Related papers: A note on Apery's constant is transcendental

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of…

We show that there is an absolute constant $c>0$ such that $|A+\lambda\cdot A|\geq e^{c\sqrt{\log |A|}}|A|$ for any finite subset $A$ of $\mathbb{R}$ and any transcendental number $\lambda\in\mathbb{R}$. By a construction of Konyagin and…

We study a lower bound for the constant of the Szemer\'edi-Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies $I({\mathcal P},{\mathcal L})\ge (c+o(1)) |{\mathcal P}|^{2/3}|{\mathcal…

The average value of log s(n)/n taken over the first N even integers is shown to converge to a constant lambda when N tends to infinity; moreover, the value of this constant is approximated and proven to be less than 0. Here s(n) sums the…

We study real numbers defined by multidimensional automatic arrays weighted by multiplicatively independent bases. Let $a_1, \dots, a_r\geq 2$ be integers such that $\log a_1, \dots, \log a_r$ are $\mathbb Q$-linearly independent. Given…

Let $p>3$ be a prime, and let $a$ be a rational $p$-adic integer, using WZ method we establish the congruences modulo $p^3$ for $$\sum_{k=0}^{p-1} \binom ak\binom{-1-a}k\binom{2k}k\frac {w(k)}{4^k},$$ where $$w(k)=1,\frac 1{k+1},\frac…

Let $a>1$. Denote by $l_a(p)$ the multiplicative order of $a$ modulo $p$. We look for an estimate of sum of $\frac{l_a(p)}{p-1}$ over primes $p\leq x$ on average. When we average over $a\leq N$, we observe a statistic of $C\mathrm{Li}(x)$.…

Given a positive integer $n$ we let $A_k(n)$ be the number of positive integers $a$ such that $\frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}$ for some $m_1,m_2,\ldots,m_k\in {\mathbb N}$. We show that $x(\log x)^3\ll…

Let $0<s<1$ and $p>1$ be such that $ps<N$. Assume that $\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\ge…

Let $P^{\left(\frac 12\right)}(n)$ denote the middle prime factor of $n$ (taking into account multiplicity). More generally, one can consider, for any $\alpha \in (0,1)$, the $\alpha$-positioned prime factor of $n$, $P^{(\alpha)}(n)$. It…

Using a clear and straightforward approach, we prove new ternary (base 3) digit extraction BBP-type formulas for polylogarithm constants. Some known results are also rediscovered in a more direct and elegant manner. A previously unproved…

It is proved that, for all odd integer $s \geqslant s_0(\varepsilon)$, there are at least $\big( c_0 - \varepsilon \big) \frac{s^{1/2}}{(\log s)^{1/2}} $ many irrational numbers among the following odd zeta values:…

We prove transcendence of the Hecke-Mahler series $\sum_{n=0}^\infty f(\lfloor n\theta+\alpha \rfloor) \beta^{-n}$, where $f(x) \in \mathbb{Z}[x]$ is a non-constant polynomial $\alpha$ is a real number, $\theta$ is an irrational real…

We consider the family B-tilde of bounded nonvanishing analytic functions f(z) = a_0 + a_1 z + a_2 z^2 + ... in the unit disk. The coefficient problem had been extensively investigated, and it is known that |a_n| <= 2/e for n=1,2,3, and 4.…

The classical inequality of Bohr asserts that if a power series converges in the unit disk and its sum has modulus less than or equal to $1$, then the sum of absolute values of its terms is less than or equal to $1$ for the subdisk…

We will show that the number of integers $\leq x$ that can be written as the square of an integer plus the square of a prime equals $\frac{\pi}{2} \cdot \frac {x}{\log x}$ minus a secondary term of size $x/(\log x)^{ 1+\delta+o(1)}$, where…

In the 1980s, Koecher and, independently, Leshchiner found an elegant formula for the generating function of odd zeta values. In this short note, we derive a $q$-analogue of this formula, which provides a $q$-version of the accelerated…

In this paper we refine Ball-Rivoal's theorem by proving that for any odd integer $a$ sufficiently large in terms of $\epsilon>0$, there exist $[ \frac{(1-\epsilon)\log a}{1+\log 2}]$ odd integers $s$ between 3 and $a$, with distance at…

Let $n$ be a positive integer and let $0 < \alpha < n.$ In this paper, we continue our study of the integral equation $$ u(x) = \int_{R^{n}} \frac{u(y)^{(n+\alpha)(n-\alpha)}{|x - y|^{n-\alpha}}dy.$$ We mainly consider singular solutions in…

For the solution $\{u_n\}_{n=0}^\infty$ to the polynomial recursion $(n+1)^5u_{n+1}-3(2n+1)(3n^2+3n+1)(15n^2+15n+4)u_n -3n^3(3n-1)(3n+1)u_{n-1}=0$, where $n=1,2,...$, with the initial data $u_0=1$, $u_1=12$, we prove that all $u_n$ are…