Effective Zero-Free Regions for Zeta and the Prime Number Theorem Error Term

Intuition Beginner

The prime number theorem says the weighted prime count $\psi (x)$ grows like $x$. But "grows like" hides a second question that is just as interesting: how far off is the approximation? If you replace $\psi (x)$ by $x$, what size of error are you making? The answer is controlled entirely by where the zeros of the zeta function sit. Push the zeros away from the vertical line at the right edge of the critical strip, and the error shrinks. The further you can guarantee the zeros stay away, the better your estimate of the primes.

Think of each zero as a radio station broadcasting a wave that disturbs the smooth count. A zero close to the edge broadcasts a loud, slowly fading wave; a zero far from the edge broadcasts a quiet one. A zero-free region is a promise: a strip hugging the right edge of the critical strip where no station broadcasts at all. The wider that silent strip, the quieter the total noise, and the more accurately $x$ predicts $\psi (x)$.

The first such promise, proved by de la Vallée Poussin in 1896, is thin near the top of the strip and grows thinner as you go up, but it never closes. It is enough to pin the error down to something that fades faster than any power of $\log x$. Decades later, Vinogradov and Korobov widened the silent strip and improved the error again. The race to widen this region is the race to understand the primes more precisely.

Visual Beginner

Picture the critical strip as a vertical band, from the line at horizontal position $0$ to the line at position $1$. The substantive zeros live somewhere inside. Now shade a thin sliver just to the left of the right edge: a curved boundary that starts a fixed distance from the edge near the bottom and presses closer and closer to the edge as you climb. Inside that sliver, no zeros are allowed. That shaded sliver is the zero-free region.

height $t$	de la Vallée Poussin gap $c/\log (t+2)$	Vinogradov-Korobov gap (shape)
$10$	$0.20$	wider
$1000$	$0.10$	wider still
$10^6$	$0.07$	much wider

The gap is the distance from the right edge to the nearest allowed zero. Both regions narrow as you climb, but the Vinogradov-Korobov gap narrows much more slowly. A wider gap means a smaller error in counting primes.

Worked example Beginner

See how the size of the error depends on the gap, with concrete numbers.

Step 1. Suppose the zeros are kept away from the right edge by a fixed gap $\delta$, so the worst zero sits at horizontal position $1-\delta$. The wave from that zero has size about $x^{1-\delta }$ compared with the main term $x$. The ratio of noise to signal is then $x^{1-\delta }/x=x^{-\delta }$.

Step 2. Plug in $\delta =0.1$ and $x=10^6$. The ratio is $x^{-\delta }=(10^6{)}^{-0.1}=10^{-0.6}\approx 0.25$. So a fixed gap of $0.1$ would leave an error around a quarter of the main term — not good enough.

Step 3. The real region does better because the gap is not fixed: for the de la Vallée Poussin region the optimal balance between the gap and the height of the zeros you must control produces an error of the form $x\cdot \exp (-c\sqrt{\log x})$. For $x=10^6$, $\log x\approx 13.8$, so $\sqrt{\log x}\approx 3.7$, and with $c=1$ the factor $\exp (-3.7)\approx 0.025$ — ten times smaller than the fixed-gap estimate.

What this tells us: a fixed gap gives only a power saving $x^{-\delta }$, but the true shape of the de la Vallée Poussin region — a gap that shrinks with height — gives the better factor $\exp (-c\sqrt{\log x})$, which beats every fixed power of $\log x$ while still being weaker than the power saving the Riemann hypothesis would give.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $s=\sigma +it$ with $\sigma ,t$ real, $\zeta$ is the Riemann zeta function of 06.01.16 with its Euler product for $\sigma >1$, $\Lambda$ is the von Mangoldt function and $\psi (x)={\sum }_{n\le x}\Lambda (n)$ the Chebyshev function of 21.12.01, $\rho =\beta +i\gamma$ denotes a substantive (critical-strip) zero, $\log$ is the natural logarithm, and $\mathrm{Li}(x)={\int }_2^{x}dt/\log t$. Constants $c,c^{\prime }$ are positive and absolute; their values may change between statements. We follow Davenport ^{[Davenport §13-14]}.

Definition (zero-free region). A zero-free region for $\zeta$ is a set $R\subseteq \mathbb{C}$ on which $\zeta (s)\ne 0$. An effective zero-free region is one specified by an explicit inequality with a numerically determinable constant. The classical (de la Vallée Poussin) region is $$ \mathcal{R}_{\mathrm{dlVP}} = \Big{ s = \sigma + it : \sigma > 1 - \frac{c}{\log(|t| + 2)} \Big} $$ for a suitable absolute $c>0$.

Definition (Vinogradov-Korobov region). The Vinogradov-Korobov region is $$ \mathcal{R}{\mathrm{VK}} = \Big{ s = \sigma + it : \sigma > 1 - \frac{c}{(\log(|t| + 3))^{2/3} (\log\log(|t| + 3))^{1/3}} \Big}, $$ a strictly wider region than $\mathcal{R}{\mathrm{dlVP}}$forlarge$|t|$,obtainedfromexponential-sumboundsfor$\zeta$neartheline$\sigma = 1$.

Definition (PNT error term). Writing $\psi (x)=x+E(x)$, the function $E(x)$ is the prime number theorem error term. A zero-free region $\sigma >1-\eta (t)$ for a positive decreasing $\eta$ yields a bound on $E(x)$ through the explicit formula of 21.12.01; the Riemann hypothesis is the statement $\beta =\frac{1}{2}$ for every substantive zero, equivalent to $E(x)=O(x^{1/2+\epsilon })$ for all $\epsilon >0$.

The notation $\sigma ,t,\rho =\beta +i\gamma$, the digamma symbol ${\Gamma }^{\prime }/\Gamma$, the counting function $N(\sigma ,T)$, and $\mathrm{Li}(x)$ are recorded in _meta/NOTATION.md. The substantive zeros are those with $0<\beta <1$; the elementary zeros at $s=-2,-4,\dots$ play no role here.

Counterexamples to common slips

• "A zero-free region is a region with no zeros at all, including the elementary ones." The regions here concern the substantive zeros in the critical strip. The elementary zeros at $s=-2,-4,\dots$ lie outside the strip and are irrelevant to the error term; the line $\sigma =1$ is the relevant edge.
• "The de la Vallée Poussin region has a fixed width." The width $c/\log (|t|+2)$ shrinks to $0$ as $|t|\to \infty$. There is no fixed vertical strip $\sigma >1-\delta$ known to be zero-free; if one existed, the error term would be a power saving $O(x^{1-\delta +\epsilon })$, far stronger than what is proved.
• "A wider zero-free region proves the Riemann hypothesis." Widening the region improves the error term, but the Riemann hypothesis demands the zeros lie on the line $\sigma =\frac{1}{2}$, a statement no zero-free region near $\sigma =1$ can reach. The region pushes zeros off a neighbourhood of $\sigma =1$, not onto $\sigma =\frac{1}{2}$.

Key theorem with proof Intermediate+

The signature result is the classical zero-free region. Its engine is an elementary trigonometric inequality that converts the simple pole of $\zeta$ at $s=1$ into a lower bound on $|\zeta |$ just left of the line $\sigma =1$, forbidding a zero too close to the line.

Theorem (de la Vallée Poussin zero-free region). There is an absolute constant $c>0$ such that $\zeta (\sigma +it)\ne 0$ whenever $$ \sigma \geq 1 - \frac{c}{\log(|t| + 2)}, \qquad |t| \geq 1. $$

Proof. For $\sigma >1$ the Euler product gives, as in 21.12.01, $$ -\frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}, \qquad \operatorname{Re}!\left(-\frac{\zeta'}{\zeta}(\sigma + it)\right) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^\sigma} \cos(t \log n). $$ Apply the inequality $$ 3 + 4\cos\theta + \cos 2\theta = 2(1 + \cos\theta)^2 \geq 0 $$ with $\theta =t\log n$. Multiplying the three real parts at heights $0,t,2t$ by $3,4,1$ and summing, every coefficient $\Lambda (n)n^{-\sigma }(3+4\cos (t\log n)+\cos (2t\log n))\ge 0$, so $$ 3\left(-\frac{\zeta'}{\zeta}(\sigma)\right) + 4\operatorname{Re}!\left(-\frac{\zeta'}{\zeta}(\sigma + it)\right) + \operatorname{Re}!\left(-\frac{\zeta'}{\zeta}(\sigma + 2it)\right) \geq 0. \tag{$\ast$} $$

Bound each term for $\sigma$ slightly larger than $1$. The pole of $\zeta$ at $s=1$ gives $$ -\frac{\zeta'}{\zeta}(\sigma) < \frac{1}{\sigma - 1} + A $$ for an absolute constant $A$. From the partial-fraction expansion of ${\zeta }^{\prime }/\zeta$ established in 21.12.01, for $1<\sigma \le 2$, $$ -\operatorname{Re}\frac{\zeta'}{\zeta}(\sigma + it) < A_1 \log(|t| + 2) - \sum_\rho \operatorname{Re}\frac{1}{\sigma + it - \rho}, $$ and each summand $\mathrm{Re}\frac{1}{\sigma +it-\rho }=\frac{\sigma -\beta }{(\sigma -\beta {)}^2+(t-\gamma {)}^2}\ge 0$ since $\beta <1<\sigma$. Dropping all but one zero ${\rho }_0={\beta }_0+i{\gamma }_0$ in the $t$-line and dropping the entire (nonnegative) sum in the $2t$-line gives $$ \operatorname{Re}!\left(-\frac{\zeta'}{\zeta}(\sigma + it)\right) < A_1\log(|t|+2) - \frac{1}{\sigma - \beta_0}, \qquad \operatorname{Re}!\left(-\frac{\zeta'}{\zeta}(\sigma + 2it)\right) < A_2\log(|t|+2), $$ where we now set $t={\gamma }_0$. Substituting into $(\ast )$, $$ 0 \leq \frac{3}{\sigma - 1} + 3A - \frac{4}{\sigma - \beta_0} + A_3 \log(|\gamma_0| + 2). $$

It remains to optimise the auxiliary parameter $\sigma$. Write $L=\log (|{\gamma }_0|+2)$ and choose $\sigma =1+\delta /L$ for a small constant $\delta$ to be fixed. Then $3/(\sigma -1)=3L/\delta$ and $$ \frac{4}{\sigma - \beta_0} \leq \frac{3L}{\delta} + A_4 L. $$ Hence $1-{\beta }_0\ge \sigma -{\beta }_0-(\sigma -1)\ge \frac{4}{3L/\delta +A_{4}L}-\frac{\delta }{L}$. Choosing $\delta$ small enough that the right side is $\ge c/L$ for an absolute $c>0$ yields ${\beta }_0\le 1-c/\log (|{\gamma }_0|+2)$. Since ${\rho }_0$ was an arbitrary zero with $|{\gamma }_0|=|t|\ge 1$, no zero lies in $\sigma \ge 1-c/\log (|t|+2)$. $\square$

Bridge. This region builds toward the prime number theorem with a quantitative error and appears again in 21.12.01, whose explicit formula $\psi (x)=x-{\sum }_{\rho }{x}^{\rho }/\rho +O(1)$ converts a bound on $\beta =\mathrm{Re}\rho$ directly into a bound on $E(x)$. The foundational reason the $3+4\cos \theta +\cos 2\theta$ trick works is that it is dual to the pole at $s=1$: the pole forces $-{\zeta }^{\prime }/\zeta (\sigma )$ to blow up like $3/(\sigma -1)$, and a hypothetical zero on the line $\sigma =1$ would force the competing $-4/(\sigma -{\beta }_0)$ term to blow up faster, which the nonnegativity $(\ast )$ forbids — this is exactly the mechanism that proved $\zeta (1+it)\ne 0$ in 06.01.16, now made effective with a rate. Putting these together, the optimisation over the height $t$ of the zeros one must control against the width of the region generalises the bare nonvanishing into a quantitative strip, and the bridge is the explicit formula: a region of width $\eta (t)$ becomes an error of size $x\exp (-c\sqrt{\log x})$ once the contour is pushed to the boundary of the region.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib carries the analytically continued zeta and the boundary nonvanishing on $\mathrm{Re}(s)=1$, but not the effective region or its error-term consequence. The companion notes record the load-bearing declarations and the gap.

-- Operative imports: Mathlib.NumberTheory.LSeries.RiemannZeta,
-- Mathlib.NumberTheory.LSeries.Nonvanishing
import Mathlib.NumberTheory.LSeries.RiemannZeta

-- The boundary nonvanishing: ζ(s) ≠ 0 on the closed half-plane Re(s) ≥ 1 (s ≠ 1).
-- This is the QUALITATIVE input the effective region refines.
#check @riemannZeta_ne_zero_of_one_le_re
-- 1 ≤ s.re → s ≠ 1 → riemannZeta s ≠ 0

-- The von Mangoldt L-series identity -ζ'/ζ = ∑ Λ(n) n^{-s} (Re s > 1),
-- whose real part feeds the 3 + 4cosθ + cos2θ inequality:
#check @LSeries.vonMangoldt
-- LSeries (fun n => Λ n) s = - deriv riemannZeta s / riemannZeta s

-- The effective region itself — a statement of the shape
--   ∀ σ t, 1 ≤ |t| → σ ≥ 1 - c / Real.log (|t| + 2) → riemannZeta (σ + t*I) ≠ 0
-- with a verified constant c — is NOT in Mathlib. Nor is the contour-shift
-- consequence ψ(x) = x + O(x * Real.exp (-c * Real.sqrt (Real.log x))).

The gap named in Mathlib gap analysis is the quantitative refinement: Mathlib proves $\zeta \ne 0$ on the line but does not yet carry a region of positive width with an explicit constant, nor the Borel-Carathéodory and three-circles bounds on ${\zeta }^{\prime }/\zeta$ that make the constant effective, nor the exponential-sum machinery behind the Vinogradov-Korobov region.

Advanced results Master

Theorem 1 (bounds on ${\zeta }^{\prime }/\zeta$ near $\sigma =1$). The effectiveness of the region rests on a polynomial-in-$\log t$ bound for ${\zeta }^{\prime }/\zeta$ to the right of $\sigma =1$. From the partial-fraction expansion of 21.12.01 and the Borel-Carathéodory theorem applied to $\log \zeta$ on a disc centred at $2+it$, together with the Hadamard three-circles theorem controlling $\log |\zeta |$ across radii, one obtains ^{[Titchmarsh Ch. 3]} $$ \left| \frac{\zeta'}{\zeta}(s) \right| \ll \log^2(|t| + 2) \qquad \text{for } \sigma \geq 1 - \frac{c}{\log(|t| + 2)},\ |t| \geq 2, $$ after excising small discs around any zeros. This bound is what lets the contour in Perron's formula be pushed to the boundary of ${\mathcal{R}}_{\mathrm{dlVP}}$ with controlled integrand.

Theorem 2 (PNT with classical error). Shifting the contour in the explicit formula to the boundary of ${\mathcal{R}}_{\mathrm{dlVP}}$ and optimising the height $T=\exp (\sqrt{\log x})$ yields ^{[Davenport §18]} $$ \psi(x) = x + O!\big(x\exp(-c\sqrt{\log x})\big), \qquad \pi(x) = \operatorname{Li}(x) + O!\big(x\exp(-c\sqrt{\log x})\big). $$ The saving $\exp (-c\sqrt{\log x})$ beats every fixed power of $\log x$ yet falls short of any power $x^{-\delta }$; the square root is the geometric mean of the region width $1/\log T$ and the truncation exponent.

Theorem 3 (Vinogradov-Korobov region). The van der Corput and Vinogradov methods for exponential sums ${\sum }_{N<n\le 2N}{n}^{-it}$ give the subconvex bound $|\zeta (\sigma +it)|\ll |t{|}^{B(1-\sigma {)}^{3/2}}{\log }^{2/3}|t|$ near $\sigma =1$, which sharpens the zero-free region to ^{[Korobov 1958]} $$ \zeta(\sigma + it) \neq 0 \quad \text{for } \sigma > 1 - \frac{c}{(\log|t|)^{2/3}(\log\log|t|)^{1/3}}. $$ The corresponding error term is $\psi (x)=x+O(x\exp (-c(\log x{)}^{3/5}(\log \log x{)}^{-1/5}))$, the best unconditional bound known. The exponential-sum input is the only known route past de la Vallée Poussin; no improvement has lowered the exponent $3/5$ since 1958.

Theorem 4 (zero-density estimates). Let $N(\sigma ,T)=\#\{\rho =\beta +i\gamma :\beta \ge \sigma ,0<\gamma \le T\}$. Although individual zeros may approach $\sigma =1$, they are sparse there: there is an absolute $A$ with ^{[Titchmarsh Ch. 9]} $$ N(\sigma, T) \ll T^{A(1 - \sigma)}\log^B T, \qquad \tfrac12 \leq \sigma \leq 1. $$ The density hypothesis conjectures $A=2$ for $\sigma \ge \frac{1}{2}$. Zero-density estimates substitute for the Riemann hypothesis in many applications — primes in short intervals, primes in arithmetic progressions — because they bound the number of exceptional zeros even when their location cannot be pinned down.

Theorem 5 (equivalence with RH). The Riemann hypothesis $\beta =\frac{1}{2}$ for all substantive zeros is equivalent to the error bound $\psi (x)=x+O(x^{1/2}{\log }^{2}x)$, and to $\pi (x)=\mathrm{Li}(x)+O(x^{1/2}\log x)$ ^{[Davenport §18]}. More precisely, $\psi (x)-x=O(x^{\Theta +\epsilon })$ holds for every $\epsilon >0$ if and only if every zero satisfies $\beta \le \Theta$, where $\Theta ={\sup }_{\rho }\beta$ is the supremum of the real parts. RH is the case $\Theta =\frac{1}{2}$; the de la Vallée Poussin region only gives $\Theta \le 1$ effectively, with the saving hidden in the rate at which the region approaches the line.

Synthesis. The effective zero-free region is the foundational reason the prime number theorem comes with a usable error term rather than a bare asymptotic. The central insight is that a single elementary inequality, $3+4\cos \theta +\cos 2\theta =2(1+\cos \theta {)}^2\ge 0$, weighting three copies of $-{\zeta }^{\prime }/\zeta$ at heights $0,t,2t$, converts the pole of $\zeta$ at $s=1$ into a forbidden zone for zeros near the line — this is exactly the qualitative nonvanishing $\zeta (1+it)\ne 0$ of 06.01.16, now sharpened into a quantitative width $c/\log (|t|+2)$. Putting these together with the explicit formula of 21.12.01 generalises the asymptotic $\psi (x)\sim x$ into $\psi (x)=x+O(x\exp (-c\sqrt{\log x}))$, where the region width and the truncation height meet at their geometric mean. The bridge is the same logarithmic derivative throughout: its pole gives the main term, its zeros give the error, and the region controls how loud the zeros are. The Vinogradov-Korobov sharpening is dual to this picture — bounding the size of $\zeta$ via exponential sums rather than the sign of a trigonometric polynomial — and the Riemann hypothesis is the limiting statement that the region could be pushed to $\sigma =\frac{1}{2}$, collapsing the error to the square-root barrier. Zero-density estimates are the fallback: where the region cannot locate the zeros, it counts how few sit near the edge, and that count powers the applications from short intervals to progressions.

Full proof set Master

Proposition 1 (the $3$-$4$-$1$ inequality forbids zeros on $\sigma =1$). For every real $t\ne 0$, $\zeta (1+it)\ne 0$.

Proof. For $\sigma >1$, inequality $(\ast )$ gives $3(-{\zeta }^{\prime }/\zeta (\sigma ))+4\mathrm{Re}(-{\zeta }^{\prime }/\zeta (\sigma +it))+\mathrm{Re}(-{\zeta }^{\prime }/\zeta (\sigma +2it))\ge 0$. As $\sigma \to 1^{+}$: the first term is $3/(\sigma -1)+O(1)$ from the pole. If $\zeta (1+it)=0$ with order $m\ge 1$, the partial-fraction expansion gives $\mathrm{Re}(-{\zeta }^{\prime }/\zeta (\sigma +it))=-m/(\sigma -1)+O(\log (|t|+2))$, so the second term is $-4m/(\sigma -1)+O(\log )$. The third term is $O(\log (|t|+2))$ since $\zeta$ has at worst a bounded-order zero at $1+2it$. Summing, the left side is $(3-4m)/(\sigma -1)+O(\log (|t|+2))$. For $m\ge 1$ the coefficient $3-4m\le -1<0$, so the left side $\to -\infty$ as $\sigma \to 1^{+}$, contradicting $(\ast )\ge 0$. Hence $m=0$ and $\zeta (1+it)\ne 0$. $\square$

Proposition 2 (single-zero exclusion with a rate). There is an absolute $c>0$ such that if ${\rho }_0={\beta }_0+i{\gamma }_0$ is a zero with $|{\gamma }_0|\ge 1$, then ${\beta }_0\le 1-c/\log (|{\gamma }_0|+2)$.

Proof. This is the Key theorem; we record the balance compactly. With $L=\log (|{\gamma }_0|+2)$, $t={\gamma }_0$, and $\sigma =1+\delta /L$, inequality $(\ast )$ together with the pole bound $3(-{\zeta }^{\prime }/\zeta (\sigma ))<3L/\delta +O(1)$, the single-zero bound $4\mathrm{Re}(-{\zeta }^{\prime }/\zeta (\sigma +it))<4A_{1}L-4/(\sigma -{\beta }_0)$, and the $2t$-bound $O(L)$ gives $0\le 3L/\delta -4/(\sigma -{\beta }_0)+A_{5}L$. Solving, $\sigma -{\beta }_0\ge 4/((3/\delta +A_5)L)$, so $1-{\beta }_0\ge (\sigma -{\beta }_0)-(\sigma -1)\ge \frac{4}{(3/\delta +A_5)L}-\frac{\delta }{L}$. The function $\delta \mapsto \frac{4}{3/\delta +A_5}-\delta$ is positive for small $\delta >0$; fixing such a $\delta$ gives $1-{\beta }_0\ge c/L$. $\square$

Proposition 3 (region $\Rightarrow$ error term). The region $\beta \le 1-c/\log (|\gamma |+2)$ implies $\psi (x)=x+O(x\exp (-c^{\prime }\sqrt{\log x}))$.

Proof. Use the truncated explicit formula $\psi (x)=x-{\sum }_{|\gamma |\le T}{x}^{\rho }/\rho +O(x{\log }^{2}x/T)$ from 21.12.01. By the region, each $|x^{\rho }|=x^{\beta }\le x\exp (-c\log x/\log (T+2))$ for $|\gamma |\le T$. The number of zeros up to height $T$ is $\ll T\log T$ and ${\sum }_{|\gamma |\le T}1/|\rho |\ll {\log }^{2}T$, so $$ \Big| \sum_{|\gamma| \leq T} \frac{x^\rho}{\rho} \Big| \ll x\exp!\Big(-\frac{c\log x}{\log T}\Big)\log^2 T. $$ Choose $T=\exp (\sqrt{\log x})$: then $\log T=\sqrt{\log x}$, the zero-sum is $\ll x\exp (-c\sqrt{\log x})\log x$, and the truncation term is $x{\log }^{2}x\exp (-\sqrt{\log x})$. Both are $\ll x\exp (-c^{\prime }\sqrt{\log x})$ for any $c^{\prime }<\min (c,1)$ and $x$ large, absorbing the polynomial factors. Hence $\psi (x)-x=O(x\exp (-c^{\prime }\sqrt{\log x}))$. $\square$

Proposition 4 (RH $\iff$ square-root error). Every substantive zero satisfies $\beta \le \Theta$ if and only if $\psi (x)-x=O(x^{\Theta +\epsilon })$ for every $\epsilon >0$.

Proof. ($\Rightarrow$) If $\beta \le \Theta$ for all $\rho$, run the proof of Proposition 3 with the region replaced by the fixed bound $x^{\beta }\le x^{\Theta }$: the zero-sum is $\ll x^{\Theta }{\log }^{2}T$ and the truncation term $\ll x{\log }^{2}x/T$; choosing $T=x$ gives $\psi (x)-x\ll x^{\Theta }{\log }^{2}x\ll x^{\Theta +\epsilon }$. ($\Leftarrow$) Conversely, suppose $\psi (x)-x=O(x^{\Theta +\epsilon })$ for all $\epsilon$ but some zero has ${\beta }_1>\Theta$. The Mellin transform ${\int }_1^{\infty }(\psi (x)-x)x^{-s-1}dx=-\frac{1}{s}\frac{{\zeta }^{\prime }}{\zeta }(s)-\frac{1}{s-1}$ converges for $\mathrm{Re}(s)>\Theta +\epsilon$ by the assumed bound, so $-{\zeta }^{\prime }/\zeta (s)$ is holomorphic there apart from the pole at $s=1$; but a zero at ${\beta }_1>\Theta$ forces a pole of ${\zeta }^{\prime }/\zeta$ at $s={\beta }_1+i{\gamma }_1$ with $\mathrm{Re}>\Theta$, a contradiction once $\epsilon <{\beta }_1-\Theta$. Hence $\beta \le \Theta$ for all zeros. Taking $\Theta =\frac{1}{2}$ recovers the Riemann hypothesis equivalence. $\square$

Connections Master

• The explicit formula $\psi (x)=x-{\sum }_{\rho }{x}^{\rho }/\rho -\log (2\pi )-\frac{1}{2}\log (1-x^{-2})$ and the partial-fraction expansion of ${\zeta }^{\prime }/\zeta$ are imported wholesale from 21.12.01; this unit supplies the missing quantitative ingredient — a bound on $\mathrm{Re}\rho$ — that turns that formula's qualitative $\psi (x)\sim x$ into an effective error term, so the two units are the two halves of the contour proof of the prime number theorem.

• The boundary nonvanishing $\zeta (1+it)\ne 0$, the functional equation, the pole at $s=1$, and the Hadamard-product order-one growth of $\xi$ all come from 06.01.16; the de la Vallée Poussin region is the effective refinement of that nonvanishing, replacing the bare statement "no zeros on the line" with "no zeros within $c/\log (|t|+2)$ of the line", and the Borel-Carathéodory and three-circles bounds on ${\zeta }^{\prime }/\zeta$ are entire-function-theory tools native to that complex-analysis chapter.

• The Vinogradov-Korobov sharpening depends on exponential-sum bounds for ${\sum }_n{n}^{-it}$ developed via the van der Corput and Vinogradov methods, the same circle-method and exponential-sum technology that powers the Hardy-Littlewood asymptotics of 21.16.02 and the large-sieve inequality of 21.14.01; the analytic estimation of $|\zeta |$ in the critical strip is a shared dependency across all of analytic number theory.

Historical & philosophical context Master

The prime number theorem was proved in 1896 independently by Jacques Hadamard and Charles-Jean de la Vallée Poussin, each establishing $\zeta (1+it)\ne 0$ and inverting the logarithmic derivative by a contour integral. De la Vallée Poussin's longer 1899 memoir ^{[de la Vallée Poussin 1899]} in the Mémoires couronnés of the Belgian Academy went further, isolating the effective region $\sigma >1-c/\log (|t|+2)$ and deducing the error term $O(x\exp (-c\sqrt{\log x}))$; this remained the best known error for nearly sixty years. The $3+4\cos \theta +\cos 2\theta$ device, which makes the nonvanishing effective, is due to the same period and is sometimes attributed to Mertens.

The next advance came in 1958, when N. M. Korobov ^{[Korobov 1958]} and I. M. Vinogradov ^{[Vinogradov 1958]}, working independently in the Soviet Union, applied Vinogradov's method of exponential sums to bound $\zeta$ near the line $\sigma =1$ and widen the region to $\sigma >1-c/(\log |t|{)}^{2/3}(\log \log |t|{)}^{1/3}$, improving the error exponent from $\frac{1}{2}$ to $\frac{3}{5}$. Davenport's Multiplicative Number Theory and Titchmarsh's The Theory of the Riemann Zeta-Function, revised by Heath-Brown, are the canonical modern expositions ^{[Titchmarsh Ch. 6]}. No unconditional improvement on the Vinogradov-Korobov exponent has been found since; under the Riemann hypothesis the error would drop to $O(x^{1/2}{\log }^{2}x)$, a target the zero-free region cannot reach by any presently known method.

Bibliography Master

@article{delavalleepoussin1899,
  author  = {de la Vall\'{e}e Poussin, Charles-Jean},
  title   = {Sur la fonction $\zeta(s)$ de Riemann et le nombre des nombres premiers inf\'{e}rieurs \`{a} une limite donn\'{e}e},
  journal = {M\'{e}moires couronn\'{e}s et autres m\'{e}moires publi\'{e}s par l'Acad\'{e}mie royale de Belgique},
  volume  = {59},
  year    = {1899},
  note    = {First effective zero-free region and PNT error term}
}

@article{korobov1958,
  author  = {Korobov, Nikolai M.},
  title   = {Estimates of trigonometric sums and their applications},
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  volume  = {13},
  number  = {4},
  pages   = {185--192},
  year    = {1958}
}

@article{vinogradov1958,
  author  = {Vinogradov, Ivan M.},
  title   = {A new estimate of the function $\zeta(1 + it)$},
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  year    = {1958}
}

@book{davenport2000,
  author    = {Davenport, Harold},
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}

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  year      = {1986},
  note      = {Revised by D. R. Heath-Brown; Ch. 3, 5, 6, 9}
}

@book{montgomeryvaughan2007,
  author    = {Montgomery, Hugh L. and Vaughan, Robert C.},
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}

@book{ivic1985,
  author    = {Ivi\'{c}, Aleksandar},
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  publisher = {Wiley},
  year      = {1985},
  note      = {Ch. 6 (zero-free regions), Ch. 12 (zero-density estimates)}
}