Related papers: On Primes Represented by Cubic Polynomials

We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice - the d-fold direct product of the primes - of positive…

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers.…

We prove several results regarding the distribution of numbers that are the product of a prime and a $k$-th power. First, we prove an asymptotic formula for the counting function of such numbers; this generalises a result of E. Cohen. We…

In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can…

We prove that the average of the $k$-th smallest prime quadratic non-residue modulo a prime approximates the $2k$-th smallest prime.

In this note, we generalize an ancient Greek inequality about the sequence of primes to the cases of arithmetic progressions even multivariable polynomials with integral coefficients. We also refine Bouniakowsky's conjecture [16] and…

We estimate the average of any arithmetic function $k$ over the values of any smooth polynomial in many variables provided only that $k$ has a distribution in arithmetic progressions of fixed modulus. We give several applications of this…

The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…

We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive…

We describe the limit zero distributions of sequences of polynomials with positive coefficients.

We show that both primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to $x^{5/8 - o(1)}$, using triply-well-factorable weights for the primes (we also get improvements for the well-factorable linear sieve…

In this paper we continue the investigations about unlike powers in arithmetic progression. We provide sharp upper bounds for the length of primitive non-constant arithmetic progressions consisting of squares/cubes and $n$-th powers.

In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime…

In this note we consider the links of prime ideals of certain skew polynomial rings and prove our main theorem, namely theorem [5], which states the following.Let R be a noetherian ring that is link k-symmetric and let {\sigma} be an…

We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length $k$ in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.

A geometric-arithmetic progression of primes is a set of $k$ primes (denoted by GAP-$k$) of the form $p_1 r^j + j d$ for fixed $p_1$, $r$ and $d$ and consecutive $j$, {\it i.e}, $\{p_1, \, p_1 r + d, \, p_1 r^2 + 2 d, \, p_1 r^3 + 3 d,…

We give a new proof that there are infinitely many primes, relying on van der Waerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else…

Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$…

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any $k \geq 3$ and $N$ large, there exist non-trivial $k$-term arithmetic progressions in (any positive density subset…

Let $P_1, \ldots, P_m \in K[y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y),…