Related papers: Fractional part correlations and the M\"obius func…

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 sums $\sum_{n\le X, (n,q)=1}\frac{\mu(n)}{n^s}\log^k\left(\frac{X}{n}\right)$, where $k\in\{0,1\}$, $s\in\mathbb{C}$, $\Re s>0$. Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this,…

For integer $n\geqslant 1$ and real number $z\geqslant 1$, define $M(n,z):=\sum_{d|n,\,d\leqslant z}\mu(d)$ where $\mu$ denotes the M\"obius function. Put ${\cal L}(y):=\exp\left\{(\log y)^{3/5}/(\log_2y)^{1/5}\right\}$ $(y\geqslant 3)$. We…

Let $\mu(n)$ denote the M\"obius function, define $M(x)= \sum_{n\leq x}^{}\mu (n)$. The main result of this paper is to prove that \begin{equation*} \displaystyle\lim_{x \to +\infty}\frac{M(x)}{x}=0 \end{equation*} which is equivalent to…

We improve on all the results of [13] by incorporating the finite range computations performed since then by several authors. Thus we have \begin{align*} \Bigg|\sum_{n\le X}\mu(n)\Bigg| &\le \frac{0.006688\,X}{\log X},&&\text{for } X\ge…

We study in an explicit manner the partial sums of the multiplicative inverse of the Riemann zeta function and its derivative.

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…

Let $\gcd(m,n)$ denote the greatest common divisor of the positive integers $m$ and $n$, and let $\mu$ represent the M\" obius function. For any real number $x>5$, we define the summatory function of the M\" obius function involving the…

Let $x\geq 1$ be a large integer, and let $\mu:\mathbb{N}\longrightarrow\{-1,0,1\}$ be the Mobius function. This article proposes an effective asymptotic result for the autocorrelation function $\sum_{n \leq x} \mu(n) \mu(n+t) =O\left(…

An elementary recursive relation for M$\ddot{\mathrm{o}}$bius function $\mu (n)$ is introduced by two simple ways. With this recursive relation, $\mu (n)$ can be calculated without directly knowing the factorization of the $n$. $\mu (1)…

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…

The Mertens function is defined as $M(x) = \sum_{n \leq x} \mu(n)$, where $\mu(n)$ is the M\"obius function. The Mertens conjecture states $|M(x)/\sqrt{x}| < 1$ for $x > 1$, which was proven false in 1985 by showing $\liminf M(x)/\sqrt{x} <…

Assuming the Riemann Hypothesis we establish an upper bound for the sum of the M{\" o}bius function up to $x$. Our method is based on estimating the frequency with which intervals of a given length can contain an unusual number of ordinates…

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…

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the M\"obius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the M\"obius function.…

In important work on the parity of the partition function, Ono related values of the partition function to coefficients of a certain mock theta function modulo 2. In this paper, we use M\"obius inversion to give analogous results which…

In this paper I introduce a criterion for the Riemann hypothesis, and then using that I prove $\sum_{k=1}^\infty \mu(k)/k^s$ converges for $\Re(s) > \frac{1}{2}$. I use a step function $\nu(x) = 2\{x/2\} - \{x\}$ for the Dirichlet eta…

This paper analyzes the M\"obius ($\mu(i)$) function defined on the partially ordered set of triangular numbers ($\mathcal T(i)$) under the divisibility relation. We make conjectures on the asymptotic behavior of the classical M\"obius and…

Let $\zeta(.)$ denote the Riemann zeta function and let $a(.)$ and $A(.)$ respectively denote a multiplicative function and its corresponding summatory function. We consider the correlation $$ \langle a(n)A(n-1) \rangle (T) =…

Let $<\P > \subset \N$ be a multiplicative subsemigroup of the natural numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $\P$ of primes (finite or infinite). We given an elementary proof that the partial sums $\sum_{n \in < \P >: n…