The Error Term in the PNT

The Prime Number Theorem says the Chebyshev weight \psi(x) = \sum_{p^k \le x}\log p is asymptotic to x: the ratio \psi(x)/x \to 1. That is a statement about a ratio, and it hides the question a number theorist actually cares about: how big is the gap \psi(x) - x? Write

\psi(x) = x + E(x),

and the whole subject becomes a hunt for the true size of the error term E(x). This is where the primes get interesting — and it is where the zeros of the zeta function stop being an abstraction and start doing visible work. The headline of this page is a single slogan:

Where the error comes from: the explicit formula

Why should the zeros of a function on the complex plane have anything to say about a sum over primes? The bridge is the explicit formula of Riemann and von Mangoldt, which is not an approximation but an exact identity:

\psi_0(x) = x - \sum_{\rho}\frac{x^{\rho}}{\rho} - \log(2\pi) - \tfrac12\log\!\left(1 - x^{-2}\right).

The leading x is the main term. The constant and the little logarithm are harmless. Everything that makes E(x) wobble lives in the sum over the non-trivial zeros \rho = \beta + i\gamma of \zeta. Take absolute values of a single term:

\left|\frac{x^{\rho}}{\rho}\right| = \frac{x^{\beta}}{|\rho|}.

Read that carefully — it is the crux of the entire page. Each zero contributes an oscillation whose amplitude is x^{\beta}, set entirely by the zero's real part \beta. The imaginary part \gamma only sets how fast it oscillates. So the error is a sum of waves, and the loudest wave is the one whose zero has the largest real part. Control the real parts, and you control the error.

The zero-free region does the work

If some zero had \beta = 1, its term would be of size x/|\rho| — as big as the main term itself, and the PNT would be false. The first thing anyone proved (Hadamard and de la Vallée Poussin, 1896) is that this cannot happen:

\sigma \;>\; 1 - \frac{c}{\log(|t| + 2)}, \qquad s = \sigma + it.

The forbidden zone is a sliver that hugs the line \Re(s)=1 and pinches inward as you climb (its width c/\log t shrinks toward zero). To turn this into a bound on E(x) you cut the zero-sum off at height |\gamma| \le T and use that every surviving zero has \beta \le 1 - c/\log T, so x^{\beta} \le x \cdot x^{-c/\log T} = x\,e^{-c\log x/\log T}. Balancing the error of the tail against this and optimising the free parameter T — the best choice turns out to be \log T \approx \sqrt{\log x} — collapses both pieces to the same size and yields the classical error term.

Trace the logic once more, because it is the template for everything that follows: a zero-free region of width \eta(t) = c/\log t feeds through the explicit formula to a saving factor \exp(-c\sqrt{\log x}). A wider \eta would give a bigger saving. The shape of the saving is the mirror image of the shape of the region.

Widening the region: Vinogradov–Korobov

The classical region's width c/\log t is not the end of the story. In the late 1950s, I. M. Vinogradov and N. M. Korobov independently found a subtler estimate for exponential sums that pushes the zero-free region slightly wider high up the strip:

\sigma \;>\; 1 - \frac{c}{(\log|t|)^{2/3}\,(\log\log|t|)^{1/3}}.

Because the region is wider, the same machinery hands back a larger saving. Running it through the explicit formula gives the best unconditional error term known today:

\psi(x) = x + O\!\left(x\,\exp\!\left(-c\,(\log x)^{3/5}(\log\log x)^{-1/5}\right)\right).

Compare the exponents: the classical saving is (\log x)^{1/2} in the exponent; Vinogradov–Korobov improves the 1/2 to 3/5 (with a small \log\log correction). That is genuinely better — but look how modest the gain is. Both are still of the shape "x times a factor that decays slower than any power of x." Six decades of the deepest work on exponential sums nudged \tfrac12 to \tfrac35, and we are still nowhere near the square-root that the Riemann Hypothesis promises for free.

Seeing the chasm: unconditional vs RH

The words "sub-polynomial saving" versus "square-root saving" do not really land until you see the curves. The chart below is plotted on a log–log scale: the horizontal axis is \log_{10} x and the vertical axis is \log_{10} of the size of each bound. On such a plot the main term x is the straight diagonal, and the honest square-root \sqrt{x} is the shallow line of half the slope.

Notice what the two error bounds do. The unconditional bound x\exp(-c\sqrt{\log x}) peels away from the x-line so slowly that even at x = 10^{20} it has barely detached — it is still almost as large as x itself. The RH bound \sqrt{x}\,(\log x)^2 lies far below, tracking the \sqrt{x} reference line and pulling away from x at genuine half-slope. The vertical gulf between the two curves is the value of the Riemann Hypothesis, drawn to scale.

Worked example — just how weak is the unconditional saving?

Let us put numbers to it. Strip away the x common to both and compare the two saving factors: the unconditional \exp(-c\sqrt{\log x}) against the RH-scale x^{-1/2}. (Take the illustrative constant c = 1 and natural logs; only the shape matters.) Both start near 1 and shrink, but watch the rates:

xunconditional e^{-\sqrt{\log x}}RH-scale x^{-1/2}
10^{3}\approx 7.2\times 10^{-2}\approx 3.2\times 10^{-2}
10^{6}\approx 2.4\times 10^{-2}10^{-3}
10^{12}\approx 5.2\times 10^{-3}10^{-6}
10^{24}\approx 5.9\times 10^{-4}10^{-12}
10^{100}\approx 2.6\times 10^{-7}10^{-50}

The two columns tell a stark story. The RH-scale saving plummets — every time x squares, it squares — while the unconditional saving crawls: even out at x = 10^{100} it has only reached 10^{-7}, a factor the RH bound passed before x = 10^{15}. The reason is structural: \sqrt{\log x} grows so agonisingly slowly that \exp(-\sqrt{\log x}) stays maddeningly close to 1. The unconditional error term saves you almost nothing off the trivial \psi(x)=x+O(x) — it is a sub-polynomial whisper, not the roar of a genuine power of x.

Two traps sit here, and both catch people. First: it is easy to read "\psi(x) = x + O(x\,e^{-c\sqrt{\log x}})" as though the error were small. It is not small — it is only slightly less than x. Because e^{-c\sqrt{\log x}} decays slower than x^{-\varepsilon} for every \varepsilon > 0, the unconditional error is bigger than x^{1-\varepsilon} for all fixed \varepsilon. It is a country mile from the \sqrt{x} that the Riemann Hypothesis would give.

Second: keep the big-O and the big-\Omega straight. E(x)=O(f(x)) is a ceiling — the error never exceeds f. E(x)=\Omega_{\pm}(g(x)) is a floor that keeps getting hit — the error is, infinitely often, at least g in size, on both the positive and negative side. An O-result is an upper bound you prove by pushing zeros left; an \Omega-result is a lower bound you prove by exhibiting a zero (any single zero already forces oscillation). They are different animals: don't quote one when you mean the other.

The best case: assuming the Riemann Hypothesis

Suppose every non-trivial zero really does sit on the critical line, so \beta = \tfrac12 for all of them. Then every term of the zero-sum has amplitude exactly x^{1/2}, and summing them carefully (there are about \tfrac{T}{2\pi}\log T zeros up to height T) produces the sharpest error term anyone can hope for:

And the implication runs both ways — this is not merely a consequence of RH, it is equivalent to it:

The equivalence is the governing principle at its most naked. A power-saving error term of size x^{1/2+\varepsilon} is exactly the same information as "no zero has real part exceeding \tfrac12." Prove one, you have proved the other. The distribution of the primes and the geography of the zeros are two views of a single fact.

You cannot do better: the Ω-results

Is the RH bound \sqrt{x}\,\log x greedy — could the truth be even smaller, like x^{1/3}? No. The existence of zeros (there are infinitely many, of amplitude x^{1/2}) forces the error to be genuinely large infinitely often. This is Littlewood's oscillation theorem and its refinements:

Now line the two up. Under RH the ceiling on \pi(x)-\operatorname{Li}(x) is \sqrt{x}\,\log x; unconditionally the floor is \sqrt{x}\,\log\log\log x/\log x. They differ only by a factor of about (\log x)^2/\log\log\log x — a whisker, on the scale of powers of x. The \sqrt{x} is common to both the floor and the ceiling: the RH error term is essentially best possible. No amount of cleverness will ever beat \sqrt{x}, because the zeros themselves guarantee oscillations of that amplitude. The square-root is not a limitation of our methods — it is the true size of the wobble in the primes.

The \Omega_{\pm} — with both a plus and a minus — is doing real work. Each zero \rho comes with its conjugate \bar\rho, and the pair contributes a term proportional to x^{1/2}\cos(\gamma\log x - \varphi)/|\rho| — a genuine oscillation, not a one-sided drift. Because the frequencies \gamma are incommensurable, these cosines occasionally line up in phase and reinforce, pushing the error strongly positive at some x and strongly negative at others. This is why \pi(x) - \operatorname{Li}(x), though negative for every x anyone can compute, must (Littlewood) turn positive infinitely often — the first crossing hiding somewhere out past 10^{300}. The zeros do not just bound the error; they orchestrate its dance.

Every result on this page is one line of the same translation table. A zero-free region of the form \sigma > 1 - \eta(t) converts into an error term of shape x\exp\!\big(-c\,\eta(\log x)\log x\big) after optimising the cutoff (very roughly — the honest optimisation is what turns \eta = c/\log t into \exp(-c\sqrt{\log x})). Read the table both directions:

Improve the error term and you have widened the region; widen the region and you have improved the error term. They are the same theorem wearing two hats.