On sums of Ramanujan sums
Let $c_q(n)$ denote the Ramanujan sum modulo $q$, and let $x$ and $y$ be large reals, with $x = o(y)$. We obtain asymptotic formulas for the sums $$\sum_{n \le y}(\sum_{q \le x} c_q(n))^k \qquad (k = 1, 2).$$
math.NT↗
arXiv subjects
Angel V Kumchev
Publications and source records attributed to Angel V Kumchev.
On sums of primes from Beatty sequences
Let $k \ge 2$ and $α_1, β_1, ..., α_k, β_k$ be reals such that the $α_i$'s are irrational and greater than 1. Suppose further that some ratio $α_i/α_j$ is irrational. We study the representations of an integer $n$ in the form $$ p_1 + p_2 + ... + p_k = n, $$ where $p_i$ is a prime from the Beatty sequence $$ \mathcal B_i = \left\{n \in \mathbb N : n = [ α_i m + β_i ] \text{for some} m \in \mathbb Z \right\}. $$
math.NT↗