Related papers: On a problem posed by Mahler

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a…

Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f)…

In this paper, we give a new proof of a result due to Bezivin that a D-finite Mahler function is necessarily rational. This also gives a new proof of the rational-transcendental dichotomy of Mahler functions due to Nishioka. Using our…

We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero…

E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like…

We prove that the Dirichlet $L$-functions associated with Dirichlet characters in $\mathbb{F}_{q}[x]$ are universal. That is, given a modulus of high enough degree, $L$-functions with characters to this modulus can be found that approximate…

{\em The Liouville number}, denoted $l$, is defined by $$l:=0.100101011101101111100...,$$ where the $n$th bit is given by ${1/2}(1+\gl(n))$; here $\gl$ is the Liouville function for the parity of prime divisors of $n$. Presumably the…

We study asymptotics of integrals of certain rational functions that depend on parameters in a field $K$ of characteristic zero. We use formal power series to represent the integral and prove certain identities about its coefficients…

We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.

Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a…

We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the $b$-adic expansion of a Liouville number cannot be generated by a…

Let $K$ be a field of characteristic 0, $f:\mathbb{N}\to K$ be a multiplicative function, and $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ be algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function…

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…

We show that if a Laurent series $f\in\mathbb{C}((t))$ satisfies a particular kind of linear iterative equation, then $f$ is either a rational function or it is differentially transcendental over $\mathbb{C}(t)$. This condition is more…

We consider the dynamical properties of transcendental entire functions and their compositions. We give several conditions under which Fatou set of a transcendental entire function $f$ coincide with that of $f\circ g,$ where $g$ is another…

In the present paper, we investigate special generalized q-Euler numbers and polynomials. Some earlier results of T. Kim in terms of q-Euler polynomials with weight alpha can be deduced. For presentation of our formulas we apply the method…

Let $(X,L)$ be a polarized variety over a number field. We suppose that $L$ is an hermitian line bundle. Let $M$ be a non compact Riemann Surface and $U\subset M$ be a relatively compact open set. Let $\varphi:M\to X({\Bbb C})$ be a…

About fifty years ago Mahler proved that if $\alpha>1$ is rational but not an integer and if $0<l<1$ then the fractional part of $\alpha^n$ is $>l^n$ apart from a finite set of integers $n$ depending on $\alpha$ and $l$. Answering…

We show that if $F(s)$ is a nondegenerate ordinary Dirichlet series with nonnegative coefficients and $F(k)$ is a rational number for all large enough positive integers $k$, then the denominators of those rational numbers are unbounded. In…

In this article, we investigate and establish some properties including analytic properties, contiguous relations, differential properties, differential operators, an expansion formula, and simple integrals, integral operators, some…