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Library · docs/08-why-it-is-hard.md

Why It Is Hard: An Honest Failure Catalogue

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The short version

Every technique in this repository — Mellin transforms, the functional equation, contour integration, the explicit formula — is genuinely powerful, and every one of them has a known ceiling that sits strictly below the Riemann Hypothesis. We can prove there are no zeros in a region that hugs the line Re s = 1 and pinches shut as you go up; we cannot widen it to a fixed strip. We can prove that a positive proportion — currently a bit over 5/12 of them — lie exactly on the critical line; we cannot get to all, and the gap between "positive proportion" and "all" is not a matter of pushing harder. Sieve and elementary methods hit the parity problem, which is a theorem about their limits, not a lack of ingenuity. Numerics are useless as evidence in a way that has been demonstrated repeatedly, most brutally by the Mertens conjecture. And two families of counterexamples — Davenport–Heilbronn / Epstein on one side, Beurling generalised primes on the other — show that the functional equation alone cannot imply RH and the Euler product alone cannot either. If your argument uses only one of them, there is a concrete function it also applies to, and that function has zeros in the wrong place.

That last point is the single most useful fact in this document. Everything else is commentary.

1. The scoreboard: what transform methods actually deliver

1.1 Zero-free regions

THEOREM (Hadamard, de la Vallée Poussin, 1896). zeta(1 + it) != 0 for all real t. This gives the Prime Number Theorem.

THEOREM (de la Vallée Poussin). There is c > 0 such that zeta(s) != 0 in

    sigma  >=  1 - c / log(|t| + 2)

THEOREM (Vinogradov and Korobov, independently, 1958). The region can be widened to

    sigma  >=  1 - c / ( (log|t|)^(2/3) (log log|t|)^(1/3) )

The shape of this region has not been improved since 1958. Only the constant has moved — the standard explicit version is due to Ford (2002), commonly quoted with c = 1/57.54; check the paper before relying on that number.

The engine behind all of these is the "3-4-1" inequality 3 + 4 cos(theta) + cos(2 theta) = 2(1 + cos theta)^2 >= 0, applied to log zeta(s) = sum_{p,k} p^(-ks)/k. That series is the Euler product. Remember this; §4 is about what happens without it.

Now look at the scale. RH asks for a zero-free region of fixed width 1/2. What we have is a region whose width tends to 0. At t = 10^12, 1/log t ≈ 0.036 and the Vinogradov–Korobov width is ≈ 0.073. At t = 10^100 they are 0.0043 and 0.015. These are not 90% of the way to RH; in the relevant sense they are 0% of the way, because no amount of improvement to a shrinking width ever produces a fixed strip. This is a qualitative wall, not a quantitative one.

1.2 Zero-density estimates

Let N(sigma, T) count zeros with Re rho >= sigma and 0 < Im rho <= T. RH says N(sigma, T) = 0 for every sigma > 1/2. The realistic goal is to prove there are few.

THEOREM (Ingham, 1940). N(sigma, T) << T^(3(1-sigma)/(2-sigma)) (log T)^5 for 1/2 <= sigma <= 1.

CONJECTURE (Density Hypothesis). N(sigma, T) << T^(2(1-sigma)+eps).

THEOREM (Guth–Maynard, arXiv:2405.20552, May 2024; I believe it has appeared in the Annals of Mathematics — verify the venue). N(sigma, T) <= T^(30(1-sigma)/13 + o(1)) for 3/4 <= sigma <= 1. At the critical case sigma = 3/4 this replaces Ingham's exponent 3/5 = 0.6 with 15/26 ≈ 0.5769 — the first improvement there since 1940. The advertised arithmetic payoff is asymptotics for primes in intervals of length x^(17/30 + o(1)), improving the previous x^(7/12).

Note what a density estimate is: permission for exceptional zeros to exist, provided they are rare. Even the full Density Hypothesis — far beyond what is known — would not exclude a single off-line zero. And a single one is fatal for the sharpest applications: if Theta = sup{Re rho}, then

    psi(x) - x  =  Omega_pm( x^(Theta - eps) )      for every eps > 0

so the error term in the Prime Number Theorem is governed by the supremum of the real parts, which one bad zero determines all by itself. docs/04-explicit-formula.md and zeta/explicit.py show exactly where this x^rho sensitivity enters.

1.3 A positive proportion on the line — and why that is not "all"

THEOREM (Hardy, 1914). Infinitely many zeros lie on the critical line.

THEOREM (Hardy–Littlewood, 1921). At least cT of them up to height T.

THEOREM (Selberg, 1942). A positive proportion of all zeros lie on the line.

THEOREM (Levinson, 1974). More than 1/3 (about 34.7%).

THEOREM (Conrey, 1989). More than 2/5 (the paper's precise proportion is commonly quoted as just under 41%; check it before citing a decimal).

THEOREM (Pratt, Robles, Zaharescu, Zeindler; arXiv 2018, published in Research in the Mathematical Sciences — 2019 or 2020 depending on which record you trust). More than 5/12 ≈ 41.67%.

Here is the part people misread. N(T) ~ (T/2pi) log(T/2pi) grows without bound, so "41.67% are on the line" leaves infinitely many zeros unaccounted for — roughly 58% of an unbounded quantity. Even a hypothetical theorem giving 99.999% would leave an infinite exceptional set, and by §1.2 the error term in the PNT is decided by whichever exceptional zero has the largest real part. Proportion theorems say nothing whatsoever about Theta. They are progress on a different question.

Why can't the method just be pushed to 100%? Levinson's technique is a mollified second moment: you multiply zeta by a Dirichlet polynomial of length x = T^theta chosen to damp its fluctuations, then count sign changes. The proportion you extract increases with theta. Levinson had theta < 1/2; Conrey reached theta < 4/7 by importing bounds on sums of Kloosterman sums (Deshouillers–Iwaniec); every subsequent gain has come from new arithmetic input of the same kind, not from better bookkeeping. It is generally stated — I believe the precise result is due to Farmer in the early 1990s, and you should check the reference before quoting it — that letting theta -> infinity would give 100%. That is exactly the point: 5/12 is a proxy for "how much cancellation in twisted moments we currently know how to prove", and each increment is a hard theorem in its own right. There is no visible route by which finitely many such increments reach theta = infinity.

2. The parity problem

Sieve methods start from an axiom set: a sequence A, and estimates for how much of A survives sifting by each prime. Selberg's parity obstruction is the fact that these axioms cannot distinguish integers with an even number of prime factors from those with an odd number. Selberg exhibited sequences satisfying perfectly good sieve axioms in which every surviving element has an even number of prime factors — so no argument using only those axioms can ever produce a prime. Bombieri's asymptotic sieve (1976) is the standard formalisation of where the limit sits.

This is not a side issue for RH; it is central. RH is equivalent to

    M(x) = sum_{n <= x} mu(n)  =  O( x^(1/2 + eps) )       for every eps > 0

and mu(n) on squarefree n is exactly (-1)^(number of prime factors). Proving RH by elementary or sieve-theoretic means requires detecting precisely the cancellation the parity problem says those methods are blind to. It is why sieves deliver almost-primes (Chen: infinitely many primes p with p + 2 having at most two prime factors) rather than primes.

Two honest caveats. First, "elementary" does not mean "weak" — the Erdős–Selberg elementary proof of the PNT (1948–49) is a standing rebuke to that idea. Second, parity can be broken, but only by injecting information the axioms do not contain: Friedlander–Iwaniec (Annals, 1998) proved there are infinitely many primes of the form a^2 + b^4, and Heath-Brown (2001) did x^3 + 2y^3, both by supplying bilinear ("Type II") estimates from outside the sieve. The lesson is structural: you get past parity by adding new arithmetic, never by rearranging the sieve.

3. Why numerical verification cannot help

3.1 What has been checked

THEOREM (Platt–Trudgian, Bull. London Math. Soc. 53 (2021), 792–797). Every zero with 0 < Im rho <= 3·10^12 has Re rho = 1/2, verified rigorously with interval arithmetic. By the Riemann–von Mangoldt formula that is about 1.24·10^13 zeros — zeta.zeros.riemann_von_mangoldt(3e12) returns 1.236·10^13 instantly; N_of_T computes the exact count, but only at heights your laptop can actually reach. Gourdon's earlier (2004) computation reached the first 10^13 zeros, at height about 2.45·10^12, to a lower standard of rigour. Odlyzko has computed zeros in windows far higher, around the 10^22-nd zero.

The verification method is in zeta/zeros.py (verify_rh_up_to): count the zeros in the strip with N(T) = 1 + theta(T)/pi + S(T) — an identity, not an asymptotic, so it counts off-line zeros too — count sign changes of Hardy's Z(t) on the line, and check the two agree. If they do, every zero below T is simple and on the line.

3.2 The relevant quantity crawls

Fluctuations in the zero counting function are carried by S(T) = (1/pi) arg zeta(1/2 + iT).

THEOREM (Selberg). S(T) is asymptotically normally distributed with variance ~ (1/(2 pi^2)) log log T.

log log is the slowest function in serious mathematics. At T = 10^13 the typical size of S(T) is sqrt( log log T / (2 pi^2) ) ≈ 0.415. At T = 10^22 it is ≈ 0.446. To merely double it relative to height 10^13, you need T ≈ 10^(3.5·10^5) — a number with about 350,000 decimal digits. Every computation ever performed lives in the first flat inch of a curve that has to climb forever. (Three lines of mpmath; check it rather than believing me.)

You can already watch a small-scale version of this fail. Gram's law — "the n-th zero lies between consecutive Gram points" — holds for the first 125 Gram points and then breaks: zeta.zeros.gram_law_violations(0, 200) returns [126, 134, 195]. A pattern with 125 consecutive confirmations was still false.

3.3 Littlewood, Skewes, and the death of numerical intuition

Every computation of pi(x) ever performed has found pi(x) < Li(x).

THEOREM (Littlewood, 1914). pi(x) - Li(x) changes sign infinitely often; more precisely pi(x) - Li(x) = Omega_pm( x^(1/2) (log log log x) / log x ).

So the numerical pattern is known to be a lie about the asymptotics, and the first crossover is grotesque: Skewes bounded it by e^(e^(e^79)) assuming RH (1933) and by e^(e^(e^(e^7.705))) unconditionally (1955). Modern work locates a crossing region near 1.398·10^316 (Bays–Hudson, 2000, later refined slightly by Chao–Plymen and by Demichel) — still hopelessly beyond computation, and not proved to be the first crossing. Compare zeta.explicit.li and zeta.explicit.pi_true: you can see the bias in the data, and the bias is not the truth.

3.4 Mertens: an overwhelming numerical pattern that was simply false

The Mertens conjecture |M(x)| < sqrt(x) was proposed on strong numerical evidence and would have implied RH (and the simplicity of the zeros).

THEOREM (Odlyzko–te Riele, J. reine angew. Math. 357 (1985), 138–160). It is false. The proof is non-constructive — no explicit counterexample is known to this day. Explicit upper bounds for the first counterexample have been pushed down to roughly exp(1.59·10^40) (commonly attributed to Kotnik–te Riele, 2006).

This is the cleanest available warning. A statement stronger than RH, supported by everything anyone could compute, was wrong — and wrong at a scale no computation will ever reach.

4. The counterexamples that should govern your intuition

4.1 Functional equation alone: Davenport–Heilbronn

THEOREM (Davenport–Heilbronn, "On the zeros of certain Dirichlet series", J. London Math. Soc. 11 (1936), two papers). There is an explicit Dirichlet series — a particular linear combination of two Dirichlet L-functions mod 5, with coefficients chosen so the phases in the functional equation cancel — which:

Bombieri–Ghosh, "Around the Davenport–Heilbronn function" (Russian Math. Surveys 66 (2011)) study it in depth: a positive proportion of its zeros lie on the line, and RH is still false for it.

The one thing it lacks is an Euler product.

Therefore: any proposed proof of RH that uses only the functional equation, the order of growth, the reality of xi(1/2 + it), the Hadamard product, or the symmetry rho -> 1 - rho of the zero set, is wrong. All of those hold for the Davenport–Heilbronn function. This is the fastest available test of a proposed proof, and it kills the large majority of them. If you cannot point at the step where your argument would break for Davenport–Heilbronn, you have not proved anything. docs/03-functional-equation.md derives, carefully and at length, exactly the structure that is not enough.

The same lesson arrives independently from Epstein zeta functions. For a positive definite binary quadratic form Q, the series zeta(s, Q) = sum' Q(m,n)^(-s) satisfies a clean Riemann-type functional equation. When the class number of the discriminant is 1, zeta(s, Q) factors as zeta times an L-function — it inherits an Euler product, and it behaves. When the class number exceeds 1, Davenport and Heilbronn proved it has infinitely many zeros in Re s > 1. The functional equation is the same in both cases; the arithmetic is not.

4.2 Euler product alone: Beurling generalised primes

Now run the experiment the other way. A Beurling system is an arbitrary sequence 1 < p_1 <= p_2 <= ... of "generalised primes" together with the multiplicative semigroup of "generalised integers" they generate, with counting function N_B(x). Its zeta function has an Euler product by construction. What it does not have is the specific arithmetic of Z — in particular, no functional equation.

THEOREM (Diamond–Montgomery–Vorhauer, "Beurling primes with large oscillation", Math. Ann. 334 (2006), 1–36). There is a Beurling system with N_B(x) = kappa·x + O(x^theta) — generalised integers as regularly distributed as you could reasonably ask for — whose zeta function has infinitely many zeros on the curve sigma = 1 - a/log t, and whose prime counting function oscillates: pi_B(x) = li(x) + Omega( x exp(-c sqrt(log x)) ).

Read that carefully. With an Euler product and PNT-quality regularity of the integers but nothing else, the classical de la Vallée Poussin zero-free region is optimal — not merely unimproved. So the Euler product by itself cannot even deliver Vinogradov–Korobov, let alone RH.

4.3 The two-sided test

Put §4.1 and §4.2 together and you get a filter that costs nothing to apply:

    uses the functional equation only     ->  Davenport-Heilbronn / Epstein refute it
    uses the Euler product only           ->  Beurling systems refute it
    must use BOTH, entangled              ->  the actual difficulty

No known technique entangles them at RH strength. That, stated plainly, is why the problem is open.

5. The graveyard, stated respectfully

Claimed proofs of RH appear at a rate of dozens per year on arXiv and elsewhere; Matthew Watkins has long maintained a public catalogue of proposed proofs and disproofs. The overwhelming majority fail the §4.3 test within the first paragraph.

Serious mathematicians are not exempt. Louis de Branges — who proved the Bieberbach conjecture in 1984, a first-rate theorem — has announced approaches to RH repeatedly over several decades. Conrey and Li ("A note on some positivity conditions related to zeta- and L-functions", IMRN 2000) gave examples showing that the positivity conditions his method requires are not satisfied in the setting relevant to zeta. De Branges has disputed the relevance of those examples; as of this writing no proof along those lines has been accepted. Michael Atiyah presented a claimed proof at the Heidelberg Laureate Forum in September 2018; it was not accepted by the community.

None of this warrants mockery. It warrants internalising what specialists say a proof must contain. Conrey's survey "The Riemann Hypothesis" (Notices of the AMS, March 2003) is the best short statement of the state of play, and it is explicit that any successful method must be sensitive to the Euler product, precisely because of Davenport–Heilbronn. Bombieri's official Clay Mathematics Institute problem description makes the same structural point. Read both before writing anything.

6. If you want to work on this: what is actually tractable

Genuinely open, genuinely approachable, roughly ordered by how fast a non-specialist can start:

  1. Rigorous verification and explicit constants. Interval-arithmetic zero verification (Platt–Trudgian style) and explicit versions of classical estimates feed directly into real theorems — Helfgott's proof of the ternary Goldbach conjecture (2013) relies on verified zero computations for Dirichlet L-functions. Unglamorous, permanently useful. Start from zeta/zeros.py.
  1. Statistics of zeros. Montgomery's pair correlation (1973) and Odlyzko's numerics support the GUE CONJECTURE; nearly everything here is open and the numerical side is accessible. docs/06-hilbert-polya-and-gue.md and zeta/statistics.py.
  1. The de Bruijn–Newman constant. Lambda <= 1/2 (de Bruijn, 1950); Lambda >= 0 is a THEOREM (Rodgers–Tao; announced 2018, published 2020), so RH is now exactly Lambda = 0. Polymath15 (2018) drove the upper bound to Lambda <= 0.22, sharpened to Lambda <= 0.2 by Platt–Trudgian (2021) — see docs/05 §3 for both, and note neither is strict. Polymath15 was a real, open, collaborative project in which a competent programmer could and did contribute compute and code. See docs/05-de-bruijn-newman.md and zeta/heatflow.py.
  1. The Guth–Maynard large-values method (2024). New, actively being extended, and its analogues for Dirichlet L-functions and other families are open. The liveliest technical frontier on the density side.
  1. Moments of zeta. The second (Hardy–Littlewood, 1918) and fourth (Ingham, 1926) moments are THEOREMs; the Keating–Snaith random-matrix CONJECTURE (2000) and the CFKRS refinement (2005) predict all of them. Sharp upper bounds under RH (Soundararajan, 2009; sharpened by Harper) and unconditional lower bounds (Radziwiłł–Soundararajan) exist. The sixth moment is open.
  1. Function-field analogues. RH for curves over finite fields is a THEOREM (Weil, 1948; Deligne, 1974 for the general Weil conjectures). Its proof uses positivity from intersection theory with no known number-field counterpart; understanding exactly why it does not transfer is a research programme in itself. Katz–Sarnak (1999) established the random-matrix symmetry predictions in that setting.
  1. L-function analogues. Test any idea against Dirichlet L-functions, modular L-functions, the Selberg class. If a phenomenon is about zeta specifically and not about the whole family, that is information — and if it is about the whole family, §4.1 is waiting for you.
  1. Computable equivalences. The Nyman–Beurling criterion, and Báez-Duarte's reformulation, express RH as a Hilbert-space distance that can be approximated numerically. Neither has produced a proof, but both are concrete and programmable — docs/07-equivalences-and-criteria.md §5 states them precisely and works out, from the conjectured C/log N asymptotic, why the numerics are hopeless (no d_N computation is implemented in this repo; doing one and watching it refuse to converge is a suggested experiment there).

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