Related papers: Primes in tuples and Romanoff's theorem

We obtain some results related to Romanoff's theorem.

Let $\mathcal{A}=\{a_{n}\}_{n=1}^{\infty}$ and $\mathcal{B}=\{b_{n}\}_{n=1}^{\infty}$ be two sequences of positive integers (not necessarily distinct). Under some restrictions on $\mathcal{A}$ and $\mathcal{B}$, we obtain a lower bound for…

We obtain optimal lower bounds for moments of theta functions. On the other hand, we also get new upper bounds on individual theta values and moments of theta functions on average over primes. The upper bounds are based on bounds of…

Romanov proved that a positive proportion of the integers have a representation as a sum of a prime and a power of an arbitrary fixed positive integer. Rieger proved the analogous result for number fields. We will determine an explicit…

Following the work of Castillo-Hall-Oliver-Pollack-Thompson who extended Maynard-Tao theorem on admissible tuples to number fields and function fields for tuples with monic linear forms, here we obtain the Maynard-Tao theorem for admissible…

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

We generalize Romanoff's theorem. Also, we obtain a result on sums related to Euler's totient function.

We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken…

Let $K\geq 2$ be a natural number and $a_i,b_i\in\mathbb{Z}$ for $i=1,\ldots,K-1$. We use the large sieve to derive explicit upper bounds for the number of prime $k$-tuplets, i.e., for the number of primes $p\leq x$ for which all $a_ip+b_i$…

We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.

We obtain a lower bound for \[ \#\{x/2< p_{n}\leq x:\ p_n \equiv\ldots\equiv p_{n+m}\equiv a\text{ (mod $q$)},\ p_{n+m} - p_{n}\leq y\}, \] where $p_{n}$ is the $n^{\text{th}}$ prime.

We obtain lower bounds for the $l_1$-norm of the Fourier transform of functions on $\mathbb{Z}_p^d$.

In the paper, some lower bounds for polygamma functions are refined.

We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a…

For a prime $r$, we obtain lower bounds on the proportion of $r$-regular elements in classical groups and show that these lower bounds are the best possible lower bounds that do not depend on the order of the defining field. Along the way,…

In this paper we establish a number of new estimates concerning the prime counting function \pi(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in…

Verifying a conjecture of Gromov we establish a generalized Margulis Lemma for manifolds with lower Ricci curvature bound. Among the various applications are finiteness results for fundamental groups of compact $n$-manifolds with upper…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

We bound from below the number of shifted primes p+s<x that have a divisor in a given interval (y,z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the…

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.