相关论文: The summatory function of the M\"obius function in…

We give an asymptotic formula for the divisor sum $\sum_{c<n\leq N}\tau\left((n-b)(n-c)\right)$ for integers $b<c$ of the same parity. Interestingly, the coefficient of the main term does not depend on the discriminant as long as it is a…

We define an S function as the sum of the asymptotic error terms of digamma function of an arithmetic series, $S(a) \equiv \sum_{n=1}^\infty \left[\ln\frac{n}{a} - \frac{a}{2n}-\psi\left(\frac{n}{a}\right)\right]$, and show a few properties…

Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated but without proof that $$\sum_{n\leq x}d^2(n)=xP(\log x)+E(x), $$ where $P(y)$ is a cubic polynomial in $y$ and $$ E(x)=O(x^{{3\over 5}+\epsilon}), $$ where $\epsilon$ is a…

For a fixed integer $k$, we define the multiplicative function \[D_{k,\omega}(n) := \frac{d(n)}{k^{\omega(n)}}, \]where $d(n)$ is the divisor function and $\omega (n)$ is the number of distinct prime divisors of $n$. The main purpose of…

Let $\ba=(a_1,a_2,\ldots,a_k)$, where $a_j \ (j=1,\ldots,k)$ are positive integers such that $a_1 \leq a_2 \leq \cdots \leq a_k$. Let $d(\ba;n)=\sum_{n_1^{a_1}\cdots n_k^{a_k}=n}1$ and $\Delta(\ba;x)$ be the error term of the summatory…

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…

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…

We provide upper bounds for the mean square integral $$ \int_X^{2X}(\Delta_k(x+h) - \Delta_k(x))^2 dx \qquad(h = h(X)\gg1, h = o(x) {\roman{as}} X\to\infty) $$ where $h$ lies in a suitable range. For $k\ge2$ a fixed integer, $\Delta_k(x)$…

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$…

We present several formulae for the large $t$ asymptotics of the Riemann zeta function $\zeta(s)$, $s=\sigma+i t$, $0\leq \sigma \leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical…

We prove an asymptotic for the sum of $\zeta^{(n)} (\rho)X^{\rho}$ where $\zeta^{(n)} (s)$ denotes the $n$th derivative of the Riemann zeta function, $X$ is a positive real and $\rho$ denotes a non-trivial zero of the Riemann zeta function.…

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.…

Given an integer $n \ge 2$, its prime factorization is expressed as $n= \prod_{i=1}^s p_i^{a_i}$. We define the function $f(n)$ as the smallest positive integer such that $f(n)!$ is divisible by $n$. The main objective of this paper is to…

The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $\mu^{2}$ where $\mu$ is the M\"{o}bius function is used to formulate this divisor sum and…

We show that, for the M\"obius function $\mu(n)$, we have $$ \sum_{x < n\leq x+x^{\theta}}\mu(n)=o(x^{\theta}) $$ for any $\theta>0.55$. This improves on a result of Ramachandra from 1976, which is valid for $\theta>7/12$. Ramachandra's…

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 $\mu(n)$ 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…

In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.

We investigate a modified M\"obius $\mu$-function which is related to an infinite product of shifted Riemann zeta-functions. We prove conditional and unconditional upper and lower bounds for its summatory function, and, finally, we discuss…

Let $j \ge1$, $k\ge 0$ be real numbers and $\varphi(n)$ be the Euler function. In this paper, we study the asymptotical behaviour of the summation function $$S_{j,k}(x):=\sum_{n\le x}\frac{\varphi\left ( \left [ \frac{x}{n} \right ]^{j}…