We prove a Bombieri-Vinogradov type theorem for the number of representations of an integer $N$ in the form $N=p_1^g+p_2^g+...+p_s^g$ with $p_1,p_2,...,p_s$ prime numbers such that $p_1\equiv l\left(\mathrm{mod}k\right)$, under suitable hypothesis on $s=s\left(g\right)$ for every integer $g\ge 2$.

For $\epsilon \>0$ and any sufficiently large odd $n$ we show that for almost all $k\le R:=n^{1/5-\epsilon }$ there exists a representation $n=p_1+p_2+p_3$ with primes $p_i\equiv b_i$ mod $k$ for almost all admissible triplets $b_1,b_2,b_3$ of reduced residues mod $k$.

Let $N$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $n\in [N-6U,N+6U]$ can be represented as $$n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3,\left|p_1^2-{\displaystyle \frac{N}{6}}\right|\le U,\left|p_i^3-{\displaystyle \frac{N}{6}}\right|\le U,i=2,3,4,5,6,$$ where $U=N^{1-\delta +\epsilon }$ with $\delta \le 8/225$.

We demonstrate the Batyrev-Manin Conjecture for the number of points of bounded height on hypersurfaces of some toric varieties whose rank of the Picard group is 2. The method used is inspired by the one developed by Schindler for the case of hypersurfaces of biprojective spaces and by Blomer and Brüdern for some hypersurfaces of multiprojective spaces. These methods are based on the Hardy-Littlewood circle method. The constant obtained in the final asymptotic formula is the one conjectured by Peyre....