Related papers: The image of Carmichael's $\lambda$-function

Let $\lambda$ denote the Liouville function. We show that the logarithmic mean of $\lambda(\lfloor \alpha_1n\rfloor)\lambda(\lfloor \alpha_2n\rfloor)$ is $0$ whenever $\alpha_1,\alpha_2$ are positive reals with $\alpha_1/\alpha_2$…

We study the Lambert series $\mathscr{L}_q(s,x) = \sum_{k=1}^\infty k^s q^{k x}/(1-q^k)$, for all $s \in \mathbb{C}$. We obtain the complete asymptotic expansion of $\mathscr{L}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields…

We present an integral expression of the Catalan numbers, based on F\'eaux' integral representation of $\log\left[\Gamma(x)\right]$, $\Gamma$ being the usual Gamma function. The obtained formula may be the starting point of the derivation…

Based on a Problem and its solution published on the pages of SIAM Review, we give an interesting integral representation for the Lambert $W$ function in this short note. In particular, our result yields a new integral representation for…

We provide an elementary proof of the left side inequality and improve the right inequality in \bigg[\frac{n!}{x-(x^{-1/n}+\alpha)^{-n}}\bigg]^{\frac{1}{n+1}}&<((-1)^{n-1}\psi^{(n)})^{-1}(x)…

We establish discrete and continuous log-concavity results for a biparametric extension of the $q$-numbers and of the $q$-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our…

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville…

The Catalan transform of a sequence (a_{n})_{n>=0} is the sequence (b_{n})_{n>=0} with b_{n} = Sum[k/(2n-k) (2n-k)-choose-(n-k) a_{k},k=0..n]. Here we show that the Catalan transform of the Catalan numbers has a simple interpretation: it…

In an interesting article entitled "A curious formula related to the Euler Gamma function", Bakir Farhi posed the open question of whether it was possible to obtain an expression of $$…

Let $\mathcal{U(\alpha, \lambda)}$, $0<\alpha <1$, $0 < \lambda <1$ be the class of functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda$$ in the unit disc ${\mathbb…

Let $\lambda$ denote the Liouville function. The Chowla conjecture asserts that $$ \sum_{n \leq X} \lambda(a_1 n + b_1) \lambda(a_2 n+b_2) \dots \lambda(a_k n + b_k) = o_{X \to \infty}(X) $$ for any fixed natural numbers $a_1,a_2,\dots,a_k$…

Let $\Lambda$ be the von Mangoldt function and $r_{\textit{HL}}(n) = \sum_{m_1 + m_2^2 = n} \Lambda(m_1),$ be the counting function for the Hardy-Littlewood numbers. Let $N$ be a sufficiently large integer. We prove that…

Let $C_1(H)$ denote the space of all trace class operators on an arbitrary complex Hilbert space $H$. We prove that $C_1(H)$ satisfies the $\lambda$-property, and we determine the form of the $\lambda$-function of Aron and Lohman on the…

A $\lambda$-calculus is introduced in which all programs can be evaluated in probabilistic polynomial time and in which there is sufficient structure to represent sequential cryptographic constructions and adversaries for them, even when…

We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$ and we define the counting function $$ F(m,r):=\#\left\{n\in[2^m, 2^{m+1}):\ell(n)\le m+r\right\}, $$ where $m$ is a positive integer and $r\ge 0$ is a real…

The study of the Mittag-Leffler function and its various generalizations has become a very popular topic in mathematics and its applications. In the present paper we prove the following estimate for the $q$-Mittag-Leffler function:…

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…

Let $f$ be a Rademacher random multiplicative function. Let $$M_f(u):=\sum_{n \leq u} f(n)$$ be the partial sum of $f$. Let $V_f(x)$ denote the number of sign changes of $M_f(u)$ up to $x$. We show that for any constant $c > 2$, $$V_f(x) =…

We present an algorithm to compute values L(s) and derivatives of L-functions of motivic origin numerically to required accuracy. Specifically, the method applies to any L-series whose Gamma-factor is a product of any number of…

We consider a sum of the derivatives of Dirichlet $L$-functions over the zeros of Dirichlet $L$-functions. We give an asymptotic formula for the sum.