Related papers: On Improving Roth's Theorem in the Primes

Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or,…

Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $P \in E(\mathbb{Q})$ be a point of infinite order. The number of elliptic primes $p \leq x$ for which $\langle P\rangle=E(\mathbb{F}_p)$ is expected to be…

We generalize current known distribution results on Shanks--R\'enyi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function…

The Prime Number Theorem asserts that the density of primes less than or equal to $N$ is asymptotically equal to $1/\log N$. The density of prime triples in coprime triples in $\mathbb{Z}^3_+$ is determined to be $3\zeta (3)/\log N$, where…

Let $p$ be a prime number, and let $S$ be the numerical semigroup generated by the prime numbers not less than $p$. We compare the orders of magnitude of some invariants of $S$ with each other, e. g., the biggest atom $u$ of $S$ with $p$…

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density…

We show that if A is a subset of {1,...,N} containing no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).

We obtain asymptotic bounds on the number of natural numbers less than $X$ satisfying $\gcd{\left(n,\lfloor P(n) \rfloor\right)}=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such…

We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25\cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic…

For a positive integer $n$, we denote by $F(n)$ the distance from $n$ to the nearest prime number. We prove that every sufficiently large positive integer $N$ can be represented as the sum $N=n_1+n_2$, where $$ F(n_i) \geqslant (\log…

Let $a_0\in\{0,\dots,9\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their decimal expansion. The proof is an application of the Hardy-Littlewood circle method to a binary problem, and rests on…

Rubinstein and Sarnak investigated systems of inequalities of the form pi(x;q,a_1) > ... > pi(x;q,a_r), where pi(x;q,a) denotes the number of primes up to x that are congruent to a mod q. They showed, under standard hypotheses on the zeros…

For any $m = 3 \left( 2n + 1 \right) with \ n \in \mathbb{N^*} ,$ the prime counting function $\pi(m) = 4 + \left \vert A_4(m) \right \vert + 2 \left \vert A_6(m) \right \vert $ where $A_6(m) $ and $ A_4(m) $ are the sets of Twin Primes and…

We establish an explicit bound for the least prime occurring in the Chebotarev density theorem without any restriction. Let $L/K$ be any Galois extension of number fields such that $L\not=\mathbb{Q}$, and let $C$ be a conjugacy class in the…

We prove some new log-free density theorems for zeros of Dirichlet L-functions (which accordingly are more sharp than earlier ones near to the boundary line of the critical strip). The results can be applied in several problems of prime…

Let $B$ be a set of natural numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B = \{bb'| \, b, b' \in B\}$ cannot be greater than $O(n \log n)$ which matches the lower bound…

Let $\mathcal{L}$ be the closure of the set of all real numbers $\alpha$, such that there exist infinitely many integers $n$, such that $\alpha=\log\frac{d(n+1)}{d(n)}$, where $d$ is the number of divisors of $n$. We give improved lower…

Let $q$ be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset $A\subset{\mathbb F}_q^n$ will contain many non-trivial three-term arithmetic progressions, whenever…

We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can…

We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if $A\subseteq \{1,\ldots,N\}$ has no non-trivial three-term arithmetic progressions then $\lvert A\rvert \leq \exp(-c(\log N)^{1/12})N$ for some…