arXiv subjects
Publications and source records attributed to Paweł Lewulis.
We prove a generalisation of Vinogradov's theorem by finding for $m\geqslant 3$ and fixed positive integers $c_1, \dots ,c_m, r_1, \dots , r_m$ the asymptotics of the number of sequences $(n_1, \dots ,n_m) \in \mathbf{N}^{m}$ such that $c_1n_1 + \dots + c_m n_m = N$ and $Ω(n_i) = r_i$ for every $i=1, \dots ,m$ under the assumption that at least three of the $r_i$ are equal to $1$.
Let $k\geq 2$ and $\mathcal{P} (n) = (A_1 n + B_1 ) \cdots (A_k n + B_k)$ where all the $A_i, B_i$ are integers. Suppose that $\mathcal{P} (n)$ has no fixed prime divisors. For each choice of $k$ it is known that there exists an integer $\varrho_k$ such that $\mathcal{P} (n)$ has at most $\varrho_k$ prime factors infinitely often. We used a new weighted sieve set-up combined with a device called an $\varepsilon$-trick to improve the possible values of $\varrho_k$ for $k\geq 7$. As a by-product of our approach, we improve the conditional possible values of $\varrho_k$ for $k\geq 4$, assuming the generalized Elliott--Halberstam conjecture.
Let $k \geq 3$ and let $L_i(n) = A_in + B_i$ be some linear forms such that $A_i$ and $B_i$ are integers. Define ${\mathcal{P}(n) = \prod_{i=1}^k L_i(n)}$. For each $k$ it is known that $Ω(\mathcal{P} (n) ) \leq ρ_k$ infinitely often for some integer $ρ_k$. We improve the possible values of $ρ_k$ for $4 \leq k \leq 10$ assuming $GEH$. We also show that we can take $ρ_5=14$ unconditionally. As a by-product of our approach we reprove the $ρ_3=7$ result which was previously obtained by Maynard who used techniques specifically designed for this case.
Instantaneous derivation of the Thomas precession.
We find a lower bound for the number of Chen primes in the arithmetic progression $a \bmod q$, where $(a,q)=(a+2,q)=1$. Our estimate is uniform for $q \leq \log^M x$, where $M>0$ is fixed.