In article <2073967333.59072.1251644984349.JavaMail.email@example.com> Luca <firstname.lastname@example.org> writes: ... > (1) zeta(1-s) = F(s) zeta(s) > (where with F(s) I mean to represent the part that would be too complex > to write here without using LaTEX, and that you can instead read at eqn > 13 of mathworld.wolfram.com/RiemannZetaFunction.html), then using it to > prove that zeros of the Riemann Zeta function always occur in pairs. > If everybody assumes that from (1) it must follow that if zeta(s)=0 ==> > zeta(1-s)=0, I imagine that they are somehow confident that F(s) not= 0.
But that is false. zeta(s) = 0 for the even negative integers, but not for any other real value, and indeed in those cases F(s) = 0. However, for non-real s, F(s) != 0, as the only part of F(s) that could be 0 is cos(pi.s/2), which has only zeros on the real axis. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/