The Riemann Hypothesis

Where this starts

For a complex number s with real part greater than 1 — that is, with Re(s) > 1 — the series below converges, and its sum is the Riemann zeta function ζ(s).

\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}

convergent for Re(s) > 1

Euler's identity is the statement that this sum equals a product running over the primes, with one factor for each prime, and it holds precisely because every integer factors into primes in exactly one way. We call such a product an Euler product, which we will discuss in part (c) below. That single identity is why a question about primes can be asked as a question about a function of a complex variable. Riemann's 1859 memoir continued ζ(s) to the whole plane and observed that the distribution of primes is governed by the location of its zeros.

All the interesting zeros lie in the critical strip, and the hypothesis is that they lie on a single line down its middle. It is unproved. It is also not one conjecture but a family. The same statement can be made for the zeta function of a number field (the Extended Riemann Hypothesis), for a Dirichlet L-function (the Generalized Riemann Hypothesis), and for the L-function of an elliptic curve over ℚ, or of any automorphic form (the Grand Riemann Hypothesis).

These names are not used consistently in the literature. Many authors call every case beyond ζ itself the Generalized Riemann Hypothesis, and use GRH for all of what is split into three panels here; the finer labels are a convenience of this page, not a standard. The four statements are in any case very far from independent. A Dedekind zeta function of an abelian field factors into Dirichlet L-functions, so ERH for those fields is exactly RH together with GRH for the factors; it is the non-abelian number fields that involve Artin L-functions and the Artin holomorphy conjecture, and the Grand Riemann Hypothesis contains the rest as special cases. Pick one above.

Variant one

The Riemann zeta function

The conjecture

Every zero of ζ(s) in the strip 0 < Re(s) < 1 satisfies Re(s) = ½.

The function on a right half-plane

Defined first by a series that only converges on a half-plane, then extended everywhere else by analytic continuation — the continued function is what the hypothesis is about.

\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}

valid for Re(s) > 1

The continuation is holomorphic on all of ℂ except for a single simple pole at s = 1 with residue 1. The pole is the continuation of the logarithmic divergence of the harmonic series; read through the Euler product it is also tied to Euler's theorem that ∑_p 1/p diverges, which is a good deal sharper than saying there are infinitely many primes.

Coefficients

The most featureless coefficients possible: every one of them is 1.

Everything arithmetic in ζ(s) comes from the denominator of 1/n^s, not from the numerators. The other three variants earn their difficulty by putting real arithmetic into these coefficients.

Euler product

\zeta(s)=\prod_{p\ \mathrm{prime}}\frac{1}{1-1/p^{s}}

valid for Re(s) > 1

Expanding each factor as a geometric series and multiplying out reproduces the terms 1/n^s exactly once for each n — the product is unique factorization, written analytically.

The critical strip

The Euler product shows there are no zeros with Re(s) > 1. The functional equation below reflects that half-plane onto Re(s) < 0, where the only zeros of ζ(s) are the trivial ones at s = −2, −4, −6, … Everything unresolved is squeezed into the strip between those two half-planes, 0 ≤ Re(s) ≤ 1.

That functional equation is not a symmetry of ζ(s) itself but of a modified function. Multiplying ζ(s) by a gamma factor and a power of π produces the completed zeta function Λ(s) — “completed” because the extra factor is the missing Euler factor at infinity, the one the product over primes leaves out.

\Lambda(s)=\pi^{-s/2}\Gamma\!\left(\frac{s}{2}\right)\zeta(s)

the completed zeta function

It is Λ that satisfies the symmetry, and the symmetry is exact:

\Lambda(s)=\Lambda(1-s)

the functional equation, reflecting s about the critical line

Γ(s) has poles at 0, −1, −2, … (all simple), so Γ(s/2) has poles at s = 0, −2, −4, …. The trivial zeros of ζ(s) are exactly what cancels those poles. Away from them the gamma factor is finite and non-zero, so the non-trivial zeros of ζ(s) are precisely the zeros of Λ(s), with the same multiplicities. That is why every statement below can be made about either one.

Zeros in the strip

Imaginary parts of the first zeros above the real axis; each has a mirror image below. These are read off Odlyzko's table, which carries them to 1000 decimal places.

The function on the plane

The view below needs a word of warning, because it is not a graph of ζ(s) on the critical line. The function ζ(½ + it) for real t is complex-valued, so it has no graph over the reals. What is real there is Λ, by the functional equation: Λ(½ + it) = Λ(½ − it), and since Λ takes conjugate values at conjugate points, that forces Λ(½ + it) to equal its own conjugate. Dividing by the modulus of the gamma factor rescales it without moving any zero, and gives Z:

Z(t)=\frac{\Lambda\!\left(\tfrac12+it\right)}{\left|\pi^{-s/2}\Gamma(s/2)\right|}

evaluated at s = ½ + it. The function Z(t) is real, so |Z(t)| = |ζ(½ + it)|, and Z(t) changes sign exactly at the imaginary parts of the zeros on the critical line. For ζ(s) this is Hardy's Z-function.

So the crossings you see are genuinely the zeros; the curve is just not ζ(s) itself.

The same construction works for every L-function on this page, but it needs one more ingredient in general. A completed L-function satisfies Λ(s) = W Λ(1 − s) for a root number W of modulus 1, and when W ≠ 1 the value Λ(½ + it) need not be real at all. What is real is the ratio Λ(½ + it) / √W, and it does not matter which square root you pick — the two choices differ by a sign, so one is real exactly when the other is. Every example on this page happens to have W = ±1, since the characters are real and the curves are over ℚ, but the plotting code divides by √W for a general W.

What is known

Infinitely many zeros are on the line. Hardy, 1914. This does not rule out infinitely many off it.
A positive proportion are on the line. Levinson (1974) proved more than a third; Conrey (1989) more than two fifths; Pratt, Robles, Zaharescu and Zeindler (2020) more than five twelfths, which is the current published record. A 2026 preprint of Alpöge and Furman claims the substantially stronger proportion two thirds, simple and on the line, together with five sixths distinct — not yet refereed, so treat it as a claim rather than a fact.
No zeros on Re(s) = 1. Proved independently by Hadamard and de la Vallée Poussin in 1896, and equivalent to the prime number theorem. The best known zero-free region on the right half of the critical strip is still essentially that due to Vinogradov–Korobov, and is far from the line.
Verified very far up. Platt and Trudgian (2021) rigorously verified that every non-trivial zero β + iγ with 0 < γ ≤ 3 × 10¹² has β = ½. That the zeros down to imaginary part −3 × 10¹² in the bottom part of the critical strip all have real part ½ follows by symmetry, and there are no real zeros in the critical strip at all, since ζ(s) < 0 for 0 < s < 1.
It is worth a million dollars. One of the seven Clay Millennium Problems, and the only one also on Hilbert's 1900 list.

The hypothesis is equivalent to a sharp error term in the prime counting function: π(x) = li(x) + O(√x log x). Loosely, RH says the primes are as evenly distributed as they could possibly be.

Dedekind zeta functions

The conjecture

For every number field K, every zero of ζ_K(s) in the strip 0 < Re(s) < 1 satisfies Re(s) = ½.

Replace ℤ by the ring of integers of a number field and the whole construction goes through — with one complication that is the reason to care. In a general ring of integers, numbers do not factor uniquely. The example here is K = ℚ(√−5), the standard smallest-radicand example in which unique factorization of elements fails. (It is not the smallest by discriminant: ℚ(√−15) has discriminant −15 and class number 2 as well.)

6=2\cdot 3=\left(1+\sqrt{-5}\right)\left(1-\sqrt{-5}\right)

two genuinely different factorizations into irreducibles

The function on a right half-plane

Kummer and Dedekind's repair was to factor ideals instead of numbers, and for ideals factorization is unique again. So the zeta function of K counts ideals, graded by their norm.

\zeta_K(s)=\sum_{I\neq 0}\frac{1}{N(I)^{s}}

I runs over the non-zero ideals of the ring of integers; Re(s) > 1

How far unique factorization fails is measured by the class number h, the size of the group of ideals modulo principal ones. For ℚ(√−5), h = 2; for ℚ(√−23), h = 3; for ℚ(√−3), h = 1 and factorization does work. All three are available in the plot below.

Coefficients

Here a_n counts the ideals of norm exactly n. A zero means no ideal has that norm.

a₁₁ = 0 because 11 stays prime in ℤ[√−5]: the only ideal above it has norm 11² = 121. a₃ = 2 because 3 splits into two distinct primes, each of norm 3. a₂ = 1 because 2 ramifies — the prime (2, 1+√−5) appears squared.

Euler product

\zeta_K(s)=\prod_{P\neq 0}\frac{1}{1-1/N(P)^{s}}

P runs over the nonzero prime ideals — not the prime numbers p

Grouping the prime ideals by the rational prime underneath them gives the identity that makes this variant computable. Each rational prime either splits into two primes of norm p, stays inert as a single prime of norm p², or ramifies — and which one happens is recorded by the quadratic character χ_D of the discriminant.

\zeta_K(s)=\zeta(s)\,L(s,\chi_D)

for any quadratic field of discriminant D

This is worth pausing on, because it settles what ERH does and does not add. For an abelian field, the Dedekind zeta function is a product of Dirichlet L-functions, so ERH for it is exactly RH together with GRH for those factors — no new content. The real weight of ERH sits on the non-abelian fields, where the factors are Artin L-functions that are not even known to be entire.

The critical strip

Same strip, same line, and the same two-step construction: first complete the function, then state the symmetry. The trivial zeros here fill in all the negative integers — the even ones inherited from ζ and the odd ones from L(s, χ_D).

\Lambda_K(s)=|d_K|^{s/2}(2\pi)^{-s}\Gamma(s)\zeta_K(s)

the completed Dedekind zeta function

The conductor here is the absolute discriminant |d_K|, and there is a single Γ-factor because an imaginary quadratic field has one complex place and no real ones. As before, it is the completed function that satisfies the symmetry:

\Lambda_K(s)=\Lambda_K(1-s)

the functional equation

There is still a simple pole at s = 1, and its residue is not an accident — it is the analytic class number formula, which ties the analysis directly to the arithmetic the field started with.

\lim_{s\to 1}(s-1)\zeta_K(s)=\frac{2\pi h}{w\sqrt{|d_K|}}

h the class number, w the roots of unity

Zeros in the strip

Because ζ_K = ζ · L(s, χ_D), the zeros are the two lists interleaved. Below they are shown separately so you can see which is which; in the plot they are one set of points on one line.

The merged column stops where it does on purpose. Interleaving two finite lists only stays correct up to the last entry of the shorter one; past that, a merge would quietly turn into “the remaining tabulated zeros of ζ”, skipping the zeros of L(s, χ_D) that belong between them. So the merge is cut at that height, and the plot marks no zeros above it either. What is listed is the correctly interleaved tabulated critical-line zeros — complete in the sense of matching every crossing a fine scan of Z finds below that height, which is not the same as a proof that none were missed.

The function on the plane

What is known

The class number formula is a theorem. Dirichlet for quadratic fields, Dedekind in general. It is what turns L(1, χ_D) into an arithmetic quantity: for ℚ(√−5) it equals π/√5 = 1.404962946…, and the 2 hiding inside it is the class number.
The quotient is entire. Aramata and Brauer proved that ζ_K / ζ is entire whenever K/ℚ is Galois. For non-Galois K this is still open in general — it is Dedekind's conjecture, and it follows from Artin's holomorphy conjecture.
ERH makes class groups cheap to compute. Bach proved that under ERH the class group is generated by the prime ideals of norm at most 12(log |d_K|)². Unconditionally one falls back on the Minkowski bound, which is enormously larger. That bound is one reason many large class group computations are run under ERH; unconditional certification is possible, but generally far more expensive.
And sharpens Chebotarev. Under ERH the Chebotarev density theorem gives much cleaner bounds for the norm of the smallest prime with a prescribed Frobenius (Lagarias–Odlyzko). Unconditionally, effective bounds still exist — Lagarias, Montgomery and Odlyzko proved one for the least prime ideal in 1979 — but they are substantially weaker, because they must accommodate the possibility of an exceptional real zero.
Checked numerically, field by field. As with ζ there is no proof, only computation: LMFDB stores computed zeros for a large number of number fields. Whether a given dataset also carries a rigorous completeness certificate is a separate question, and should not be assumed.

Dirichlet L-functions

The conjecture

For every Dirichlet character χ, every zero of L(s, χ) in the strip 0 < Re(s) < 1 satisfies Re(s) = ½.

The restriction to the open strip is doing quiet work. If χ is imprimitive, induced by a character of smaller conductor, its L-function differs from the primitive one by finitely many Euler factors (1 − χ₀(p)p^−s), and each of those contributes zeros on the line Re(s) = 0. Stating the conjecture about the interior of the strip leaves those untouched. The three characters used here are all primitive, so the question does not arise for them.

Where ζ counts all integers equally, these functions sort them by remainder. The example here is the quadratic character of conductor 3, the one attached to the field ℚ(√−3).

The function on a right half-plane

Let χ be the non-trivial character modulo 3: it sends n to +1 when n ≡ 1, to −1 when n ≡ 2, and to 0 when 3 divides n. It is the Kronecker symbol (−3 / ·), and it is odd: χ(−1) = −1.

L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}

valid for Re(s) > 1

Because χ is non-trivial its partial sums stay bounded, so partial summation already gives convergence, and hence holomorphy, on Re(s) > 0. That alone does not continue it to the whole plane. Writing it as a finite combination of Hurwitz zeta functions does — and in that combination the pole each piece has at s = 1 cancels, because ∑χ(a) = 0. So L(s, χ) is entire, which is exactly what makes the next fact possible.

That cancellation is not just theory here: it is literally how this page evaluates the function, and subtracting the pieces after dividing rather than before gives ∞ − ∞ at s = 1.

Coefficients

Periodic with period 3, and that is the whole of it.

Euler product

L(s,\chi)=\prod_{p\neq 3}\frac{1}{1-\chi(p)/p^{s}}

valid for Re(s) > 1

A prime p > 3 contributes a factor that depends only on whether p is 1 or 2 mod 3 — equivalently, by quadratic reciprocity, on whether −3 is a square mod p, which is whether p splits in ℚ(√−3).

The critical strip

Same strip, same line, a different gamma factor: because χ is odd the completed function uses (s+1)/2 rather than s/2, which moves the trivial zeros to the negative odd integers −1, −3, −5, …

\Lambda(s,\chi)=\left(\frac{3}{\pi}\right)^{(s+1)/2}\Gamma\!\left(\frac{s+1}{2}\right)L(s,\chi)

the completed Dirichlet L-function, conductor 3

The root number of this character is +1, so the functional equation carries no extra factor and Λ is symmetric about the critical line exactly as ζ's is:

\Lambda(s,\chi)=\Lambda(1-s,\chi)

the functional equation

Zeros in the strip

Computed in Magma and agreeing with LMFDB's entry 1-3-3.2-r1-0-0 to every digit it prints.

The function on the plane

The other two options are the characters of conductor 20 and 23 — the ones attached to the two fields in the ERH panel. Notice how much lower the first zero sits as the conductor grows: 8.04 for conductor 3, 2.36 for conductor 20, 2.87 for conductor 23. Zeros are packed more densely when the conductor is larger.

What is known

L(1, χ) ≠ 0. Dirichlet, 1837 — and this non-vanishing is the proof that every arithmetic progression a, a+q, a+2q, … with gcd(a, q) = 1 contains infinitely many primes.
Its value is a class number. For this character the analytic class number formula gives L(1, χ) = 2πh / (w√3) = π/(3√3) = 0.604599788…, using h = 1 and w = 6 roots of unity in ℚ(√−3). Hover the plot at s = 1 and you are reading off a class number.
Siegel zeros are the gap. Unconditionally, one cannot uniformly exclude the possibility that some real primitive Dirichlet character has an exceptional real zero extremely close to s = 1. For any one character written down explicitly the question can in principle be settled by computation; it is across the whole family that it stays open. GRH would kill it instantly; without GRH the best bounds (Siegel's) are ineffective, meaning they contain a constant no one can compute.
A great deal is conditional on Riemann hypotheses — but not all on this one. GRH for Dirichlet L-functions is enough to make Miller's primality test deterministic in polynomial time: it bounds how far you must go through Miller–Rabin bases before an odd composite is certain to have its compositeness exposed (Miller, 1976; Bach, 1990). Miller's own bound on the first witness was O((log n)²) with no explicit constant, so the test was polynomial time in theory but not yet usable; Bach showed the constant may be taken to be 2, which is what made it a practical algorithm. Hooley's conditional proof of Artin's primitive root conjecture (1967) needs more — the Riemann hypothesis for the Dedekind zeta functions of the fields ℚ(ζ_n, a^1/n), which are not abelian in general. Effective Chebotarev for a general extension likewise needs ERH for its Dedekind zeta function, or for the relevant Artin L-functions, rather than Dirichlet GRH alone. The hypotheses in this family are not interchangeable, and it is worth keeping track of which one a given theorem actually uses.

Automorphic L-functions

The conjecture

For every unitary cuspidal automorphic representation π of GL_n over a number field, every non-trivial zero of its standard L-function, in the unitary normalization, has real part ½.

This version contains the other three as special cases. ζ is the standard L-function of the trivial GL₁ character over ℚ; more generally ζ_K is the corresponding GL₁ example over a number field K; Dirichlet L-functions come from the other GL₁ characters; and a modular elliptic curve gives a GL₂ example. The example here is the curve of conductor 11 — the smallest conductor there is.

“Modular” is doing real work in that sentence. Every elliptic curve over ℚ is modular, so over ℚ the restriction costs nothing. Over a general number field the curve still has a Hasse–Weil L-function — the Euler product is defined regardless — but automorphy is not known for every such curve, so that L-function cannot always be placed unconditionally inside the automorphic framework this conjecture is stated in, and with it go the analytic continuation and functional equation. The examples below are over ℚ.

A warning about normalization. For an elliptic curve the natural variable puts the center at s = 1 and the strip at ½ < Re(s) < 3/2; this is what the plot below uses. Replacing s by s + ½ slides everything back to the familiar picture with the line at ½. Both conventions are in constant use.

The function on a right half-plane

Take the elliptic curve 11a1 (LMFDB 11.a2), the curve y² + y = x³ − x² − 10x − 20, of conductor 11. For each prime of good reduction count its points mod p and record the deficit.

L(E,s)=\sum_{n=1}^{\infty}\frac{a_n}{n^{s}}

where a_p = p + 1 − #E(𝔽_p) at primes of good reduction; at bad primes a_p ∈ {0, ±1} is determined by the reduction type. The remaining a_n are then determined by the Euler product below. valid for Re(s) > 3/2

That the series continues to the whole plane at all is the modularity theorem: L(E, s) equals the L-function of a weight-2 newform of level 11. This is the hardest input on the page, and it was only proved for all elliptic curves over ℚ in 2001.

Coefficients

Genuinely arithmetic, and no longer bounded: Hasse's theorem pins |a_p| ≤ 2√p, and how they fluctuate inside that range is itself a deep question (Sato–Tate).

Euler product

L(E,s)=\prod_{p\nmid N}\frac{1}{1-a_p/p^{s}+p/p^{2s}}\prod_{p\mid N}\frac{1}{1-a_p/p^{s}}

degree 2 at good primes, degree at most 1 at bad primes — degree 1 for this curve

The quadratic factor is what “degree 2” means, and it is why the zeros are packed asymptotically about twice as densely as ζ's — the degree controls the leading term in the zero-counting formula, though at low height the conductor matters too.

The bad primes are worth a second look. There a_p is +1 for split multiplicative reduction, −1 for non-split, and 0 for additive — so the local factor has degree 1 in the first two cases and degree 0 in the third, where it collapses to 1. Both curves here are semistable, with square-free conductor and multiplicative reduction at their one bad prime: a₁₁ = +1 for 11a1 and a₃₇ = −1 for 37a1.

The critical strip

\Lambda(s)=N^{s/2}(2\pi)^{-s}\Gamma(s)L(E,s)

the completed L-function of E, whose conductor is N

Here the functional equation does carry a factor. The root number ε = ±1 is determined by the curve, and the reflection is about s = 1 rather than ½, because this is the arithmetic normalization:

\Lambda(s)=\varepsilon\,\Lambda(2-s)

ε = +1 for 11a1, −1 for 37a1

That sign decides the parity of the order of vanishing at the center. When ε = −1 the completed function is odd about s = 1 and therefore must vanish there, which is why 37a1 has a zero sitting exactly at the center. When ε = +1 the order is even — but even includes 2, 4, …, so the sign alone does not say the function is non-zero. That 11a1 has no central zero is a separate computation: L(E, 1) = 0.2538418608…, which you can read off the plot by hovering at s = 1.

Zeros in the strip

From LMFDB, cross-checked against Magma. Note the entry at t = 0 for 37a1: a zero at the central point s = 1 itself.

The function on the plane

What is known

The functions exist. For elliptic curves over ℚ, analytic continuation and the functional equation follow from modularity: Wiles and Taylor–Wiles (1995) for the semistable case, completed by Breuil, Conrad, Diamond and Taylor (2001). For the standard L-functions of cuspidal representations of GL_n it is Godement–Jacquet; the trivial GL₁ representation is the familiar case with a pole.
The center is understood in low rank. Birch and Swinnerton-Dyer predict that the order of vanishing at s = 1 equals the rank of E(ℚ). When the analytic order is 0 or 1 this is a theorem — Gross–Zagier and Kolyvagin. 11a1 has rank 0 and L(E, 1) = 0.2538418608…; 37a1 has rank 1 and vanishes there.
No zero-free region reaching the central line. Non-vanishing on the boundary Re(s) = 3/2 is known, and quantitative zero-free regions just inside it are available. What is open is whether every non-trivial zero in the strip lies on Re(s) = 1 — not the weaker claim that nothing at all is known inside.
The statistics look right. Numerical data for high zeros agree strikingly with the eigenvalue statistics of random unitary matrices, and for low-lying zeros in families there are rigorous results matching the Katz–Sarnak predictions for suitably restricted test functions. The full conjectural picture is not a theorem, but it is evidence for the spectral interpretation that a proof would presumably need.
Maass forms are the other GL₂ case. For a Maass cusp form on the modular surface, Grand RH is equally open, and it sits beside a second unproved statement about the same objects: the Ramanujan–Petersson conjecture, where the best bound is Kim–Sarnak's 7/64. Those L-functions need spectral parameters that this page does not carry, so the live example here stays with the elliptic curves.

Compute it yourself

The numerical values and critical-line crossings shown on this page can be reproduced with the scripts below, which locate zeros by detecting sign changes of a real-valued function on the line.

What is shown here is a short illustration of the method, over modest heights. The full scripts in the repository scan the complete range of every table on this page. The tabulated ordinates themselves are Odlyzko's and LMFDB's; the scripts reproduce them rather than replace them.

What that does and does not establish. For the examples and ranges shown, the ordinates it finds agree with the Odlyzko and LMFDB data listed above — a numerical reproduction of those zeros. It is not a proof that the list is complete: a sign change detects a zero of odd order, so it can miss a zero of even multiplicity, or an even number of zeros falling inside one step, and a zero exactly at the central point has to be tested separately (the script does). Nor does searching the line say anything about zeros off it. Establishing completeness rigorously needs a zero-counting argument — Turing's method, which is what LMFDB's certified zeta-zero dataset uses.

The Riemann Hypothesis

The Riemann zeta function

The function on a right half-plane

Coefficients

Euler product

The critical strip

Zeros in the strip

The function on the plane

What is known

Further reading

Dedekind zeta functions

The function on a right half-plane

Coefficients

Euler product

The critical strip

Zeros in the strip

The function on the plane

What is known

Further reading

Dirichlet L-functions

The function on a right half-plane

Coefficients

Euler product

The critical strip

Zeros in the strip

The function on the plane

What is known

Further reading

Automorphic L-functions

The function on a right half-plane

Coefficients

Euler product

The critical strip

Zeros in the strip

The function on the plane

What is known

Further reading

Compute it yourself