Could someone please tell me what is known about the real non-trivial zeros of Riemann zeta function inside critical strip? Do we know there is none?
I want to know what is known about the above questions without the assumption of Riemann hypothesis.
Any hint or help would be appreciated. Thanks in advance.
There are no real nontrivial zeroes of $\zeta(s)$ in $(0,1)$.
This can be easily shown as follows. Let $\eta(s)=\sum_{n=1}^\infty(-1)^n/n^s$, which converges in $\mathop{\rm Re}s>0$. When $\mathop{\rm Re}s>1$, we have that $$\eta(s)=\sum_{n=1}^\infty\frac1{n^s}-2\sum_{n=1}^\infty\frac1{(2n)^s}=\left(1-\frac1{2^{1-s}}\right)\zeta(s).$$ Since $\eta(s)$ is holomorphic on $\mathop{\rm Re}s>0$, by analytic continuation this equality holds on $\mathop{\rm Re}s>0$ as well. Now, when $s$ is real and $0<s<1$, $\eta(s)=\sum_{n=0}^\infty1/(2n+1)^s-1/(2n+2)^s$ and each term is positive, so $\eta(s)>0$, in particular $\eta(s)\ne0$, thus $\zeta(s)$ cannot vanish.