18 Zero free region for the zeta function

This chapter is not yet integrated into the main blueprint.

A zero of the Riemann zeta function is a complex number $ρ = β + i γ$ for which $ζ (ρ) = 0$ . The zeta function has a infinite number of zeros of the form $ρ = - 2 n$ for integer $n \geq 1$ ; these are known as trivial zeros and are well understood. There are also an infinite number zeros inside the “critical strip" $0 < ℜ z < 1$ , called non-trivial zeros. The locations of the non-trivial zeros have deep consequences for many fields of mathematics. A well-known conjecture regarding the non-trivial zeros is the Riemann hypothesis.

Heuristic 18.1 Riemann hypothesis

If $ρ$ is a non-trivial zero of the Riemann zeta function, then $ℜ ρ = 1 / 2$ .

This conjecture remains far out of reach. Instead, currently what are known are zero free regions.

Definition 18.2 Zero free region

A zero-free region of the Riemann zeta function is a set $D \subset C$ for which $ζ (s) \neq 0$ for all $s \in D$ .

Lemma 18.3 Basic properties of zero free regions

The following properties hold:

(Symmetry about the real axis) If $ζ (σ + i t) \neq 0$ then $ζ (σ - i t) \neq 0$ .
(Symmetry about the critical line $ℜ s = 1 / 2$ ) For $0 \leq σ \leq 1$ , if $ζ (σ + i t) \neq 0$ then $ζ (1 - σ + i t) \neq 0$ .
(Non vanishing for $ℜ s > 1$ ) If $ℜ s > 1$ then $ζ (s) \neq 0$ .

Proof ▶

Claim (i) follows directly from the property $\overset{―}{ζ (s)} = ζ (\overset{―}{s})$ . Claim (ii) follows from the functional equation

ζ (s) = 2^{s} π^{s - 1} \sin (π s / 2) Γ (1 - s) ζ (1 - s)

and claim (iii) follows from the Euler product formula

ζ (s) = \prod_{p} (1 - p^{- s})^{- 1}, (ℜ s > 1) .

In light of Lemma 18.3, for the rest of the chapter we will focus on the quadrant

D \subseteq {z \in C : ℜ z > 1 / 2, ℑ z > 0} .

The first zero free region for non-trivial zeros proved was on $D = {z \in C : ℜ z = 1}$ , and was used to prove the prime number theorem $π (x) \sim x / \log x$ as $x \to \infty$ .

Theorem 18.4 Non-vanishing on the 1-line

One has $ζ (1 + i t) \neq 0$ for any real $t$ .

18.1 Relation to upper bound on zeta in the critical strip

Using estimates of $ζ (σ + i t)$ close to the line $σ = 1$ , one can extend the zero free region slightly inside the critical strip.

Lemma 18.5 Relation to growth exponents of zeta

Suppose $f (t) > 0$ and $0 < g (t) \leq 1$ are real-valued functions for $t \geq 0$ , with $f (t)$ non-decreasing and tending to infinity with $t$ , and $g (t)$ non-increasing. Suppose further that $f (t) / g (t) = o (\exp (f (t)))$ . If

ζ (σ + i t) ≪ \exp (f (t)) (1 - g (t) \leq σ \leq 2, t \geq 0)

then $ζ (σ + i t) \neq 0$ for

σ \geq 1 - A \frac{g (2 t + 1)}{f (2 t + 1)}

where $A > 0$ is an absolute constant.

Proof ▶

The following zero free region of classical type was proved independently by de la Vallée Poussin and Hadamard.

Theorem 18.6 Classical zero free region

One has $ζ (σ + i t) \neq 0$ if

σ \geq 1 - \frac{A}{\log t} .

for an absolute constant $A > 0$ and $t$ sufficiently large.

Proof ▶

Apply Lemma 18.5 with $g (t) = 1 / 2$ , $f (t) = \log (t + 2)$ and the convexity bound $μ (σ) \leq (1 - σ) / 2$ .

This classical result has been improved in a number of works, most of which make crucial use of non-trivial estimates of certain types of exponential sums.

Theorem 18.7 Littlewood zero free region

One has $ζ (σ + i t) \neq 0$ if

σ \geq 1 - \frac{A \log \log t}{\log t}

for an absolute constant $A > 0$ and $t$ sufficiently large.

Proof ▶

Follows from the zeta bound corresponding to

μ (1 - \frac{k}{2^{k} - 2}) \leq \frac{1}{2^{k} - 2}

for integer $k \geq 3$ , which is generated by the van der Corput exponent pair $A^{k - 2} B (0, 1) = (\frac{1}{2^{k} - 2}, 1 - \frac{k - 1}{2^{k} - 2})$ . However, one needs to make explicit the $o (1)$ term in the bound $ζ (σ + i t) ≪ t^{μ (σ) + o (1)}$ . In particular, by [ 217 , Theorem 5.14 ] , one has

ζ (1 - \frac{k}{2^{k} - 2} + i t) ≪ t^{1 / (2^{k} - 2)} \log t .

Taking

k = ⌊ \frac{1}{\log 2} \log (\frac{\log t}{\log \log t}) ⌋

and using the Phragmén Lindelöf principle, one has

ζ (σ + i t) ≪ (\log t)^{5}, (σ \geq 1 - \frac{(\log \log t)^{2}}{\log t}),

so we may take $f (t) = 5 \log \log t$ and $g (t) = (\log \log t)^{2} / \log t$ in Lemma 18.5.

Theorem 18.8 Chudakov zero free region

One has $ζ (σ + i t) \neq 0$ if

σ \geq 1 - \frac{1}{(\log t)^{3 / 4 + o (1)}}

for $t$ sufficiently large.

Theorem 18.9 Korobov-Vinogradov zero free region

One has $ζ (σ + i t) \neq 0$ if

σ \geq 1 - \frac{A}{(\log t)^{2 / 3} (\log \log t)^{1 / 3}}

for an absolute constant $A > 0$ and $t$ sufficiently large.

18.2 Relation to the error term in the prime number theorem

Lemma 18.10 Relation to prime number theorem error term

Suppose $ζ (σ + i t) \neq 0$ for $σ \geq 1 - η (t)$ where $η (t)$ is a positive and decreasing function. Then

\sum_{n \leq x} Λ (n) - x ≪ x \exp (- A ω (x)), (x \to \infty)

for an absolute constant $A > 0$ , where

ω (x) := inf_{t \geq 1} (η (t) \log x + \log t) .

Proof ▶

Applying Lemma 18.10, one obtains the error term estimates in the prime number theorem given in Table 18.1.

Table 18.1 Error bounds on the prime number theorem. Here

A

represents an absolute, positive constant, which may be different at each occurrence.

Bound on $(ψ (x) - x) / x$	Associated zero-free region	Reference
$\exp (- A (\log x)^{1 / 2})$	$σ \geq 1 - \frac{A}{\log t}$	Theorem 18.6
$\exp (- A (\log x \log \log x)^{1 / 2})$	$σ \geq 1 - \frac{A \log \log t}{\log t}$	Theorem 18.7
$\exp (- A \frac{(\log x)^{3 / 5}}{(\log \log x)^{1 / 5}})$	$σ \geq 1 - \frac{A}{(\log t)^{2 / 3} (\log \log t)^{1 / 3}}$	Theorem 18.9