相关论文: Distribution of $\theta-$powers and their sums

We improve upon a result of Steinerberger (2024) by demonstrating that for any fixed $k \in \mathbb{N}$ and sufficiently large $n$, there exist integers $1 \leq a_1, \dots, a_k \leq n$ satisfying: \begin{align*} 0 < \left\| \sum_{j=1}^{k}…

This is a review of several results related to distribution of powers and combination of powers modulo 1. We include a proof that given a sequence of real numbers $\theta_n$, it is possible to get an $\alpha$ (given $\lambda \ne 0$), or a…

Let $\|\cdot\|$ denote the minimum distance to an integer. For $0<\gamma< 1$, $\theta>0$ and $(\alpha, \beta) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}$ we study when \begin{equation*} \|\alpha p^{\gamma}+\beta \|<p^{-\theta},…

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon}…

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a couple of infinite families of explicit formulas for $S_k(n)$. One of the families…

Let $a>1$ be an integer. Denote by $l_a(n)$ the multiplicative order of $a$ modulo integer $n\geq 1$. We prove that there is a positive constant $\delta$ such that if $x^{1-\delta}\log^3 x = o(y)$, then $$ \frac1y \sum_{a<y} \frac1x…

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denote the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $\frac{13}{14}<\gamma<1$, there…

On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $\frac12+i\gamma$ of the Riemann zeta function, we show that the sequence \[ \Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad…

Let $\lfloor t\rfloor$ denote the integer part of $t\in\mathbb{R}$ and $\|x\|$ the distance from $x$ to the nearest integer. Suppose that $1/2<\gamma_2<\gamma_1<1$ are two fixed constants. In this paper, it is proved that, whenever $\alpha$…

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log…

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{\theta \in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n \theta) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N…

Let $q$ be an integer $\geq 2$ and let $S_q(n)$ denote the sum of digits of $n$ in base $q$. For \[ \alpha=[0;\overline{1,m}],\ m\geq 2, \] let $S_{\alpha}(n)$ denote the sum of digits in the Ostrowski $\alpha$-representation of $n$. Let…

In this paper we study the Theta splitting function $\Theta(s+1)$, a function defined on the positive integers. We study the distribution of this function for sufficiently large values of the integers. As an application we show that…

Let $2 \leq y \leq x$ such that $\beta := \frac{\log x}{\log y} \rightarrow \infty$. Let $\omega_y(n)$ denote the number of distinct prime factors $p$ of $n$ such that $p \leq y$, and let $\mu_y(n) := \mu^2(n)(-1)^{\omega_y(n)}$, where…

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $\alpha\in\mathbb{R}\backslash\mathbb{Q},\,\beta\in\mathbb{R}$ and $0<\theta<10/1561$, there…

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $(q,z)\in \mathbb{C}^2$, $|q|<1$. We show that for any $0<\delta _0<\delta <1$, there exists $n_0\in \mathbb{N}$ such that for any $q$ with…

Let $P_{r}$ denote an integer with at most $r$ prime factors counted with multiplicity. In this paper we prove that for some $\lambda < \frac{1}{12}$, the inequality $\{\sqrt{p}\}<p^{-\lambda}$ has infinitely many solutions in primes $p$…

In this paper, we improve the results in the author's previous paper \cite{Usu22}, which deals with the quantitative problem on Littlewood's conjecture. We show that, for any $0<\gamma<1$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set…

Let $\theta$ and $\rho$ be real numbers with $0 \le \theta, \rho < 1$ and $\theta$ irrational. We show that the Hecke-Mahler series $$ F_{\theta, \rho} (z_1, z_2) = \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 \theta + \rho \rfloor} \,…

Given $k, \ell \in {\bf N}^+$, let $x_{i,j}$ be, for $1 \le i \le k$ and $0 \le j \le \ell$, some fixed integers, and define, for every $n \in {\bf N}^+$, $s_n := \sum_{i=1}^k \prod_{j=0}^\ell x_{i,j}^{n^j}$. We prove that the following are…