtain an estimate which improves a recent result by L. Zhao

The Lang-Trotter Conjecture on Average

For an elliptic curve $E$ over \ratq and an integer $r$ let $\pi_E^r(x)$ be the number of primes $p\le x$ of good reduction such that the trace of the Frobenius morphism of E/\fie_p equals $r$. We consider the quantity $\pi_E^r(x)$ on average over certain sets of elliptic curves. More in particular, we establish the following: If $A,B>x^{1/2+\epsilon}$ and $AB>x^{3/2+\epsilon}$, then the arithmetic mean of $\pi_E^r(x)$ over all elliptic curves $E$ : $y^2=x^3+ax+b$ with a,b\in \intz, $|a|\le A$ and $|b|\le B$ is $\sim C_r\sqrt{x}/\log x$, where $C_r$ is some constant depending on $r$. This improves a result of C. David and F. Pappalardi. Moreover, we establish an ``almost-all'' result on $\pi_E^r(x)$.Comment: 18 page

Addendum to "On the pλp^{\lambda} problem"

We prove that the conditions $\lambda<5/19$ and $L\le T^{1/2}$ in Theorems 3 and 4 of our recent paper "On the $p^{\lambda}$ problem" can be omitted.Comment: 7 page