arXiv subjects
Publications and source records attributed to R. Garunkstis.
Given any complex number $a$, we prove that there are infinitely many simple roots of the equation $ζ(s)=a$ with arbitrarily large imaginary part. Besides, we give a heuristic interpretation of a certain regularity of the graph of the curve $t\mapsto ζ({1\over 2}+it)$. Moreover, we show that the curve $t\mapsto (ζ({1\over 2}+it),ζ'({1\over 2}+it))$ is not dense in $C^2$.
Let $d$ be a real number, let $s$ be in a fixed compact set of the strip $1/2<σ<1$, and let $L(s, χ)$ be the Dirichlet $L$-function. The hypothesis is that for any real number $d$ there exist 'many' real numbers $τ$ such that the shifts $L(s+iτ, χ)$ and $L(s+idτ, χ)$ are 'near' each other. If $d$ is an algebraic irrational number then this was obtained by T. Nakamura. Ł. Pańkowski solved the case then $d$ is a transcendental number. We prove the case then $d\ne0$ is a rational number. If $d=0$ then by B. Bagchi we know that the above hypothesis is equivalent to the Riemann hypothesis for the given Dirichlet $L$-function. We also consider a more general version of the above problem.