Related papers: Ap\'ery Limits: Experiments and Proofs

A well known theorem of Lagrange states that the simple continued fraction of a real number $\alpha$ is periodic if and only if $\alpha$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two…

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Ap\'ery sums). The study of non-alternating sums leads to an investigation of different types…

In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both…

The recent technique for estimating lower bounds of the prime counting function $\pi(x)=#\{p \leq x: p\text{ prime}\}$ by means of the irrationality measures $\mu(\zeta(s)) \geq 2$ of special values of the zeta function claims that $\pi(x)…

These notes explore three amazing formulas proved by Abel in his 1826 Paris memoir on what we now call Abelian integrals. We discuss the first two formulas from the point of view of symbolic computation and explain their connection to…

For $c \in \mathbb{Q}$, consider the quadratic polynomial map $\varphi_c(x)=x^2-c$. Flynn, Poonen and Schaefer conjectured in 1997 that no rational cycle of $\varphi_c$ under iteration has length more than $3$. Here we discuss this…

In this paper we deduce a lower bound for the rank of a family of $p$ vectors in $\R^k$ (considered as a vector space over the rationals) from the existence of a sequence of linear forms on $\R^p$, with integer coefficients, which are small…

This paper is devoted to study the existence of solutions and the monotone method of second-order periodic boundary value problems when the lower and upper solutions $\alpha$ and $\beta$ violate the boundary conditions $…

We study a certain class of arithmetic functions that appeared in Klurman's classification of $\pm 1$ multiplicative functions with bounded partial sums, c.f., Comp. Math. 153 (8), 2017, pp. 1622-1657. These functions are periodic and…

We show that for each positive integer $a$ there exist only finitely many prime numbers $p$ such that $a$ appears an odd number of times in the period of continued fraction of $\sqrt{p}$ or $\sqrt{2p}$. We also prove that if $p$ is a prime…

For an irrational $\alpha\in \mathbb{R}$, we consider additive problems with the set of primes satisfying $\lVert\alpha p\rVert\leq \frac{1}{p^\tau}$ for some fixed $\tau>0$. In particular, we show that there exist infinitely many…

This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…

Many authors have investigated the congruence relations amongst the coefficients of power series expansions of modular forms $f$ in modular functions $t$. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and…

We consider measures supported on sets of irrational numbers possessing many consecutive partial quotients satisfying a condition based on the previous partial quotients. We show that under mild assumptions, such sets will always support…

In this note, I present a simple PSLQ code for finding null linear combinations, with the best rational coefficients, of mathematical constants, within some prescribed precision. As an example, I explore approximate expressions for the…

We analyse a certain family of cellular integrals, which are period integrals on the moduli space $\mathcal{M}_{0,8}$ of curves of genus zero with eight marked points, and give rise to simultaneous rational approximations to $\zeta(3)$ and…

In this short note we establish some properties of all those motivic measures which can be exponentiated. As a first application, we show that the rationality of Kapranov's zeta function is stable under products. As a second application, we…

In [P] R. Pellikaan introduced a two variable zeta-function for a curve over a finite field and proved that it is a rational function. Here we show that its denominator is absolutely irreducible. This is motivated by work of J. Lagarias and…

We use a method of translation to recover Borweins' quadratic and quartic iterations. Then, by using the WZ-method, we obtain some initial values which lead to the limit $1/\pi$. We will not use the modular theory nor either the Gauss'…

An alternative form of Fermats equation[1] is proposed. It represents a portion of the identity that includes three terms of Fermats original equation. This alternative form permits an elementary and compact proof of the first case of…