What are some consequences of zero free strip of the Riemann zeta function?

Question

A weaker version of the Riemann hypothesis is the claim that if $\zeta(s) = 0$ then $Re(s) \leq 1 - h$ for some constant $h> 0$. What would the consequences be of a result of this type?

Answer 1

If this were proven, it 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.

Answer 2

Many number-theoretic functions have error bounds contingent on the supremum of the real parts of zeroes of the Riemann zeta function. Call this $\Theta$. If we write $f(x)=\Omega_{\pm}(g(x))$ for the notion that \begin{align*} \limsup_{x\to\infty}\frac{f(x)}{g(x)}&>0,\text{ and}\\ \liminf_{x\to\infty}\frac{f(x)}{g(x)}&<0, \end{align*} then (e.g. in Montgomery and Vaughan, Section 15.1), for any $\epsilon>0$, \begin{align*} \psi(x)-x&=\Omega_{\pm}(x^{\Theta-\epsilon}),\\ \pi(x)-\operatorname{li}(x)&=\Omega_{\pm}(x^{\Theta-\epsilon}),\\ M(x)&=\Omega_{\pm}(x^{\Theta-\epsilon}). \end{align*} Here, $\psi(x)=\log\operatorname{lcm}(1,2,\dots,n)$ is the Chebyshev Psi function, $\pi(x)$ is the prime counting function, $\operatorname{li}(x)$ is the logarithmic integral, and $M(x)$ is the Mertens function. In other words, the maximal error of these summatory functions from the standard "simple" approximations is (somewhat) precisely determined by how far the zeroes of $\zeta$ may stray from the line $\Re s=1/2$. (In fact, sharper results involving $\Omega_{\pm}(x^\Theta)$ directly can be shown if there is a zero $\rho$ with $\Re\rho=\Theta$). So, knowing that $\Theta<1$ gives quite strong information about the distribution of prime numbers, and as $\Theta\to1/2$ such information approximates the best possible bounds, at least as far as functions of the form $x^\alpha$ are concerned.