arXiv subjects

Titus Hilberdink

Publications and source records attributed to Titus Hilberdink.

8 recordsLinked to original sources

arXiv2026

What lies between polynomial and exponential growth?

In this paper we give an alternative exposition of a recent paper regarding the classification of growth rates of real functions. We take a different point of view, focussing on understanding possible growth rates between polynomial and exponential. In order to be able to explicitly name a range of such functions, we first need to extend our basic functions. We do this via a 'tower' of Abel functions. With these one can classify functions in a natural way with polynomials and exponentials in consecutive classes. We show there are large gaps between these classes which indicate that it is mostly unknown what lies between polynomial and exponential growth, especially if the "Continuum Hypothesis for classes" is true.

math.CA↗

arXiv2024

Classifying Functions via growth rates of repeated iterations

In this paper we develop a classification of real functions based on growth rates of repeated iteration. We show how functions are naturally distinguishable when considering inverses of repeated iterations. For example, $n+2\to 2n\to 2^n\to 2^{\cdot^{\cdot^2}}$ ($n$-times) etc. and their inverse functions $x-2, x/2, \log x/\log 2,$ etc. Based on this idea and some regularity conditions we define classes of functions, with $x+2$, $2x$, $2^x$ in the first three classes. We prove various properties of these classes which reveal their nature, including a `uniqueness' property. We exhibit examples of functions lying between consecutive classes and indicate how this implies these gaps are very `large'. Indeed, we suspect the existence of a continuum of such classes.

math.CA↗

arXiv2024

A Mean Value Theorem for general Dirichlet Series

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=ρx + O(x^β)$ for some $ρ>0$ and $β<1$. We prove that $\frac1T\int_0^T |f(σ+it)|^2\, dt \to \sum_{j=1}^\infty a_j^2n_j^{-2σ}$ for $σ>\frac{1+β}{2}$ and obtain an upper bound for this moment for $β<σ\le \frac{1+β}{2}$. We provide a number of examples indicating the sharpness of our results.

math.NT↗

arXiv2024

Spectral asymptotics for a family of arithmetical matrices and connection to Beurling primes

We consider the family of arithmetical matrices given explicitly by $$E=\left\{\frac{[n,m]^t}{(nm)^{(ρ+t)/2}}\right\}_{n,m=1}^\infty$$ where $[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters $ρ$ and $t$ satisfy $t>0$, $ρ>t+1$. We prove that $E$ is a compact self-adjoint operator on $\ell^2(\mathbb N)$ with infinitely many of both positive and negative eigenvalues. Furthermore, we prove that the ordered sequence of positive eigenvalues of $E$ obeys the asymptotic relation $$λ^+_n(E)=\frac{\varkappa}{n^{ρ-t}}(1+o(1)), \quad n\to\infty,$$ with some $\varkappa>0$ and the negative eigenvalues obey the same relation, with the same asymptotic coefficient $\varkappa$. We also indicate a connection of the spectral analysis of $E$ to the theory of Beurling primes.

math.SP↗

arXiv2021

Spectral asymptotics for a family of LCM matrices

We consider the family of arithmetical matrices given explicitly by $$E(σ,τ)=\left\{\frac{n^σm^σ}{[n,m]^τ}\right\}_{n,m=1}^\infty,$$ where $[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters $σ$ and $τ$ satisfy $ρ:=τ-2σ>0$, $τ-σ>\frac12$ and $τ>0$. We prove that $E(σ,τ)$ is a compact self-adjoint positive definite operator on $\ell^2({\mathbb N})$, and the ordered sequence of eigenvalues of $E(σ,τ)$ obeys the asymptotic relation $$λ_n(E(σ,τ))=\frac{\varkappa(σ,τ)}{n^ρ}+o(n^{-ρ}), \quad n\to\infty,$$ with some $\varkappa(σ,τ)>0$. We give an application of this fact to the asymptotics of singular values of truncated multiplicative Toeplitz matrices with the symbol given by the Riemann zeta function on the vertical line with abscissa $σ<1/2$. We also point out a connection of the spectral analysis of $E(σ,τ)$ to the theory of generalised prime systems.

math.SP↗

arXiv2020

On certain sums concerning the gcd's and lcm's of $k$ positive integers

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordellès (2007), and extend certain related results of Hilberdink and Tóth (2016). We also formulate some conjectures and open problems.

math.NT↗

arXiv2016

On the average value of the least common multiple of $k$ positive integers

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$ belongs to a large class of multiplicative arithmetic functions, including, among others, the functions $f(n)=n^r$, $φ(n)^r$, $σ(n)^r$ ($r>-1$ real), where $φ$ is Euler's totient function and $σ$ is the sum-of-divisors function. The proof is by elementary arguments, using the extension of the convolution method for arithmetic functions of several variables, starting with the observation that given a multiplicative function $f$, the function of $k$ variables $f([n_1,\ldots,n_k])$ is multiplicative.

math.NT↗

arXiv2016

Gál-type GCD sums beyond the critical line

We prove that \[ \sum_{k,{\ell}=1}^N\frac{(n_k,n_{\ell})^{2α}}{(n_k n_{\ell})^α} \ll N^{2-2α} (\log N)^{b(α)} \] holds for arbitrary integers $1\le n_1<\cdots < n_N$ and $0<α<1/2$ and show by an example that this bound is optimal, up to the precise value of the exponent $b(α)$. This estimate complements recent results for $1/2\le α\le 1$ and shows that there is no "trace" of the functional equation for the Riemann zeta function in estimates for such GCD sums when $0<α<1/2$.

math.NT↗