The zeta function has infinitely many zeros in $0<\Re{s}<1/2$?

Question

The following paragraph appears on page 42 in the book Rational Number Theory in the 20th Century: From PNT to FLT (Par Wladyslaw Narkiewicz):

The fact that the strip $0<\Re{s}<1/2$ contains infinitely many zeros of the zeta-function follows from the formula for the number of zeros lying in the rectangle $0<\Re{s}<1/2$, $0<\Im{s}<T$, conjectured by Riemann and established by H. von Mangoldt in 1895: $$N(T)=\frac{1}{2\pi}T\log\left(\frac{T}{2\pi}\right)-\frac{T}{2\pi}+R(T)$$ with $R(T)=O(\log^2T)$.

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Wouldn't this contradict the Riemann hypothesis?

Answer 1

Looking at some other sources, it appears that to be a typo: It should be "$0<\Re s<1$" not "$0<\Re s<1/2$", i.e. the number of zeros in the critical strip.

In fact, no non-trivial zeros of the Riemann Zeta function occur outside the critical strip, so this restriction is superfluous i.e. the formula gives the number of zeroes in $0 <\Im s<T$. It is in this form that Wikipedia and Mathworld state the Riemann-von Mangoldt formula. Technical sources can be found in both links.

Answer 2

Yes, if it were correct, the Riemann hypothesis would be false. It's definitely a typo, however. You can see, for example, here for a fairly detailed proof; suffice to say that nothing can be done to cut the region of validity down effectively, due to the difficult nature of the $\zeta$-function's behaviour in the critical strip.