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On the roots of the equation \(\zeta (s)=a\). (English) Zbl 1357.11073

Summary: Given any complex number \(a\), we prove that there are infinitely many simple roots of the equation \(\zeta (s)=a\) with arbitrarily large imaginary part. Besides, we give a heuristic interpretation of a certain regularity of the graph of the curve \(t\mapsto \zeta ({1\over 2}+it)\). Moreover, we show that the curve \(\mathbb {R}\ni t\mapsto (\zeta ({1\over 2}+it),\zeta '({1\over 2}+it))\) is not dense in \(\mathbb {C}^2\).

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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