Related papers: Primes in tuples and Romanoff's theorem

We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter and lower bounded injectivity radius.

Many upper bounds for the moduli of polynomial roots have been proposed but reportedly assessed on selected examples or restricted classes only. Regarding quality measured in terms of worst-case relative overestimation of the maximum…

We establish sharp lower bounds for shifted (with two shifts) moments of Dirichlet $L$-function of fixed modulus under the generalized Riemann hypothesis.

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for…

In this note we prove an inequality involving primes and the product of consecutive primes.

In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due…

For a set $A$ of nonnegative integers, let $R_2(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a<a'$. Let $A_0$ be the Thue-Morse sequence and $B_0=\mathbb{N}\setminus A_0$. Let $A\subset \mathbb{N}$ and $N$ be a…

We show that there is a low T-upper bound for the class of K-trivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in the…

We establish a new lower bound for Mathieu's series and present a new derivation of its expansions in terms of Riemann Zeta functions.

Lower bounds for some explicit decision problems over the complex numbers are given.

The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.

For $\alpha >0$, let $$\mathscr{A}=\{ a_1<a_2<a_3<\cdots\}$$ and $$\mathscr{L}=\{ \ell_1, \ell_2, \ell_3,\cdots\} \quad \text{(not~necessarily~different)}$$ be two sequences of positive integers with $\mathscr{A}(m)>(\log m)^\alpha $ for…

We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.

We obtain an upper bound for the distribution of primes in the form $n^4 + k$ up to $x$, averaged over $k$ with small square-full part. As a corollary, we show that for almost all $k$, there is an expected amount of primes in the form $n^4…

We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rouch\'{e} theorem for matrix-valued functions, to generate upper, lower, and internal bounds on the eigenvalues of matrix polynomials. Variations of…

We give lower bounds on the number of effective divisors of degree $\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and…

We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the…

We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.

We show that a positive proportion of linear forms in four variables admit a solution in the primes that is as small as one would heuristically expect. Out of the linear forms that satisfy certain local solvability conditions, almost all…

We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive…