相关论文: Partial Liouville sums over divisors

In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by Bordell\'{e}s \cite{Bo} and Liu-Wu-Yang \cite{LWY}.

This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…

Summation arithmetic functions of Mertens and Liouville are investigated in the paper. It is proved that the limiting distribution of these functions is the normal. It is also shown that the estimating of standard deviation of these…

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…

In this note, we solve some sign change problems on the functions involving sums of divisors posed by Pongsriiam recently.

This paper provides a Liouville principle for integration in terms of dilogarithm and partial result for polylogarithm.

The equidistribution of the double sign patterns of the Liouville function $\lambda$ is proved unconditionally. As application, it is shown that the computable real number $$\sum_{n \geq 1} \frac{1+\lambda(n)}{2^{n}}$$ is a simply normal…

We show that the Liouville function changes sign infinitely often for $x^2+d$.

In the paper, we first prove a sufficient condition for the Riemann hypothesis which involves the order of magnitude of the partial sum of the Liouville function. Then we show a formula which is curiously related to the proved sufficient…

We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture…

We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and $k,l\in\mathbb{N}$ are…

Under the assumption of the Riemann Hypothesis, the Linear Independence Hypothesis, and a bound on negative discrete moments of the Riemann zeta function, we prove the existence of a limiting logarithmic distribution of the normalisation of…

In this paper, we study the sum of the divisor function over sets with digit restrictions.

Let $F$ be a number field, $k$ a positive integer. In this paper, we define the Mobius and Liouville functions of order $k$ in $F$. We give a formula about the partial sums of them by using elementary number theory and complex analysis.…

We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.

In the paper, a Liouville theorem for mild bounded ancient solutions to the 2D Navier-Stokes equations in half space has been proven.

We prove the analogue of an identity of Huard, Ou, Spearman and Williams and apply it to evaluate a variety of sums involving divisor functions in two variables. It turns out that these sums count representations of positive integers…

We prove a Liouville theorem for the plurisubharmonic functions on complete Kaelher manifolds. As the applications, we prove a splitting theorem for complete Kaehler manifolds with nonnegative biscetional curvature in terms of the linear…

We improve using elementary means an explicit bound on the divisor function due to Friedlander and Iwaniec. Consequently we modestly improve a result regarding a sieving inequality for Gaussian sequences.

We introduce more general concepts of Riemann-Liouville fractional integral and derivative on time scales, of a function with respect to another function. Sufficient conditions for existence and uniqueness of solution to an initial value…