We show that the Riemann zeta function ζ has only countably many self-intersections on the critical line, i.e., for all but countably many z∈Cdouble-struck the equation ζ(1/2+it)=z has at most one solution t∈Rdouble-struck. More generally, we prove that if F is analytic in a complex neighborhood of R and locally injective on Rdouble-struck, then either the set (a,b)∈Rdouble-struck2:a≠b and F(a)=F(b) is countable, or the image F(Rdouble-struck) is a loop in Cdouble-struck. © 2013 Elsevier Ltd
Banks, W., Castillo Garate, V., Fontana, L., Morpurgo, C. (2013). Self-intersections of the Riemann zeta function on the critical line. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 406(2), 475-481 [10.1016/j.jmaa.2013.04.083].
Self-intersections of the Riemann zeta function on the critical line
FONTANA, LUIGIPenultimo
;
2013
Abstract
We show that the Riemann zeta function ζ has only countably many self-intersections on the critical line, i.e., for all but countably many z∈Cdouble-struck the equation ζ(1/2+it)=z has at most one solution t∈Rdouble-struck. More generally, we prove that if F is analytic in a complex neighborhood of R and locally injective on Rdouble-struck, then either the set (a,b)∈Rdouble-struck2:a≠b and F(a)=F(b) is countable, or the image F(Rdouble-struck) is a loop in Cdouble-struck. © 2013 Elsevier Ltd
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