Riemann Hypothesis

Statement and background

The Riemann zeta function is defined for Re(s) > 1 by the Dirichlet series

ζ(s) = Σₙ₌₁^∞  1 / nˢ  =  1 + 1/2ˢ + 1/3ˢ + 1/4ˢ + …

It extends by analytic continuation to all complex s except s = 1 (a simple pole). The continuation satisfies the functional equation

ζ(s) = 2ˢ π^(s−1) sin(πs/2) Γ(1−s) ζ(1−s)

which forces the trivial zeros at s = −2, −4, −6, … (the sin factor vanishes). The non-trivial zeros all lie in the critical strip 0 < Re(s) < 1.

The Riemann Hypothesis is the conjecture that every non-trivial zero has Re(s) = 1/2 — the so-called critical line.

The first non-trivial zeros

The zeros come in complex-conjugate pairs (because ζ is real on the real axis), so we list only the upper half t > 0:

Zero #	Height t (Im part)	Real part	Notes
1	14.13472514…	0.5	First non-trivial zero, computed by Backlund (1903)
2	21.02203964…	0.5
3	25.01085758…	0.5
4	30.42487613…	0.5
5	32.93506159…	0.5
6	37.58617816…	0.5
10^13-th	~10^12	0.5	Gourdon 2004, Platt 2014

Every computed zero has been on the critical line. None has ever been found off it. But this is empirical evidence, not proof.

Historical context

1737. Euler proves ζ(s) = ∏_p (1 − p^(−s))⁻¹ for real s > 1, connecting ζ to the primes.

1859. Bernhard Riemann publishes "Über die Anzahl der Primzahlen unter einer gegebenen Größe" — eight pages, his only paper in number theory. He extends ζ to the complex plane, states the functional equation, and writes: "It is very probable that all roots [in the critical strip] are real," meaning all non-trivial zeros have Re(s) = 1/2. He notes this would imply strong control on prime counting and adds: "Of course, one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this."

1896. Jacques Hadamard and Charles-Jean de la Vallée Poussin independently prove no zeros lie on Re(s) = 1. This is enough to deduce the prime number theorem: π(x) ~ x / log x.

1900. David Hilbert lists RH as his 8th Problem at the International Congress of Mathematicians in Paris.

1914. G.H. Hardy proves infinitely many non-trivial zeros lie on the critical line. Doesn't say all.

1932. Carl Siegel publishes Riemann's previously unread research notes (the Riemann–Siegel formula) — Riemann had numerically computed the first three zeros himself.

1942. Atle Selberg proves a positive proportion of zeros lie on the critical line.

1974. Norman Levinson proves at least 1/3 of zeros are on the line.

1989. Brian Conrey raises Levinson's bound to 40%; recent work pushes past 41%.

2000. Clay Mathematics Institute announces seven Millennium Prize Problems, $1M each. RH is one.

2004. Xavier Gourdon verifies the first 10^13 non-trivial zeros lie on the critical line.

2018. Michael Atiyah, age 89, announces a claimed proof at the Heidelberg Laureate Forum. The proof relies on a "Todd function" argument that is not accepted by the community. Atiyah dies January 2019. RH remains open.

As of 2026, RH is still unproven — 167 years after Riemann's paper.

Why the line Re(s) = 1/2 matters for primes

Riemann's explicit formula writes the prime counting function as

π(x) = li(x) − Σ_ρ li(x^ρ) + lower-order terms

where the sum runs over all non-trivial zeros ρ. Each zero contributes a term of size x^(Re ρ). If RH holds (Re ρ = 1/2 for all ρ), the contributions are all O(√x), and a careful sum gives

π(x) − li(x) = O(√x · log x)         (under RH)

This is the tightest possible error bound — no smaller power of x can work, by an Omega-result of Littlewood (1914) showing the error attains size ~ √x infinitely often. Without RH, the best known bound is much weaker: roughly π(x) − li(x) = O(x · exp(−c · (log x)^(3/5))), which loses a substantial polynomial factor.

So RH is equivalent to the strongest possible bound on the error in prime counting. Hundreds of theorems in analytic number theory are stated "assume RH" and would become unconditional the moment RH is proved.

Evidence beyond computation

• Hardy 1914 — infinitely many on the line. Selberg (1942), Levinson (1974), Conrey (1989), Bui–Conrey–Young (2011), Pratt–Robles (2020) — successively stronger lower bounds on the fraction of zeros on the critical line, now above 41%.
• Density theorems. Carleson, Ingham, Huxley — zeros far from the critical line are sparse. If RH fails, the failure must be very mild.
• Montgomery–Odlyzko law (1973–1987). The spacings between consecutive zeta zeros statistically match the eigenvalue spacings of large random Hermitian matrices (GUE distribution). This is the strongest "random matrix theory" evidence and supports the Hilbert–Pólya conjecture.
• Function-field analogue. The analogue of RH for zeta functions of curves over finite fields was proved by André Weil in 1948. For varieties of any dimension, Pierre Deligne completed the proof in 1973–1974 (the Weil conjectures), winning him the Fields Medal. The function-field RH is true; the number-field RH should be true too.
• Selberg zeta function. For hyperbolic Riemann surfaces, the Selberg zeta has all its non-trivial zeros on the critical line — a consequence of the trace formula. A spectral interpretation in the number-field case would yield RH the same way.
• De Bruijn–Newman constant. Λ ≤ 0 is equivalent to RH. Brad Rodgers and Terence Tao proved (2018) Λ ≥ 0. So RH ⟺ Λ = 0. The community now believes Λ is exactly zero — a tight constraint.

What RH would unlock

• Sharp prime counting. π(x) = li(x) + O(√x log x); also the same bound for primes in arithmetic progressions (under GRH).
• Mertens function bounds. M(x) = Σ_{n ≤ x} μ(n) satisfies M(x) = O(x^(1/2+ε)) under RH — the optimal exponent (Mertens conjecture M(x) < √x was disproved by Odlyzko and te Riele 1985, but the weaker RH-bound remains plausible).
• Cramér's conjecture (related). Prime gaps p_(n+1) − p_n = O((log p_n)²) — under RH the bound is O(√p_n log p_n), still weaker than the conjectured (log p_n)².
• Polynomial-time primality. Miller's test runs in polynomial time under GRH (Miller 1976). AKS test (2002) is unconditional polynomial-time but slower in practice.
• Sharper sieve bounds. The Selberg sieve, Bombieri–Vinogradov theorem, large sieve — all have sharper conditional versions under GRH.
• Implications cascade. Hundreds of "GRH-conditional" theorems would become unconditional. Many cryptography security analyses assume GRH — proving it would tighten these analyses.

Approaches that have been tried

• Direct analytic methods. Manipulate the zeta function, its derivatives, and contour integrals. Hardy–Littlewood, Selberg, Levinson, Conrey are all in this lineage. Has yielded "infinitely many on the line" and bounds on the fraction — not the full hypothesis.
• Hilbert–Pólya / spectral interpretation. Find a self-adjoint operator whose eigenvalues are the imaginary parts of zeros. Real eigenvalues ⟹ Re(ρ) = 1/2. No such operator has been identified; Berry–Keating and Connes have proposed candidates (chaotic Hamiltonians).
• Random matrix theory. Statistical match with GUE eigenvalues is too good to be coincidence. Suggests a deep spectral origin but does not yet yield a proof.
• Function field analogue. Weil and Deligne proved RH for zeta functions over finite fields. The number-field version resists analogous methods because there is no obvious "geometric" foundation.
• De Bruijn–Newman approach. Show Λ = 0 by analytic means. Rodgers–Tao made it tight from one side (2018); the other side is RH itself.
• Operator-theoretic and physical proposals. Berry–Keating Hamiltonian H = xp, generalisations involving 1/2 + H. Connes's noncommutative geometry. All suggestive, none complete.

JavaScript — verify the first zeros numerically

// Compute |ζ(1/2 + it)| via the Riemann-Siegel formula's main sum (low precision)
// Good enough to locate the first zeros via sign changes of Z(t)
function zetaCritical(t, N = 200) {
  // Use the Dirichlet series with Euler-Maclaurin smoothing — only works for Re(s) > 1
  // Here we use a different approach: alternating series + functional equation isn't
  // demonstrated; we use the eta function and the relation eta(s) = (1 - 2^(1-s)) zeta(s)
  // for Re(s) > 0, which works on the critical line.
  let real = 0, imag = 0;
  for (let n = 1; n <= N; n++) {
    const sign = (n % 2 === 1) ? 1 : -1;
    const logn = Math.log(n);
    // n^(-(1/2 + it)) = n^(-1/2) * (cos(-t log n) + i sin(-t log n))
    const mag = Math.pow(n, -0.5);
    real += sign * mag * Math.cos(t * logn);
    imag -= sign * mag * Math.sin(t * logn);
  }
  // Divide by (1 - 2^(1 - s)), s = 1/2 + it
  const a = 1 - Math.pow(2, 0.5) * Math.cos(t * Math.log(2));
  const b =     Math.pow(2, 0.5) * Math.sin(t * Math.log(2));
  const denomSq = a * a + b * b;
  return {
    re: (real * a + imag * b) / denomSq,
    im: (imag * a - real * b) / denomSq,
  };
}

function absZetaCritical(t, N = 200) {
  const z = zetaCritical(t, N);
  return Math.sqrt(z.re * z.re + z.im * z.im);
}

// Probe near the first known zeros — should see local minima
console.log('|ζ(1/2 + 14.13i)| =', absZetaCritical(14.134).toFixed(4));   // ~ 0.0002
console.log('|ζ(1/2 + 14.00i)| =', absZetaCritical(14.000).toFixed(4));   // ~ 0.4
console.log('|ζ(1/2 + 21.02i)| =', absZetaCritical(21.022).toFixed(4));   // ~ 0.0003
console.log('|ζ(1/2 + 25.01i)| =', absZetaCritical(25.011).toFixed(4));   // ~ 0.0005

// Compare to a putative off-line point (Re = 0.6, RH says no zeros here)
function zetaAt(re, im, N = 500) {
  let r = 0, i = 0;
  for (let n = 1; n <= N; n++) {
    const sign = (n % 2 === 1) ? 1 : -1;
    const mag = Math.pow(n, -re);
    const ang = im * Math.log(n);
    r += sign * mag * Math.cos(ang);
    i -= sign * mag * Math.sin(ang);
  }
  // Divide eta by (1 - 2^(1-s))
  const a = 1 - Math.pow(2, 1 - re) * Math.cos(im * Math.log(2));
  const b =     Math.pow(2, 1 - re) * Math.sin(im * Math.log(2));
  const dSq = a*a + b*b;
  return { re: (r*a + i*b) / dSq, im: (i*a - r*b) / dSq };
}
const z06 = zetaAt(0.6, 14.134, 500);
console.log('|ζ(0.6 + 14.13i)| =', Math.sqrt(z06.re**2 + z06.im**2).toFixed(4)); // ≈ 0.6 — non-zero, as RH predicts

Common misconceptions

• "RH is just a numerical curiosity." No — RH is precisely the optimal error bound on prime counting. Hundreds of theorems in number theory are stated conditionally on RH, and resolving it would make them unconditional. The Birch–Swinnerton-Dyer conjecture, Sato–Tate, and quantitative bounds in the Langlands programme all interact with RH-type hypotheses.
• "10^13 zeros checked means it's basically proven." No. Empirical evidence at finite scale cannot rule out unreachable exotic zeros. The Skewes-style threshold for the prime-counting error switching sign is around 10^316 — far beyond computational reach. Numerical evidence is necessary but not sufficient.
• "Riemann was almost certain RH is true." He wrote it was "very probable" and noted he had set the proof aside. He explicitly said proof was elusive. Modern researchers are also split: most expect RH is true, but a vocal minority believes the lack of progress suggests it may be false (with the first off-line zero astronomically high).
• "ζ has only the zeros at −2, −4, … and on the line." Under RH, yes. Without RH, the existence of off-line zeros in the critical strip cannot be ruled out — only their density can be controlled. The question RH answers is exactly "are there any off-line zeros?"
• "GRH is the same as RH." Generalised RH applies to Dirichlet L-functions L(s, χ) — same statement, broader class. Extended RH covers Dedekind zeta of number fields. Grand RH covers automorphic L-functions. Each is harder than its predecessor; RH is the simplest case.
• "Atiyah proved it in 2018." No — Atiyah's claimed proof was not accepted by the mathematical community. He relied on a "Todd function" argument that did not hold up to scrutiny. RH remains open.