Zero-Free Region of Zeta

Establishing zero-free regions for $\zeta(s)$ near the line $\Re(s) = 1$ is crucial for the Prime Number Theorem and understanding prime distribution.

TheoremZero-Free Region (de la Vallée Poussin)

There exists a constant $c > 0$ such that $\zeta(s) \neq 0$ for:

\Re(s) \geq 1 - \frac{c}{\log(|t| + 2)}

where $s = \sigma + it$ .

The classical value is $c = 1/(3\log 2) \approx 0.48$ , though improved constants have been found.

ExampleConsequence for Prime Number Theorem

The zero-free region immediately implies the Prime Number Theorem with explicit error:

\psi(x) = x + O\left(x \exp\left(-c\sqrt{\log x}\right)\right)

Better zero-free regions give better error bounds. The Riemann Hypothesis would give the optimal bound $O(\sqrt{x} \log^2 x)$ .

DefinitionKey Inequality (Mertens-Hadamard)

For $\sigma > 1$ :

\Re\left(\frac{\zeta'(s)}{\zeta(s)}\right) < 0

and near $\sigma = 1$ :

3\zeta(\sigma) + 4\Re\left(\frac{\zeta'(\sigma+it)}{\zeta(\sigma+it)}\right) + \Re\left(\frac{\zeta'(\sigma+2it)}{\zeta(\sigma+2it)}\right) > 0

This inequality, combined with the Euler product, forces $\zeta(1+it) \neq 0$ for all $t \neq 0$ .

TheoremNon-Vanishing on the Line $\Re(s) = 1$

$\zeta(s) \neq 0$ for all $s$ with $\Re(s) = 1$ .

This is equivalent to the Prime Number Theorem:

\pi(x) \sim \frac{x}{\log x}

ExampleCounting Primes in Short Intervals

The zero-free region allows estimating primes in short intervals. For $x^{\theta} \leq y \leq x$ :

\pi(x+y) - \pi(x) \sim \frac{y}{\log x}

provided $\theta$ is large enough relative to the zero-free region.

Remark

The Riemann Hypothesis is equivalent to: $\zeta(s) \neq 0$ for all $s$ with $\Re(s) > 1/2$ .

Current technology gives zero-free regions of the form:

\Re(s) \geq 1 - \frac{c}{\log^{2/3}(|t|+2) (\log\log(|t|+3))^{1/3}}

These improvements use deep analytical techniques including exponential sums and the large sieve.

Zero-free regions translate directly into quantitative information about prime distribution. Every improvement pushes the boundary closer to the critical line and yields sharper estimates for $\pi(x)$ , $\psi(x)$ , and related prime-counting functions.