Related papers: Improving Roth's theorem in the primes

Let $N$ be an odd perfect number. Let $\omega(N)$ be the number of distinct prime factors of $N$ and let $\Omega(N)$ be the total number of prime factors of $N$. We prove that if $(3,N)=1$, then $ \frac{302}{113}\omega - \frac{286}{113}…

For n>1, let G(n)=\sigma(n)/(n log log n), where \sigma(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime…

Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions. In particular we show that \begin{align}\pi(x)=\Theta(x)+O\bigg(\frac{1}{\log x}\bigg), \nonumber…

Let $N$ be a large prime and $P, Q \in \mathbb{Z}[x]$ two linearly independent polynomials with $P(0) = Q(0) = 0$. We show that if a subset $A$ of $\mathbb{Z}/N\mathbb{Z}$ lacks a progression of the form $(x, x + P(y), x + Q(y), x + P(y) +…

Let $\overline{p}(n)$ denote the overpartition function. In this paper, our primary goal is to study the asymptotic behavior of the finite differences of the logarithm of the overpartition function, i.e., $(-1)^{r-1}\Delta^r \log \p(n)$, by…

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain $5$ elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that \[r_5(N)\ll \frac{N}{\exp((\log\log N)^{c})}.\] Our work…

Let $1<c<\frac{26088036}{12301745},c\not=2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R\in (N,2N]$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c-R\big|<\log^{-1}N…

We establish discorrelation estimates between the Piatetski-Shapiro prime set \[ \mathcal{P}_{\gamma} := \{p \text{ is prime and } p = \lfloor n^{1/\gamma} \rfloor \text{ for some } n \in \mathbb{N}\} \] and arbitrary nilsequences when…

We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| < K|A|, then A is contained in a multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at most exp(O(K^{7/4}…

Let $\mathbf{P} \subset [H_0,H]$ be a set of primes, where $\log H_0 \geq (\log H)^{2/3 + \epsilon}$. Let $\mathscr{L} = \sum_{p \in \mathbf{P}} 1/p$. Let $N$ be such that $\log H \leq (\log N)^{1/2-\epsilon}$. We show there exists a subset…

Let $\Lambda = \mathrm{SL}_2(\Bbb Z)$ be the modular group and let $c_n(\Lambda)$ be the number of congruence subgroups of $\Lambda$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log c_n(\Lambda)}{(\log n)^2/\log\log…

Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that in fact any subset of the primes of relative density tending to…

Let $p_n$ denotes the $n$-th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}-p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and…

Let $p$ be a prime number, and $h$ a positive integer such that $\gcd(p,h)=1$. We prove, without invoking Dirichlet's theorem, that the arithmetic progression $p\left(\mathbf{N}\cup \{0\}\right)+h$ contains infinitely many prime numbers.…

It is proved that \[ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N\exp\left(C\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right) \] holds for arbitrary integers $1\le n_1<\cdots < n_N$. This bound is essentially…

We prove that for every $n \in \mathbb{N}$ and $\delta>0$ there exists a word $w_n \in F_2$ of length $n^{2/3} \log(n)^{3+\delta}$ which is a law for every finite group of order at most $n$. This improves upon the main result of [A. Thom,…

Let $f(n)$ be a random completely multiplicative function such that $f(p) = \pm 1$ with probabilities $1/2$ independently at each prime. We study the conditional probability, given that $f(p) = 1$ for all $p < y$, that all partial sums of…

Let $p(n)$ denote the partition function. Desalvo and Pak proved the log-concavity of $p(n)$ for $n>25$ and the inequality $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)>\frac{p(n)}{p(n+1)}$ for $n>1$. Let $r(n)=\sqrt[n]{p(n)/n}$ and…

We show that if A is a subset of Z_4^n containing no three-term arithmetic progression in which all the elements are distinct then |A|=o(4^n/n).

Let $F$ be a finite field of odd characteristic. We prove that any set $A\subset F$ with $|A|\geq C|F|^{5/6}$ contains a nontrivial quadratic progression $(x, x+y, x+y^2), y\neq 0.$ For prime fields, this improves the previous best-known…