Related papers: An explicit upper bound for the Helfgott delta in …

It is known that $|\zeta(1+ it)|\ll (\log t)^{2/3}$. This paper provides a new explicit estimate, viz.\ $|\zeta(1+ it)|\leq 3/4 \log t$, for $t\geq 3$. This gives the best upper bound on $|\zeta(1+ it)|$ for $t\leq 10^{2\cdot 10^{5}}$.

This article proves the bound $|\zeta(\frac{1}{2} + it)|\leq 0.732 t^{\frac{1}{6}} \log t$ for $t \geq 2$, which improves on a result by Cheng and Graham. We also show that $|\zeta(\frac{1}{2}+it)|\leq 0.732 |3.3081+it|^{\frac{1}{6}} \log…

Let $G$ be a finite $p$-group and $\delta(G)$ denote the number of all non-cyclic subgroups of $G$. In this paper, an upper bound for $\delta(G)$ is obtained. Furthermore, we prove that $\delta(G)\leq \delta(M_p(1, 1, 1) \times…

In this paper we provide an explicit bound for $|\zeta(1+it)|$ in the form of $|\zeta(1+it)|\leq \min\left(\log t, \frac{1}{2}\log t+1.93, \frac{1}{5}\log t+44.02 \right)$. This improves on the current best-known explicit bound of…

In this work, we show that for all $t\geq e$, \[|\zeta(1+it)|\leq 0.6443 \log t. \] The equality is achieved when $t=17.7477$. We also use the Riemann-Siegel formula and numerical computations to show that \[|\zeta(1+it)|\leq\frac{1}{2}\log…

Let $\overline{p}(n)$ denote the overpartition function. In this paper, we obtain an inequality for the sequence $\Delta^{2}\log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^{\alpha}}$ which states that \begin{equation*} \log…

In section 1 we give an improved lower bound on Hermite's constant $\delta_{2g}$ for symplectic lattices in even dimensions ($g=2n$) by applying a mean-value argument from the geometry of numbers to a subset of symmetric lattices. Here we…

We give an improved lower bound for the average of the Erd\H{o}s-Hooley function $\Delta(n)$, namely $\sum_{n\le x} \Delta(n) \gg_\varepsilon x(\log\log x)^{1+\eta-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where…

We prove that if $\alpha$ is a non-zero algebraic number of degree $d \geq 2$ which is not a root of unity, then $dh(\alpha)>(1/4) (\log(\log (d))/\log(d))^3.

Let $f$ be a holomorphic modular form of prime level $p$ and trivial nebentypus. We show that there exists a computable $\delta>0$, such that $$ L\left(\tfrac{1}{2},\mathrm{Sym}^2 f\right)\ll p^{\tfrac{1}{2}-\delta}, $$ with the implied…

The main result of this paper is that for any $1/2 \leq s < 2 - \sqrt{2} \approx 0.5858$, there is a number $\sigma = \sigma(s) < s$ with the following property. Let $\delta > 0$ be small, assume that $A \subset [0,1]$ is a…

We prove that the operator $L_0=-(1+|x|)^\beta(-\Delta)^{\alpha/2}$ with $\alpha\in(0,2)$, $d>\alpha$ and $\beta\ge0$ generates a compact semigroup or resolvent on $L^2(\R^d;(1+|x|)^{-\beta}\,dx)$, if and only if $\beta>\alpha$. When…

Let $f$ be the Hecke eigenform for the modular group $SL_2(\mathbb{Z})$, and $L(s, \text{sym}^2 f)$ be the symmetric square $L$-function associated with $f$. For $\frac{1}{2}<\sigma<1$, define $m(\sigma)$ as the supremum of all numbers $m$…

Let $p$ be a prime. Let $f$ be a holomorphic modular form of level $p$ with trivial nebentypus. We prove the bound $L\left(\text{sym}^2f, \frac{1}{2} + it\right) \ll_{f,\epsilon} p^{1/2+\epsilon}t^{3/4-1/12 + \epsilon}$. This bound is…

Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset $P$ of width $2$. This constant is defined as $\delta(P) = \max_{(x, y)\in P^2}\min\{\mathbb{P}(x\prec y),…

Let $A\subset [1, 2]$ be a $(\delta, \sigma)$-set with measure $|A|=\delta^{1-\sigma}$ in the sense of Katz and Tao. For $\sigma\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox \delta^{-c}|A|, $$ for…

Let $d(n)$ be the number of divisors of $n$, let $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. Several…

Let $\theta(x) = \sum_{p\leq x} \log p$. We show that $\theta(x)<x$ for $2<x< 1.39\cdot 10^{17}$. We also show that there is an $x<\exp(727.951332668)$ for which $\theta(x) >x.$

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…

We deal with the distribution of the fractional parts of $p^{\lambda}$, $p$ running over the prime numbers and $\lambda$ being a fixed real number lying in the interval $(0,1)$. Roughly speaking, we study the following question: Given a…