Related papers: Chebyshev's bias for irrational factor function

Let $\{p_n\}_{n\ge 1}$ be the sequence of primes and $\vartheta(x) = \sum_{p \leq x} \log p$, where $p$ runs over the primes not exceeding $x$, be the Chebyshev $\vartheta$-function. In this note we derive lower and upper bounds for…

We use discrete Morse theory to determine the M\"obius function of generalized factor order. Ordinary factor order on the Kleene closure A* of a set A is the partial order defined by letting u\leq w if w contains u as a subsequence of…

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta…

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…

This article examines the value distribution of $S_{N}(f, \alpha) := \sum_{n=1}^N f(n\alpha)$ for almost every $\alpha$ where $N \in \mathbb{N}$ is ranging over a long interval and $f$ is a $1$-periodic function with discontinuities or…

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a…

The best polynomial approximation and Chebyshev approximation are both important in numerical analysis. In tradition, the best approximation is regarded as more better than the Chebyshev approximation, because it is usually considered in…

Iterated function systems (IFS) can be a surprisingly useful tool for studying structure in data. Here we present results stemming from a 2013 computational study by the author using IFS. The results include fractal patterns that reveal…

In order to study the analytic properties of the Goldbach generating function we consider a smooth version, similar to the Chebyshev function for the Prime Number Theorem. In this paper, we obtain explicit numerical estimates for the…

For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length…

The study of iterations of functions over a finite field and the corresponding functional graphs is a growing area of research with connections to cryptography. The behaviour of such iterations is frequently approximated by what is know as…

In this paper we consider error sums of the form \[\sum_{m=0}^{\infty} \varepsilon_m\Big( \,b_m\alpha - \frac{a_m}{c_m}\,\Big) \,,\] where $\alpha$ is a real number, $a_m$, $b_m$, $c_m$ are integers, and $\varepsilon_m=1$ or $\varepsilon_m…

The challenge to measure exposures regularly forces financial institutions into a choice between an overwhelming computational burden or oversimplification of risk. To resolve this unsettling dilemma, we systematically investigate replacing…

In the paper Sato (2006) there are introduced two families of improper random integrals and the corresponding two convolution semigroups of infinitely divisible laws on $\Rset^d$. Theorem 3.1 gives a relation (a factorization property)…

A fractional binomial distribution, introduced by Hino and Namba (2024) via the generalized binomial theorem, is a fractional variant of the classical binomial distribution. Building upon previous work that established limit theorems, such…

There have been a plethora of investigations carried out in studying inequalities for the Fourier coefficients of weakly holomorphic modular forms, for example, on the partition function. Recently, Bringmann, Kane, Rolen, and Tripp studied…

We derive the joint asymptotic distribution of empirical quantiles and expected shortfalls under general conditions on the distribution of the underlying observations. In particular, we do not assume that the distribution function is…

The well known Erdos-Turan law states that the logarithm of an order of a random permutation is asymptotically normally distributed. The aim of this work is to estimate convergence rate in this theorem and also to prove analogous result for…

It is well known that $li(x)>\pi(x)$ (i) up to the (very large) Skewes' number $x_1 \sim 1.40 \times 10^{316}$ \cite{Bays00}. But, according to a Littlewood's theorem, there exist infinitely many $x$ that violate the inequality, due to the…

In this paper we have introduced a generalized version of alpha beta skew normal distribution in the same line of Sharafi et al. (2017) and investigated some of its basic properties. The extensions of the proposed distribution have also…