nt.number theory – What are some consequences of zero free strip of the Riemann zeta function?

I don’t feel familiar enough with Eisenstein series to point out the exact line which is wrong, but the argument in page 7 which claims to show that all zeroes of the Riemann zeta function cannot have real part between $frac{1}{2} – h$ and $frac{1}{2}$ works equally well at proving that no zero of the Riemann zeta function has real part $frac{1}{2}$, which is of course false.

I believe that the part that is false is the fact that if $zeta left( ς right) = 0$ then $E(z, 1 − ς) = 0$, and in fact the functional equation for $E^{*} (z, s)$ seems a bit off to me, but if someone who knows a bit more than me cares to check then we can know for sure.


Anyway, this result, if it were true, would be a huge breakthrough in number theory. The most direct improvement would of course be a power savings in the error term of $lvert pi (x) – mathrm{Li} (x) rvert$, but there are many more applications for such things. For example, this would imply that $zeta left( sigma + i t right) = mathcal{O} left( lvert t rvert^{varepsilon} right)$ for $sigma geq 1 – 2 h$, which is a significant improvement on Heath-Brown’s bound $zeta left( sigma + i t right) = mathcal{O} left( lvert t rvert^{frac{1}{2} (1 – sigma)^{frac{3}{2}} + varepsilon} right)$, which (if I recall correctly) is the currently best known bound for values close to $sigma = 1$. Bounds like this are very useful in all types of applications, where integrals containing the zeta function appear.

Of course, this is just one of many and varied consequences.