A Diophantine approximation problem with two primes and one k-power of a prime

We refine a result of the last two Authors on a Diophantine approximation problem with two primes and a k-th power of a prime which was only proved to hold for 1<k<4/3. We improve the k-range to 1<k 643 by combining Harman's technique on the minor arc with a suitable estimate for the L4-norm of the relevant exponential sum over primes Sk. In the common range we also give a stronger bound for the approximation

A Diophantine problem with prime variables

We study the distribution of the values of the form $\lambda_1 p_1 + \lambda_2 p_2 + \lambda_3 p_3^k$, where $\lambda_1$, $\lambda_2$ and $\lambda_3$ are non-zero real number not all of the same sign, with $\lambda_1 / \lambda_2$ irrational, and $p_1$, $p_2$ and $p_3$ are prime numbers. We prove that, when $1 < k < 4 / 3$, these value approximate rather closely any prescribed real number.Comment: submitte

Sums of four prime cubes in short intervals

We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$, where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones previously known.Comment: Unconditional result improved by using a Robert-Sargos estimate (lemmas 6-7); more detailed proof of Lemma 5 inserted. Correction of a typo. 10 page

Sum of one prime and two squares of primes in short intervals

Assuming the Riemann Hypothesis we prove that the interval $[N, N + H]$ contains an integer which is a sum of a prime and two squares of primes provided that $H \ge C (\log N)^{4}$, where $C > 0$ is an effective constant.Comment: removed unconditional case; other minor changes inserte

Ces\`aro average in short intervals for Goldbach numbers

We prove that a suitable explicit formula for the Cesaro-averaged number of representations of an integer as a sum of two primes holds in short intervals.Comment: revised version; to appear in Proc. AM

A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets

Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces $X$ and $Y$ whenever a map $f:X\to Y$ with strong connectivity conditions on the fibers is given. We apply similar techniques in o-minimal expansions of fields to compare the o-minimal homotopy of a definable set $X$ with the homotopy of some of its bounded hyperdefinable quotients $X/E$. Under suitable assumption, we show that $\pi_{n}(X)^{\rm def}\cong\pi_{n}(X/E)$ and $\dim(X)=\dim_{\mathbb R}(X/E)$. As a special case, given a definably compact group, we obtain a new proof of Pillay's group conjecture "$\dim(G)=\dim_{\mathbb R}(G/G^{00}$)" largely independent of the group structure of $G$. We also obtain different proofs of various comparison results between classical and o-minimal homotopy.Comment: 24 page

The number of Goldbach representations of an integer

We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then $$\sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + O(N \log^{3}N),$$ where $\rho=1/2+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function $\zeta(s)$