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Yes, (2) holds for all large enough $n$. According to Wikipedia (see also this question) $\frac{p_n}n=\log n+\log\log n-1+\frac{\log\log n}{\log n}(1+o(1))$, so $$\frac{n+p_{n+1}}{n+1}=\log(n+1)+\log\ …

Is it true that for every completely multiplicative function $f:\mathbb N\to\mathbb C$ with $|f(n)|=1$ for all $n$, the logarithmic average $$\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac1nf …

Tao and Ziegler extended the Green-Tao theorem to the polynomial setting. As a very special case we get that any subset of the primes with positive relative density contains a difference which is a sq …

A result of Green and Tao (initially conditional on two conjectures which were eventually settled by them and Ziegler) states that for any $s\in\mathbb N$, $$\lim_{w\to\infty}\limsup_{N\to\infty}\sup_ …

It is well known that for each $m\in\mathbb{N}$ $$\lim_{N\to\infty}\frac1N\sum_{n=1}^Ne^{2\pi i\sqrt{nm}}=0$$ My question is whether there is some uniformity in the variable $m$. More precisely, is it …

A positive integer is multiplicatively even (odd) if, when decomposed into primes, the sum of the exponents is even (odd). A subset of the integers is thick if it contains arbitrarily long intervals $ …

I am interested in the sums $$g(a,k)=\sum_{n=0}^{p-1}e^{2\pi i a n^k/p}$$ where $p\equiv1\mod k$ is a prime and $a$ is coprime with $p$. When $k=2$, it is a classical fact that $g(a,2)=\chi(a)g(1,2)$ …

For a positive integer $n$, let $\Omega(n)$ be the number of primes dividing $n$, counted with multiplicities (eg $\Omega(5)=1$, $\Omega(8)=\Omega(12)=3$). For a real number $x$, let $\{x\}\in[0,1)$ …

I am interested in the set $A$ of all positive integer numbers such that when factored into primes, the sum of the exponents is odd (I think of $A$ as the multiplicative odd numbers). I want to know …