Functional distribution for a collection of Lerch zeta functions
Let \(L(\lambda ,\alpha ,s)=\sum _{n=0}^{\infty }\frac{e^{2\pi {\kern 1pt} i\lambda n{\kern 1pt} } }{(n+\alpha )^{s} } \) be the Lerch zeta function. Motivated by some results of \textit{A. Laurinčikas} [Lith. Math. J. 37, No. 3, 275--280 (1997); translation from Liet. Mat. Rink. 37, No. 3, 367--375 (1997; Zbl 0938.11045)], the author formulates here the following conjecture: Let \(0<\alpha <1\) be a transcendental real number and let also \(\lambda _{{\kern 1pt} 1} ,\lambda _{{\kern 1pt} 2} ,\ldots,\lambda _{{\kern 1pt} r} \) be distinct real numbers in the interval \([0,1)\). Then the joint universality theorem holds for the set of Lerch zeta functions \(\left\{L(\lambda _{j} ,\alpha ,s)|1\leq j\leq r\right\}\) on the strip \(1/2<\mathrm{Re} (s)<1\) . \noindent The author presents some results which justify this conjecture. For example, it is proved that the joint universality theorem holds for the set \(\left\{L(\lambda _{j} ,\alpha ,s)\right\}\)for almost all real numbers \(\lambda _{j} \).
scientific article; zbMATH DE number 6383637