arXiv subjects
Publications and source records attributed to Kalle Leppälä.
We study the $k$:th raw moment of a variable $R$ following the binomial distribution $\text{B}(n, p)$, where $n/k \rightarrow β> 0$. It is known that $\mathbb{E}(R^k)$ is bounded both from below and from above by functions of the form $k^k Ψ^k$. We solve the asymptotically optimal value of $Ψ$ as a function of $p$ and $β$.
We give explicit and asymptotic lower bounds for the quantity $|e^{s/t}-M/N|$ by studying a generalized continued fraction expansion of $e^{s/t}$. In cases $|s|\geq 3$ we improve existing results by extracting a large common factor from the numerators and the denominators of the convergents of that generalized continued fraction.
We study how the asymptotic irrationality exponent of a given generalized continued fraction \[ \K_{n=1}^\infty \frac{a_n}{b_n}\,,\quad a_n, b_n\in \mathbb{Z}^+, \] behaves as a function of growth properties of partial coefficient sequences $(a_n)$ and $(b_n)$.
Let $\mathbb{I}$ denote an imaginary quadratic field or the field $\mathbb{Q}$ of rational numbers and $\mathbb{Z}_{\mathbb{I}}$ its ring of intergers. We shall prove an explicit Baker type lower bound for $\mathbb{Z}_{\mathbb{I}}$-linear form of the numbers \begin{equation}\label{1} 1,\ e^{α_1},...,\ e^{α_m},\quad m\ge 2, \end{equation} where $α_0=0$, $α_1,...,α_m$, are $m+1$ different numbers from the field $\mathbb{I}$. Our work gives gives some improvements to the existing explicit versions of of Baker's work about exponential values at rational points. In particilar, dependences on $m$ are improved.