Related papers: Summatory Mobius Function, and Summatory Liouville…

We obtain asymptotic formulas with remainder terms for the hyperbolic summations $\sum_{mn\le x} f((m,n))$ and $\sum_{mn\le x} f([m,n])$, where $f$ belongs to certain classes of arithmetic functions, $(m,n)$ and $[m,n]$ denoting the gcd and…

In this paper we consider $L^p$ Liouville type theorems for harmonic functions on gradient Ricci solitons. In particular, assume that $(M,g)$ is a gradient shrinking or steady K\"ahler-Ricci soliton, then we prove that any pluriharmonic…

We prove simple theorems concerning the maximal order of a large class of multiplicative functions. As an application, we determine the maximal orders of certain functions of the type $\sigma_A(n)= \sum_{d\in A(n)} d$, where A(n) is a…

In this paper, we apply the combinatorial results on counting permutations with fixed pinnacle and vale sets to evaluate the special values of the spectral zeta functions of Sturm-Liouville differential operators. As applications, we get a…

One of the most challenging problems in the domain of 2-D image or 3-D shape is to handle the non-rigid deformation. From the perspective of transformation groups, the conformal transformation is a key part of the diffeomorphism. According…

We present an infinite set of higher equations of motion in N=2 supersymmetric Liouville field theory. They are in one to one correspondence with the degenerate representations and are enumerated in addition to the U(1) charge \omega by the…

We present an overview of the meaningful aggregation functions mapping ordinal scales into an ordinal scale. Three main classes are discussed, namely order invariant functions, comparison meaningful functions on a single ordinal scale, and…

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…

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\rm SL}_2(\mathbb Z)$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper…

In this paper, usual Sturm-Liouville problems are extended for symmetric functions so that the corresponding solutions preserve the orthogonality property. Two basic examples, which are special cases of a generalized Sturm-Liouville…

We consider $\omega(n)$ and $\Omega(n)$, which respectively count the number of distinct and total prime factors of $n$. We survey a number of similarities and differences between these two functions, and study the summatory functions…

We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two…

We show that the ordinates of the nontrivial zeros of certain $L-$functions attached to half-integral weight cusp forms are uniformly distributed modulo one.

In this note we describe weight functions that exhibit a transitional behavior between weak and strong correlation with the Liouville function. We also describe a binary problem which may be considered as an interpolation between Chowla's…

In 1953 LeVeque proved the existence of $U_m$-numbers by showing that for some specially defined Liouville number $\lambda$, the $m$th root $\lambda^{1/m}$ is in $U_m$. In this article we study the following question: let $u$ be an…

Approximative properties of linear summation methods of Fourier series are considered in the Orlicz type spaces ${\mathcal S}_{M}$. In particular, in terms of approximations by such methods, constructive characteristics are obtained for…

In the explicit formula for the signed mock theta functions $\Phi^{(-)[m,s]}$ obtained from the coroot lattice of $D(2,1;a)$, functions with indefinite quadratic forms naturally take place. We compute their modular transformation properties…

Let $\lambda$ and $\mu$ denote the Liouville and M\"obius functions respectively. Hildebrand showed that all eight possible sign patterns for $(\lambda(n), \lambda(n+1), \lambda(n+2))$ occur infinitely often. By using the recent result of…

The main purpose of this article is to study higher order moments of Kummer sums weighted by $L$-functions using estimates for character sums and analytic methods. The results of this article complement a conjecture of Zhang Wenpeng (2002).…

We investigate the meromorphic quasi-modular forms and their $L$-functions. We study the space of meromorphic quasi-modular forms. Then we define their $L$-functions by using the technique of regularized integral. Moreover, we give an…