Related papers: Computations of the Mertens Function and Improved …

We provide some upper bounds for the Mertens function ($M(n)$: the cumulative sum of the M$\ddot{\mathrm{o}}$bius function) by an approach of statistical mechanics, in which the M$\ddot{\mathrm{o}}$bius function is taken as a particular…

In this paper, we derive new properties of the Mertens function and discuss a likely upper bound of the absolute value of the Mertens function $\sqrt{\log{x!}}>|M(x)|$ when $x>1$. Using this likely bound we show that we have a sufficient…

We report the finding of the new upper bound on the lowest positive integer $x$ for which the Mertens conjecture \begin{equation*} \left| \sum_{1 \leq n \leq x} \mu(n) \right| < \sqrt{x} \end{equation*} fails to hold: $x < \exp(1.017 \times…

Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function $\vert M(x) \vert$ that $ \left\vert M(x) \right\vert \sim \left[\frac{1}{\pi \sqrt{\varepsilon}(x+\varepsilon)}\right]\sqrt{x}$…

A GPU implementation of an algorithm to compute the Mertens function in O(x2/3+{\ko}) time is discussed. Results for x up to $10^{22}$, and a new extreme value for $M(x)/x^{1/2}$, -0.585768 ($M(x) \approx -1.996 \ast 10^9$ at $x \approx…

Two elementary formulae for Mertens function $M(n)$ are obtained. With these formulae, $M(n)$ can be calculated directly and simply, which can be easily implemented by computer. $M (1) \sim M (2 \times 10^7) $ are calculated one by one.…

Let $M(x)=\sum_{1\le n\le x}\mu(n)$ where $\mu$ is the M\"obius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that $M(x)=O(x^{1/2+\epsilon})$ for all $\epsilon>0$. There has been much interest and…

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…

The Mertens function, $M(x) := \sum_{n \leq x} \mu(n)$, is defined as the summatory function of the classical M\"obius function. The Dirichlet inverse function $g(n) := (\omega+1)^{-1}(n)$ is defined in terms of the shifted strongly…

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of $A(s)$ with $|\Im s| \leq T$ for some large constant $T$. What is the best way to use such finite spectral data…

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)…

Let the summatory function of the M\"{o}bius function be denoted $M(x)$. We deduce in this article conditional results concerning $M(x)$ assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the…

In classical prime number theory there are several asymptotic formulas said to be "equivalent" to the PNT. One is the bound $M(x) = o(x)$ for the sum function of the Moebius function. For Beurling generalized numbers, this estimate is not…

In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{\Omega(n)}, \end{equation*} where $\Omega(n)$ counts the number of prime factors of $n$, with multiplicity. We prove $W(x)=O(x)$, and in particular,…

We obtained the formulas for the quantities of positive, negative and zero values of the Mobius function for any real x in terms of the Mobius function values for square root of x - similar to the identities we found earlier for the Mertens…

We prove that $|\sum_{n \leq x} \mu(n) | \leq x/160\,383$ for all $x\geq 8.4 \times 10^9$ thus improving the record for this type of estimates which was $|\sum_{n \leq x} \mu(n) | \leq x/4345$ for all $x\geq 2\,160\,535$ and was due to…

The Mertens' first theorem gives us the following asymptotic formula \begin{equation*} \sum_{\substack{p\leq x\\ p~prime}}\frac{lnp}{p}=lnx+O(1), \end{equation*} and the Mertens' second theorem indicates that there exists a constant…

We discuss the multiplicity of the non-trivial zeros of the Riemann zeta-function and the summatory function $M(x)$ of the M\"obius function. The purpose of this paper is to consider two open problems under some conjectures. One is that…

In this paper, we establish new explicit bounds for the Mertens function $M(x)$. In particular, we compare $M(x)$ against a short-sum over the non-trivial zeros of the Riemann zeta-function $\zeta(s)$, whose difference we can bound using…

We show that the sum function of the M\"{o}bius function of a Beurling number system must satisfy the asymptotic bound $M(x)=o(x)$ if it satisfies the prime number theorem and its prime distribution function arises from a monotone…