In the paper, we provide an effective criterion for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous.

Let $L(s,\unicode[STIX]{x1D712})$ be the Dirichlet $L$-function associated to a non-principal primitive character $\unicode[STIX]{x1D712}$ modulo $q$ with $3\leqslant q\leqslant 400\,000$. We prove a new explicit zero-free region for $L(s,\unicode[STIX]{x1D712})$: $L(s,\unicode[STIX]{x1D712})$ does not vanish in the region $\mathfrak{Re}\,s\geqslant 1-1/(R\log (q\max (1,|\mathfrak{Im}\,s|)))$ with $R=5.60$. This improves a result of McCurley where $9.65$ was shown to be an admissible value for $R$.

Let $\Vert \cdots \Vert$ denote distance from the integers. Let $\unicode[STIX]{x1D6FC}$, $\unicode[STIX]{x1D6FD}$, $\unicode[STIX]{x1D6FE}$ be real numbers with $\unicode[STIX]{x1D6FC}$ irrational. We show that the inequality

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FC}p^{2}+\unicode[STIX]{x1D6FD}p+\unicode[STIX]{x1D6FE}\Vert <p^{-3/17+\unicode[STIX]{x1D700}}\end{eqnarray}$$

$p$

$\unicode[STIX]{x1D6FC}p$

$\unicode[STIX]{x1D6FD}=0$

Let $\unicode[STIX]{x1D6E4}\subseteq \operatorname{PSL}(2,\mathbf{R})$ be a finite-volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\unicode[STIX]{x1D6E4}$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $\text{e}^{(2/3)R}$ is known, and this has not been improved for any group. Recently, Risager and Petridis proved that in the special case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$ taking $z=w$ and averaging over $z$ in a certain way the error term can be improved to $\text{e}^{(7/12+\unicode[STIX]{x1D716})R}$. Here we show such an improvement for a general $\unicode[STIX]{x1D6E4}$; our error term is $\text{e}^{(5/8+\unicode[STIX]{x1D716})R}$ (which is better than $\text{e}^{(2/3)R}$ but weaker than the estimate of Risager and Petridis in the case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$). Our main tool is our generalization of the Selberg trace formula proved earlier.

We study a kernel function of the twisted symmetric square $L$-function of elliptic modular forms. As an application, several exact special values of the $L$-function are computed.

We generalize Skriganov’s notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov’s celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. We also establish a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erdős, and others. Finally, we use o-minimality to describe large classes of sets to which our counting results apply.

We prove an exact formula for the second moment of Rankin–Selberg $L$-functions $L(\frac{1}{2},f\times g)$ twisted by $\unicode[STIX]{x1D706}_{f}(p)$, where $g$ is a fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given level $q$. The formula is a reciprocity relation that exchanges the twist parameter $p$ and the level $q$. The method involves the Bruggeman–Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.

We show that for $p\geqslant 1$, the $p$th moment of suprema of linear combinations of independent centered random variables are comparable with the sum of the first moment and the weak $p$th moment provided that $2q$th and $q$th integral moments of these variables are comparable for all $q\geqslant 2$. The latest condition turns out to be necessary in the independent and identically distributed case.

We formulate some conjectures about the precise determination of the monodromy groups of certain rigid local systems on $\mathbb{A}^{1}$ whose monodromy groups are known, by results of Kubert, to be finite. We prove some of them.

Let $\unicode[STIX]{x1D713}:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ be a non-increasing function. A real number $x$ is said to be $\unicode[STIX]{x1D713}$-Dirichlet improvable if it admits an improvement to Dirichlet’s theorem in the following sense: the system

$$\begin{eqnarray}|qx-p|<\unicode[STIX]{x1D713}(t)\quad \text{and}\quad |q|<t\end{eqnarray}$$

$t$

$D(\unicode[STIX]{x1D713})$

$D(\unicode[STIX]{x1D713})^{c}$

$\unicode[STIX]{x1D713}$

It has been known since Vinogradov that, for irrational $\unicode[STIX]{x1D6FC}$, the sequence of fractional parts $\{\unicode[STIX]{x1D6FC}p\}$ is equidistributed in $\mathbb{R}/\mathbb{Z}$ as $p$ ranges over primes. There is a natural second-order equidistribution property, a pair correlation of such fractional parts, which has recently received renewed interest, in particular regarding its relation to additive combinatorics. In this paper we show that the primes do not enjoy this stronger equidistribution property.

The purpose of this article is to generalize some known characterizations of Banach space properties in terms of graph preclusion. In particular, it is shown that superreflexivity can be characterized by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a non-trivial finitely branching bundle graph. It is likewise shown that asymptotic uniform convexifiability can be characterized within the class of reflexive Banach spaces with an unconditional asymptotic structure by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a non-trivial $\aleph _{0}$-branching bundle graph. For the specific case of $L_{1}$, it is shown that every countably branching bundle graph bi-Lipschitzly embeds into $L_{1}$ with distortion no worse than $2$.

We prove asymptotic formulas for the number of integers at most $x$ that can be written as the product of $k~({\geqslant}2)$ distinct primes $p_{1}\cdots p_{k}$ with each prime factor in an arithmetic progression $p_{j}\equiv a_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q$, $(a_{j},q)=1$ $(q\geqslant 3,1\leqslant j\leqslant k)$. For any $A>0$, our result is uniform for $2\leqslant k\leqslant A\log \log x$. Moreover, we show that there are large biases toward certain arithmetic progressions $(a_{1}\hspace{0.2em}{\rm mod}\hspace{0.2em}q,\ldots ,a_{k}\hspace{0.2em}{\rm mod}\hspace{0.2em}q)$, and such biases have connections with Mertens’ theorem and the least prime in arithmetic progressions.

While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form

$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D70C}}R(\unicode[STIX]{x1D70C})x^{\unicode[STIX]{x1D70C}},\end{eqnarray}$$

$\unicode[STIX]{x1D70C}$

$\unicode[STIX]{x1D701}(s)$

$R(x)\in \overline{\mathbb{Q}}(x)$

$x>0$

$$\begin{eqnarray}f(x)=\mathop{\sum }_{\unicode[STIX]{x1D70C}}\frac{x^{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}}\end{eqnarray}$$

$(1,\infty )$

$f(x)$

$x<1$

$x>1$

$x<1$

$d\geqslant 1$

$f(\unicode[STIX]{x1D70B}\sqrt{d}x)$

$(0,1/\unicode[STIX]{x1D70B}\sqrt{d})$

$$\begin{eqnarray}L^{\prime }(1,\unicode[STIX]{x1D712}_{-d})\neq 0,\end{eqnarray}$$

$\unicode[STIX]{x1D712}_{-d}$

$K:=\mathbb{Q}(\sqrt{-d})$

For a positive integer $n$ let

$$\begin{eqnarray}\mathfrak{P}_{n}=\mathop{\prod }_{\substack{ p \\ s_{p}(n)\geqslant p}}p,\end{eqnarray}$$

$p$

$s_{p}(n)$

$p$

$n$

$n$

$\mathfrak{P}_{n}$

$\mathfrak{P}_{n}$

$\sqrt{n}$

$n^{20/37}$

$\sqrt{n}$

$\unicode[STIX]{x1D705}\sqrt{n}/\text{log}\,n$

$\unicode[STIX]{x1D705}$

$\unicode[STIX]{x1D705}=2$

$\mathfrak{P}_{n}$

$\mathfrak{P}_{n+1}$

$\mathfrak{P}_{n}$

$n$

$\mathfrak{P}_{n}>\mathfrak{P}_{n+1}$

Let $s\geqslant 3$ be a fixed positive integer and let $a_{1},\ldots ,a_{s}\in \mathbb{Z}$ be arbitrary. We show that, on average over $k$, the density of numbers represented by the degree $k$ diagonal form

$$\begin{eqnarray}a_{1}x_{1}^{k}+\cdots +a_{s}x_{s}^{k}\end{eqnarray}$$

$k$

Almost-flat manifolds were defined by Gromov as a natural generalization of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental group. Utilizing this, we classify the spin structures on all four-dimensional almost-flat manifolds that are not flat. Out of 127 orientable families, we show that there are exactly 15 that are non-spin, the rest are, in fact, parallelizable.

Given any positive integers $m$ and $d$, we say a sequence of points $(x_{i})_{i\in I}$ in $\mathbb{R}^{m}$ is Lipschitz-$d$-controlling if one can select suitable values $y_{i}\;(i\in I)$ such that for every Lipschitz function $f\,:\,\mathbb{R}^{m}\,\rightarrow \,\mathbb{R}^{d}$ there exists $i$ with $|f(x_{i})\,-\,y_{i}|\,<\,1$. We conjecture that for every $m\leqslant d$, a sequence $(x_{i})_{i\in I}\subset \mathbb{R}^{m}$ is $d$-controlling if and only if

$$\begin{eqnarray}\displaystyle \sup _{n\in \mathbb{N}}\frac{|\{i\in I:|x_{i}|\leqslant n\}|}{n^{d}}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$

$d$

$m=1$

Let $\unicode[STIX]{x1D703}$ be an irrational number and $\unicode[STIX]{x1D711}:\mathbb{N}\rightarrow \mathbb{R}^{+}$ be a monotone decreasing function tending to zero. Let

$$\begin{eqnarray}E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})=\{y\in \mathbb{R}:\Vert n\unicode[STIX]{x1D703}-y\Vert <\unicode[STIX]{x1D711}(n),\text{for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$

$\unicode[STIX]{x1D711}(n)$

$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$

$\unicode[STIX]{x1D711}$

$\unicode[STIX]{x1D703}$