Applications of inversion formulas to the joint \(t\)-universality of Lerch zeta functions (Q868890)

The Lerch zeta-function \(L(\lambda,\alpha,s)\), \(\lambda \in \mathbb R\), \(0<a\leq 1\), \(s=\sigma+it\), defined, for \(\sigma>1\), \[ L(\lambda,a,s)=\sum_{n=0}^{\infty}\frac{e^{2 \pi i \lambda n}}{(n+a)^s} \] is investigated. The universality in Voronin's sense for \(L(\lambda,\alpha,s)\) was obtained by \textit{A.~Laurinčikas} and \textit{R.~Garunkštis} [The Lerch Zeta-Function. Dordrecht: Kluwer (2002; Zbl 1028.11052)] and \textit{A.~Laurinčikas} and \textit{K. Matsumoto} [Nagoya Math. J. 157, 211--227 (2000; Zbl 0970.11034)]. The author proves the joint \(t\)-universality for a collection of Lerch zeta-functions and considers the relations among them. Let \(0<R<\frac{1}{4}\), \(K=\{s: | s-3/4| \leq R\}\), \(k\) be a positive integer, and \(U\) denote the set of functions \(f(s)\) continuous on \(K\) and analytic in the interior of \(K\). Let \(T>0\) be a fixed number. Then the joint \(t\)-universality for \(L(\lambda,a,s)\) are discussed in two cases: 1. The \(\lambda\)-joint \(t\)-universality for \(\{L(\lambda+n/m,a,s)\}_n\), \((n,m)=1\), \(n<m\), is the following property: for \(f_n(s) \in U\) and every \(\epsilon >0\), \[ \lim\inf_{T \to \infty}\frac{1}{T}\text{meas}\left\{\tau \in [0,T]: \max_{n=0,\dots,m-1}\max_{s \in K}\left| k^{it}L(\lambda+n/m,a,s+i\tau)-f_n(s)\right| <\varepsilon\right\}>0. \] 2. The \(a'\)-joint \(t\)-universality for \(\{L(\lambda,a+j/m,s)\}_j\), \((j,m)=1\), \(j<m\), is the following property: for \(f(s) \in U\) and every \(\epsilon >0\), \[ \lim\inf_{T \to \infty}\frac{1}{T}\text{meas}\left\{\tau \in [0,T]: \max_{j=0,\dots,m-1}\max_{s \in K}\left| k^{it}L(\lambda,a+j/m,s+i\tau)-f(s)\right| <\varepsilon\right\}>0. \] In the paper, for example, the author proves the following statement: If the set \(\{L(\lambda+n/m,ma,s)\}_n\) has \(\lambda\)-joint \(t\)-universality, then the set \(\{L(m\lambda,a+j/m,s)\}_j\) has \(a'\)-joint \(t\)-universality, \(0 \leq j \leq m-1\). If the set \(\{L(\lambda+n/m,1,s)\}_n\) has \(\lambda\)-joint \(t\)-universality, then the set \(\{L(m\lambda,l/m,s)\}_l\) has \(a'\)-joint \(t\)-universality, where \(a=1/m\) and \(ma+j=l=1,2,\dots,m\), \(1\leq l\leq m\). Also, the double joint \(t\)-universality is obtained.