相关论文: Biases in the Shanks-Renyi Prime Number Race

We consider the three dimensional array $\mathcal{A} = \{a_{i,j,k}\}_{1\le i,j,k \le n}$, with $a_{i,j,k} \in [0,1]$, and the two random statistics $T_{1}:= \sum_{i=1}^n \sum_{j=1}^n a_{i,j,\sigma(i)}$ and $T_{2}:= \sum_{i=1}^{n}…

Let $P$ and $T$ be disjoint sets of prime numbers with $T$ finite. A simple formula is given for the natural density of the set of square-free numbers which are divisible by all of the primes in $T$ and by none of the primes in $P$. If $P$…

Let $a_0=b_0=0$ and $0<a_1\leq b_1<a_2\leq b_2<\ldots\leq b_{n}$ be integers. Let $Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)$ be the number of integers between $1$ and $x$ such that all exponents in their prime factorization are in…

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the $a_i$ and $b$ are fixed coefficients, and $h$ is an arbitrary integer…

For a base $b\geq 2$ and a set of digits $\mathcal{A}\subset \{0,...,b-1\}$, let $\mathcal{P}$ denote the set of prime numbers with digits restricted to $\mathcal{A}$, when written in base-$b$. We prove that if $A\subset \mathbb{N}$ has…

We compute the one-level density of zeros of order $\ell$ Dirichlet $L$-functions over function fields $\mathbb{F}_q[t]$ for $\ell=3,4$ in the Kummer setting ($q\equiv1\pmod{\ell}$) and for $\ell=3,4,6$ in the non-Kummer setting…

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} ~(\bmod ~…

The Raney numbers are a class of combinatorial numbers generalising the Fuss--Catalan numbers. They are indexed by a pair of positive real numbers $(p,r)$ with $p>1$ and $0 < r \le p$, and form the moments of a probability density function.…

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta…

Let P(x,d,a) denote the number of primes p<=x with p=a(mod d). Chebyshev's bias is the phenomenon that `more often' P(x;d,n)>P(x;d,r) than the other way around, where n is a quadratic non-residue mod d and r is a quadratic residue mod d. If…

We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be considered as a combined scenario of Duke,…

Let $\theta\in(1,2)$, and $\mu_{\theta}$ be the Bernoulli convolution parametrized by $\theta$, that is, the measure corresponding to the distribution of the random variable $\sum_{n=1}^{\infty} a_n\theta^{-n}$, where the $a_n$ are i.i.d.…

In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $\mathbb N$ as limiting values of $q$-series as $q\to \zeta$…

Following both Ernvall-Mets\"{a}nkyl\"{a} and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic $\lambda$-invariant) for the $p$-adic zeta-function twisted by a Dirichlet character $\chi$ of any…

We say that $d_3(n)$ has exponent of distribution $\theta$ if, for every $\varepsilon>0$, the expected asymptotic holds uniformly for all moduli $q \le x^{\theta-\varepsilon}$. Nguyen proved, following earlier work of Banks, Heath-Brown,…

It is shown that there is a constant A and a density one subset S of the positive integers, such that for all q in S there is some 1<=p<q, (p, q)=1, so that p/q has all its partial quotients bounded by A.

We study the distributions of the random Dirichlet series with parameters $(s, \beta)$ defined by $$ S=\sum_{n=1}^{\infty}\frac{I_n}{n^s}, $$ where $(I_n)$ is a sequence of independent Bernoulli random variables, $I_n$ taking value $1$ with…

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do…

We consider several problems about pseudoprimes. First, we look at the issue of their distribution in residue classes. There is a literature on this topic in the case that the residue class is coprime to the modulus. Here we provide some…

Suppose that $A \subset \mathbb{R}$ has positive upper density, \[ \limsup_{|I| \to \infty} \frac{|A \cap I|}{|I|} = \delta > 0,\] and $P(t) \in \mathbb{R}[t]$ is a polynomial with no constant or linear term, or more generally a non-flat…