Related papers: A definite recursive relation and some statistical…
Let $\mu(n)$ be the M\"{o}bius function and $e(\alpha)=e^{2\pi i\alpha}$. In this paper, we study upper bounds of the classical sum $$S(x,\alpha):=\sum_{1\leq n\leq x}\mu(n)e(\alpha n).$$ We can improve some classical results of Baker and…
A simple recurrence relation for the even order moments of the Fabius function is proven. Also, a very similar formula for the odd order moments in terms of the even order moments is proved. The matrices corresponding to these formulas (and…
The purpose of this paper is to give some explicit formulas involving M\"obius functions, which may be known under the generalized Riemann Hypothesis, but unconditional in this paper. Concretely, we prove explicit formulas of partial sums…
Given a probability distribution $\mu$ a set $\Lambda (\mu)$ of positive real numbers is introduced, so that $\Lambda (\mu)$ measures the "divisibility" of $\mu$. The basic properties of $\Lambda (\mu)$ are described and examples of…
We study the values of the M\"obius function $\mu$ of intervals in the containment poset of permutations. We construct a sequence of permutations $\pi_n$ of size $2n-2$ for which $\mu(1,\pi_n)$ is given by a polynomial in $n$ of degree 7.…
Bayesian hypothesis testing is re-examined from the perspective of an a priori assessment of the test statistic distribution under the alternative. By assessing the distribution of an observable test statistic, rather than prior parameter…
We discuss certain identities involving $\mu(n)$ and $M(x)=\sum_{n\leq x}\mu(n)$, the functions of M\"{o}bius and Mertens. These identities allow calculation of $M(N^d)$, for $d=2,3,4,\ldots\ $, as a sum of $O_d \left( N^d(\log N)^{2d -…
We present a two term formula for the M\"obius function of intervals in the poset of all permutations, ordered by pattern containment. The first term in this formula is the number of so called normal occurrences of one permutation in…
Let $2 \leq y \leq x$ such that $\beta := \frac{\log x}{\log y} \rightarrow \infty$. Let $\omega_y(n)$ denote the number of distinct prime factors $p$ of $n$ such that $p \leq y$, and let $\mu_y(n) := \mu^2(n)(-1)^{\omega_y(n)}$, where…
We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis. These sums contain the M\"obius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably…
In this paper, we introduce a new method for computing generating functions with respect to the number of descents and left-to-right minima over the set of permutations which have no consecutive occurrences of a pattern that starts with 1.
The paper presents some results for reducing the computation of the M\"obius functon of a M\"obius category that arises from a combinatorial inverse semigroup to that of locally finite partially ordered sets. We illustrate the computation…
We investigate the utility to computational Bayesian analyses of a particular family of recursive marginal likelihood estimators characterized by the (equivalent) algorithms known as "biased sampling" or "reverse logistic regression" in the…
Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that…
Though the ability of human beings to deal with probabilities has been put into question, the assessment of rarity is a crucial competence underlying much of human decision-making and is pervasive in spontaneous narrative behaviour. This…
In a previous work, we developed an algorithm for the computation of incomplete Bessel functions, which pose as a numerical challenge, based on the $G_{n}^{(1)}$ transformation and Slevinsky-Safouhi formula for differentiation. In the…
We introduce a recursive algorithm of conveniently general form for estimating the coefficient of a moving average model of order one and obtain convergence results for both correct and misspecified MA(1) models. The algorithm encompasses…
By using exclusively real analysis, we give explicit estimates of some classical summatory functions involving the M\"obius function.
An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some…
The widely claimed replicability crisis in science may lead to revised standards of significance. The customary frequentist confidence intervals, calibrated through hypothetical repetitions of the experiment that is supposed to have…