Related papers: Some properties of the pseudo-Smarandache function

The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z) = z + \sum\limits_2^{\infty} a_n z^n$ on the unit disk satisfy $|a_n^2 - a_{2n-1}| \le (n-1)^2$ for all $n…

In this paper, we can show that \begin{align*} S_{\Lambda}(x)=\sum_{1\leq n\leq x}\Lambda \left(\left[\frac{x}{n}\right]\right)= \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n(n+1)}x +O\left(x^{7/15+1/195+\varepsilon}\right), \end{align*} where…

The singularities of the $\Gamma$ function, a meromorphic function on the complex plane, are known to occur at the nonpositive integers. We show, using Euler and Gauss identities, that for all positive integers $n$ and $k$, $$…

Let $m$ be a positive integer, and define $$\zeta_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\omega(n)}}{n^s}\ \ \ \ \text{and} \ \ \ \ \zeta^*_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\Omega(n)}}{n^s},$$ for $\Re(s)>1$, where…

In the integer case, the Smarandache function of a positive integer $n$ is defined to be the smallest positive integer $k$ such that $n$ divides the factorial $k!$. In this paper, we first define a natural order for polynomials in…

Consider exponential Carmichael function $\lambda^{(e)}$ such that $\lambda^{(e)}$ is multiplicative and $\lambda^{(e)}(p^a) = \lambda(a)$, where $\lambda$ is usual Carmichael function. We discuss the value of $\sum \lambda^{(e)}(n)$, where…

In this paper, we use the Riemann zeta function $\zeta(x)$ and the Bessel zeta function $\zeta_{\mu}(x)$ to study the log-behavior of combinatorial sequences. We prove that $\zeta(x)$ is log-convex for $x>1$. As a consequence, we deduce…

It has been recently discovered that a convex function can be determined by its slopes and its infimum value, provided this latter is finite. The result was extended to nonconvex functions by replacing the infimum value by the set of all…

In this paper, we show the existence of a transcendental function $f\in\mathbb{Z}\{z\}$ with coefficients that are almost all bounded such that $f$ and all its derivatives assume algebraic values at algebraic points. Furthermore, we…

We consider the family of all analytic and univalent functions in the unit disk of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. Our objective in this paper is to estimate the difference of the moduli of successive coefficients, that is $\big |…

We consider integer sequences that satisfy a recursion of the form $x_{n+1} = P(x_n)$ for some polynomial $P$ of degree $d > 1$. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form $x_n \sim A…

Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many…

In this paper, we prove the existence of unbounded sequence of eigenvalues for the fractional $p-$Laplacian with weight in $\mathbb{R}^N.$ We also show a nonexistence result when the weighthas positive integral. In addition, we show some…

In this article, we have studied transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan's identity for the Riemann zeta function. The above transformation leads to a new…

We give several new characterizations of completely monotone functions and Bernstein functions via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the…

We extend a result of Han\v{c}l, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{\alpha_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value…

Using properties of the Riemann zeta-function we propose two new large classes of evaluated series. Incidentally the first class represents integrals as generalized average on very nonuniform sequences. The second class contains inter alia…

We consider the series $\sum_{n=1}^{\infty} z^{n} (a_{n} + x)^{-s}$ where $a_{n}$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under appropriate conditions, we prove that it can be continued to a meromorphic…

Building on the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, and by defining the Zeta and related functions including the Hurwitz Zeta function…

For any irrational $\alpha > 0$ and any initial value $z_{-1} \in \mathbb{C}$, we define a sequence of complex numbers $(z_n)_{n=0}^{\infty}$ as follows: $z_n$ is $z_{n-1} + e^{2 \pi i \alpha n}$ or $z_{n-1} - e^{2 \pi i \alpha n}$,…