Numerical calculations on the zeroes of the Riemann Zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) or more have been calculated, all simple and all located on the critical line $\Re s=1/2$.

Question 1: what is the order of magnitude of the largest known zero?

Question 2: assuming RH and writing the $k$-th zero as $z_k=\frac12+i t_k$ with a non-decreasing sequence $t_k$ of positive numbers, is there an asymptotic formula for the size of $t_k$ in terms of $k$?

Question 3: if the Riemann hypothesis is wrong, up to a double logarithmic error, say if in fact, no better approximation than $$ \pi(x)=\text{Li}(x)+O\bigl(x^{1/2}(\ln x)( \ln \ln x)\bigr), $$ holds true, how far do the numerical computations should go to detect that "doubly logarithmic" quantity of zeroes off the critical line? If it is up to $10^{100}$, there is a chance that in the next 30 years a computer can detect a zero off line. If it is of the order of $10^{10000}$, RH won't be disproved by a computer before the sun becomes a red giant.

Numerical calculations on the zeroes of the Riemann zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) or more have been calculated, all simple and all located on the critical line $\Re s=1/2$.

Question 1: what is the order of magnitude of the largest known zero?

Question 2: assuming RH and writing the $k$-th zero as $z_k=\frac12+i t_k$ with a non-decreasing sequence $t_k$ of positive numbers, is there an asymptotic formula for the size of $t_k$ in terms of $k$?

Question 3: if the Riemann hypothesis is wrong, up to a double logarithmic error, say if in fact, no better approximation than $$ \pi(x)=\text{Li}(x)+O\bigl(x^{1/2}(\ln x)( \ln \ln x)\bigr), $$ holds true, how far do the numerical computations should go to detect that "doubly logarithmic" quantity of zeroes off the critical line? If it is up to $10^{100}$, there is a chance that in the next 30 years a computer can detect a zero off line. If it is of the order of $10^{10000}$, RH won't be disproved by a computer before the sun becomes a red giant.

Numerical calculations on the zeroes of the Riemann Zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) or more have been calculated, all simple and all located on the critical line $\Re s=1/2$.

Question 1: what is the order of magnitude of the largest known zero?

Question 2: writing the $k$-th zero as $z_k=\frac12+i t_k$ with a non-decreasing sequence $t_k$ of positive numbers, is there an asymptotic formula for the size of $t_k$ in terms of $k$?

Question 3: if the Riemann hypothesis is wrong, up to a double logarithmic error, say if in fact, no better approximation than $$ \pi(x)=\text{Li}(x)+O\bigl(x^{1/2}(\ln x)( \ln \ln x)\bigr), $$ holds true, how far do the numerical computations should go to detect that "doubly logarithmic" quantity of zeroes off the critical line? If it is up to $10^{100}$, there is a chance that in the next 30 years a computer can detect a zero off line. If it is of the order of $10^{10000}$, RH won't be disproved by a computer before the sun becomes a red giant.

Numerical calculations on the zeroes of the Riemann Zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) or more have been calculated, all simple and all located on the critical line $\Re s=1/2$.

Question 1: what is the order of magnitude of the largest known zero?

Question 2: assuming RH and writing the $k$-th zero as $z_k=\frac12+i t_k$ with a non-decreasing sequence $t_k$ of positive numbers, is there an asymptotic formula for the size of $t_k$ in terms of $k$?

Question 3: if the Riemann hypothesis is wrong, up to a double logarithmic error, say if in fact, no better approximation than $$ \pi(x)=\text{Li}(x)+O\bigl(x^{1/2}(\ln x)( \ln \ln x)\bigr), $$ holds true, how far do the numerical computations should go to detect that "doubly logarithmic" quantity of zeroes off the critical line? If it is up to $10^{100}$, there is a chance that in the next 30 years a computer can detect a zero off line. If it is of the order of $10^{10000}$, RH won't be disproved by a computer before the sun becomes a red giant.