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Equivalences and Criteria

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A curated catalogue of statements exactly equivalent to RH, with honest notes on which ones have ever gone anywhere.

The short version

The Riemann Hypothesis has an unusually large number of exactly equivalent reformulations, and they look nothing like each other: an error bound on prime counting, a growth bound on a sum of Möbius values, a positivity condition on a sequence of numbers, a least-squares approximation problem in L^2(0,1), an inequality about the sum of divisors of an integer, and the statement that a certain heat flow has already run exactly to its critical time. Each is a THEOREM of the form "RH ⟺ X", proved and published. None of them is easier than RH. That is the whole lesson of this document: an equivalence is a translation, not a reduction — it relocates the difficulty without shrinking it, and in every case below you can point at exactly where the difficulty went. A small number of these criteria have nevertheless been genuinely productive (the error-term version, Speiser's theorem, de Bruijn–Newman); most, including the famous elementary-looking ones, have produced no partial progress whatsoever, and I say so where that is the case.

Notation. rho = beta + i*gamma runs over non-trivial zeros; Theta = sup(beta); RH is Theta = 1/2. mu is Möbius, sigma(n) the sum of divisors, Lambda(n) von Mangoldt (as in docs/04-explicit-formula.md). The de Bruijn–Newman constant is written Lambda_dBN here to avoid the clash; docs/05-de-bruijn-newman.md calls it simply Lambda.

0. Ground rules

Two words are used with force throughout. THEOREM means proved and published. CONJECTURE means believed and unproved. Anything labelled HEURISTIC is a plausibility argument, not a proof.

Note also that "equivalent to RH" cuts both ways. If X ⟺ RH, then disproving X disproves RH. Nobody has managed that either. The one instructive near-miss on this list is the Mertens conjecture, which was a strengthening of an equivalence, and which turned out to be false — a standing warning that intuition about the size of these objects is unreliable.

1. The error term in pi(x) and psi(x) — the original

THEOREM (von Koch, 1901).

    RH   <=>   pi(x)  = li(x) + O( sqrt(x) log x )
         <=>   psi(x) = x     + O( sqrt(x) (log x)^2 )

The mechanism is derived in docs/04-explicit-formula.md §6 and will not be repeated: each zero contributes a wave of amplitude x^beta / |rho| to psi(x), so the real part of a zero is literally an exponent on x. Three cautions people get wrong:

Productive? Yes — by far the most productive item on the list. It is why "assume RH" appears in thousands of papers: it converts directly into effective bounds on almost every prime-counting quantity, and those conditional results are real mathematics. It has never suggested a route to proving RH.

2. The Mertens function M(x) — and the conjecture that was false

Let M(x) = sum_{n <= x} mu(n). The bridge to zeta is one partial summation:

    1/zeta(s)  =  s * integral_1^inf  M(x) * x^(-s-1) dx        (Re s > 1)

That integral converges, and so defines an analytic function, precisely as far left as M(x) is small. A growth bound on M is therefore a zero-free region for zeta — zeros of zeta are poles of 1/zeta — and conversely.

THEOREM. For 1/2 <= theta < 1: M(x) = O(x^{theta+eps}) for every eps > 0 if and only if zeta has no zeros with Re s > theta. In particular

    RH   <=>   M(x) = O( x^(1/2 + eps) )    for every eps > 0

Now the trap. The Mertens conjecture asserted the clean epsilon-free bound

    |M(x)| < sqrt(x)     for all x > 1                         [FALSE]

Stieltjes claimed in 1885 to have a proof of the weaker M(x) = O(sqrt(x)) but never produced one; Mertens stated the strong form in 1897 on numerical evidence. It would have implied RH and the simplicity of every zero.

THEOREM (Odlyzko and te Riele, 1985, "Disproof of the Mertens conjecture"). The Mertens conjecture is false. They showed limsup M(x)/sqrt(x) > 1.06 and liminf M(x)/sqrt(x) < -1.009 (these are the values usually quoted; Kotnik and te Riele later pushed both past 1.2 in absolute value — I am hedging on those improved constants). The proof is non-constructive: take a few thousand zeros to high precision, treat M(x)/sqrt(x) via the explicit formula as an almost-periodic sum of waves cos(gamma * log x + phase), and use lattice basis reduction to locate a log x at which enough waves align. No explicit counterexample is known and the bounds on the first one are astronomical.

Sit with this. Sieving up to x = 10^6 I get max M(x)/sqrt(x) = 1.0 (at x = 1) and min M(x)/sqrt(x) = -0.894 (at x = 5), with M(10^6) = 212. The numerical evidence was overwhelming and the conjecture was wrong. docs/08-why-it-is-hard.md §3.4 develops this as the cautionary tale it is. Note carefully, though, that RH itself survived the disproof untouched — only the over-strong strengthening died.

Still open: whether M(x) = O(sqrt(x)) holds without the epsilon. That would imply RH plus simple zeros. It is widely believed false; work of Ng and others suggests, under a linear-independence hypothesis on the gamma, that M(x)/sqrt(x) is unbounded. That is a CONJECTURE and I am hedging on its precise form.

Productive? Not as an attack route. mu(n) is hard to control precisely because RH is what controls it; the implication runs the wrong way for progress.

3. Riesz and Hardy–Littlewood — the same idea, smoothed

THEOREM (M. Riesz, 1916). Define the Riesz function

    Riesz(x)  =  sum_{k >= 1}  (-1)^(k+1) * x^k / ( (k-1)! * zeta(2k) )

Then RH <=> Riesz(x) = O(x^(1/4 + eps)) for every eps > 0.

THEOREM (Hardy and Littlewood, 1918, Acta Mathematica 41 — as it is usually stated; verify the exact form against the paper).

    RH   <=>   sum_{k >= 1} (-x)^k / ( k! * zeta(2k+1) )  =  O( x^(-1/4) )    as x -> inf

These are §2 in disguise, and the disguise comes off in two lines. Expand 1/zeta(2k) = sum_n mu(n) n^(-2k), swap the order of summation, and use sum_{k>=1} (-1)^(k+1) y^k/(k-1)! = y * exp(-y):

    Riesz(x)  =  sum_{n >= 1}  mu(n) * (x/n^2) * exp(-x/n^2)

and the Hardy–Littlewood series collapses the same way to sum_n (mu(n)/n) * (exp(-x/n^2) - 1). I checked both identities numerically: the Riesz power series and its Möbius form agree to 7–8 digits at x = 1, 10, 100 (limited by my truncation of the Möbius sum), and the Hardy–Littlewood pair agrees to 10 digits at x = 1, 10, 50. So these criteria are the Mertens criterion with a Gaussian-type smoothing — and the smoothing is what buys the clean exponent 1/4, which is sqrt of sqrt(x) because the natural variable is n^2.

Values: Riesz(1) = 0.0439818, Riesz(10) = -0.7806756, Riesz(100) = -0.1519372.

Productive? No, and the reason is worth seeing concretely. Evaluating Riesz(x) from its defining series is a cancellation catastrophe: at x = 100 the largest single term is about 10^44 while the sum is -0.152, so roughly 45 decimal digits cancel — and that count grows linearly in x. Any numerical probe of the x^{1/4} growth needs working precision proportional to x, so the asymptotic regime is unreachable in practice. Exactly true, computationally inert.

4. Li's criterion — positivity of a sequence

THEOREM (Xian-Jin Li, 1997). Define, for n >= 1,

    lambda_n  =  (1/(n-1)!) * d^n/ds^n [ s^(n-1) * log xi(s) ]  evaluated at s = 1
              =  sum_rho [ 1 - (1 - 1/rho)^n ]        (zeros paired rho <-> 1-rho)

Then RH <=> lambda_n >= 0 for every n >= 1.

Why. The Möbius map s -> w = 1 - 1/s sends the half-plane Re s > 1/2 onto the open unit disk and the critical line onto the unit circle (check: s = 1 -> w = 0, s = 1/2 -> w = -1). Since zeros come in pairs rho, 1-rho, RH says every w_rho lies on the circle, rather than one strictly inside and its partner strictly outside. Now lambda_n = sum_rho (1 - w_rho^n), and if some zero is off the line its w has |w| > 1, so the term -Re(w^n) oscillates with exponentially growing amplitude -- the delicate part of the proof is showing it cannot be forever cancelled by the infinitely many other terms, i.e. that some lambda_n really does go negative. Bombieri and Lagarias (1999) distilled exactly this into a clean statement about arbitrary multisets of complex numbers, with no zeta in it.

Checks I ran. There is a closed form for the first coefficient:

    lambda_1  =  1 + gamma/2 - log(4 pi)/2  =  0.0230957089661...

Under RH, sum_rho 1/|rho|^2 = sum_rho 1/(rho(1-rho)) = 2*lambda_1 = 2 + gamma - log(4 pi). Summing 2/(1/4 + gamma^2) over the first 1000 zeros gives 0.0447523; the crude density-based tail estimate adds 0.0014397, for 0.0461920 against the exact 0.0461914... — agreement to about 6e-7. Truncated sums for lambda_1 ... lambda_4 over those 1000 zeros are 0.0224, 0.0895, 0.201, 0.357: all positive, increasing, and all slight *under*estimates since the omitted tails are positive.

Productive? No, and the shape of the trouble is visible in the numbers. lambda_1 = 0.023 is barely positive, and it is positive only through a delicate near-cancellation between gamma and log(4 pi). Worse, evaluating lambda_n requires either the zeros themselves — in which case you have assumed what you wanted to prove — or an arithmetic expression involving prime sums whose individual terms grow. Under RH one expects lambda_n to grow like (n/2) log n; nobody can prove non-negativity unconditionally for large n.

How weak a detector is it? Measurable, and the answer is discouraging. The Davenport–Heilbronn function F has the same shape (F(s) = F(1-s), real coefficients, real Hardy-style Z) and a zero provably off the line, so some lambda_n for F must eventually go negative — note this leans on the Bombieri–Lagarias multiset statement above rather than on Li's original, since F has no Euler product and is outside the Selberg class, and the multiset form needs neither. Running the same contour extraction on F — validated first against zeta.li.li_coefficients on xi, where it agrees bit-identically — gives lambda_n > 0 for every n <= 24, in fact uniformly larger than zeta's. The reason is in the off-line zero itself: with rho = 0.80851718... + 85.69934848...i, the mirror zero 1 - rho has Re < 1/2 and |1 - 1/(1-rho)| = 1.00004200616..., so the exponentially growing term grows at rate 4.2e-5 per step and needs n ~ 2.4e4 merely to double, against a background growing like (n/2) log n. So observing lambda_n >= 0 for zeta over any comparable range distinguishes nothing: a function that violates RH passes the identical test. This sharpens §11's "the equivalences restate rather than reduce" into a quantitative statement about one of them. scripts/18_dh_li_coefficients.py.

5. Nyman–Beurling, and Báez-Duarte's strengthening

This is the criterion that comes closest to being a finite problem.

Write {y} for the fractional part, and define on (0,1), for 0 < theta <= 1:

    rho_theta(x)  =  { theta / x }

THEOREM (Nyman, 1950; Beurling, 1955). RH holds if and only if the constant function 1 lies in the closed linear span of { rho_theta : 0 < theta <= 1 } inside L^2(0,1). Beurling proved the L^p refinement: zeta has no zeros in Re s > 1/p if and only if that span is dense in L^p(0,1).

Why. Take Mellin transforms along Re s = 1/2. The transform of rho_theta is essentially -theta^s * zeta(s)/s, and the transform of the constant 1 is 1/s. So approximating 1 by a combination sum_k c_k * rho_{theta_k} amounts to finding a Dirichlet-polynomial-like D(s) with zeta(s) D(s) ≈ 1 in mean square on the critical line — that is, 1/zeta is approximable by Dirichlet polynomials on the critical line. If zeta had a zero at beta > 1/2, then 1/zeta has a pole strictly inside the relevant half-plane and no such approximation can exist. That is the whole content, and it is a clean statement: RH ⟺ 1/zeta is reachable from the right.

THEOREM (Báez-Duarte, 2003). You only need theta = 1/k for positive integers k. Define

    d_N^2  =  inf over c_1,...,c_N  of  || 1 - sum_{k=1}^N c_k * rho_{1/k} ||^2  in L^2(0,1)

Then d_N is non-increasing (extra basis vectors can only help), and

    RH   <=>   d_N -> 0   as N -> infinity

So RH becomes: does this explicitly computable, monotonically decreasing sequence of least-squares residuals tend to zero? The Gram matrix entries are elementary integrals; not a single zero appears.

CONJECTURE (Báez-Duarte, Balazard, Landreau and Saias, around 2000).

    d_N^2  ~  C / log N       with     C = sum_rho 1/|rho|^2

and under RH that constant is 2 + gamma - log(4 pi) = 0.0461914..., the same number verified in §4. I believe they also proved a matching lower bound of this shape; hedge on the exact statement.

Productive? Structurally beautiful, computationally hopeless — and it is worth being precise about why. If d_N^2 ≈ 0.0462/log N, then at N = 100 you have d_N ≈ 0.10, nowhere near zero, and to reach d_N^2 = 0.001 you would need log N ≈ 46, i.e. N ≈ 10^20. No finite computation can distinguish "tends to 0 like 1/log N" from "tends to a small positive limit". On top of that the Gram matrix is severely ill-conditioned, so the least-squares problem itself resists high-precision solution. This is the sharpest illustration of the theme of this document: exactly equivalent to RH, and it tells you nothing you can act on.

6. Robin, Lagarias, Nicolas — RH as an inequality about divisors

Unconditional background: THEOREM (Gronwall, 1913). limsup_n sigma(n)/(n log log n) = e^gamma = 1.7810724.... So e^gamma is exactly the right constant and the only question is whether the limsup is ever exceeded.

THEOREM (Robin, 1984).

    RH   <=>   sigma(n)  <  e^gamma * n * log log n      for every n > 5040

I computed the failures directly. For 3 <= n <= 20000 the inequality fails at exactly

    3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84,
    120, 180, 240, 360, 720, 840, 2520, 5040

— 26 values, largest 5040 (plus degenerate n = 1, 2, where log log n is not positive). The margin at the top is thin: sigma(5040) = 19344 against e^gamma * 5040 * log log 5040 = 19237.06, a ratio of 1.00556. Robin also proved the unconditional companion sigma(n) <= n log log n * (e^gamma + 0.6483/(log log n)^2) for n >= 3 — I am confident in the shape, and treat 0.6483 as "commonly cited" — and that if RH is false there are infinitely many counterexamples, occurring among the superabundant numbers.

THEOREM (Lagarias, 2002, Amer. Math. Monthly, "An elementary problem equivalent to the Riemann hypothesis"). With H_n = 1 + 1/2 + ... + 1/n,

    RH   <=>   sigma(n)  <=  H_n + exp(H_n) * log(H_n)     for all n >= 1

with equality only at n = 1. Verified: at n = 1 both sides are exactly 1; at n = 5040, 19344 against 19836.32; at n = 55440, 232128 against 241179.92.

This is Robin in elementary clothing, and the disguise comes off cleanly: H_n = log n + gamma + O(1/n), so exp(H_n) ≈ e^gamma * n and log H_n ≈ log log n. The extra H_n term plus the lower-order corrections supply exactly enough slack to absorb all 26 exceptional values, which is why Lagarias' form needs no "n > 5040" clause at all. That is a genuinely elegant piece of bookkeeping.

Related: Nicolas (1983, I believe — verify the date) proved RH equivalent to N_k/phi(N_k) > e^gamma * log log N_k for every primorial N_k = 2*3*5*...*p_k. I checked this for the first 17 primorials and the gap widens steadily (at p_k = 59: 7.4749 versus 6.9319).

Productive? No — and this is the most misleading family on the list. These statements are elementary to state and are routinely presented as "RH for people who don't know complex analysis". But the proof of each equivalence runs through the explicit formula and the distribution of primes; the elementary appearance is entirely cosmetic, and nothing about the extremal behaviour of sigma is tractable by elementary means at the required precision. Be suspicious of anyone claiming an elementary attack from this direction.

7. Weil positivity — the one with a proof in a parallel universe

Take the explicit formula of docs/04-explicit-formula.md in its symmetric distributional form. For a suitable even test function h with Fourier transform g, it reads schematically

    sum_rho h(gamma_rho)  =  (archimedean/Gamma terms)  +  (pole terms)
                             -  2 * sum_{n >= 2} (Lambda(n)/sqrt(n)) * g(log n)

where rho = 1/2 + i*gamma_rho — so gamma_rho is real exactly when rho lies on the critical line.

THEOREM (Weil, 1952; refined 1972). RH holds if and only if the functional W(h) = sum_rho h(gamma_rho) is >= 0 for every h of positive-definite type — every h arising as a self-convolution g * g~ of a nice test function.

Why it is a criterion. If all gamma_rho are real and h >= 0 on the real line, the sum is non-negative automatically. If some zero is off the line its gamma_rho is genuinely complex, h extends to the complex plane, and one can engineer an h making that term very negative — off-line zeros come in quadruples (docs/03-functional-equation.md), and the pair off the real gamma-axis can be made to dominate.

Why people care. Weil proved the exact analogue for curves over finite fields (announced 1940, full proof 1948), where the positivity is not analysis at all but the Castelnuovo–Severi / Hodge index inequality for intersection numbers on the surface C x C. This is the only criterion here with a complete proof in a genuinely parallel setting, by genuinely geometric means. Connes (1999) recast it as a trace formula on a space of adele classes. Both stories are told properly in docs/06-hilbert-polya-and-gue.md (§4 Connes, §6 function fields), including a precise account of what is missing over Spec Z.

Productive? The most structurally suggestive item on this list, and the origin of essentially all geometric programmes. Also the least checkable: W(h) >= 0 admits no meaningful numerical test, and the missing ingredient — an arithmetic surface playing the role of C x C — has resisted decades. High explanatory value, zero computational value, no proof.

Detector Power and the Gaussian/Fejér Gap. If Weil's criterion is viewed computationally as a detector for off-line zeros, its sensitivity depends entirely on the choice of the test-function family h. The gap in sensitivity between admissible choices spans thousands of orders of magnitude. For the Gaussian family h(r) = exp(-a r^2), an off-line zero at height T with shift delta produces a maximal negative dip bounded by 4 * exp(-pi(T^2 - delta^2)/(2T delta)). At the height of gamma_1 = 14.13 and delta=0.01, the Gaussian dip is 10^{-964} — the detector is completely blind. In contrast, band-limited test functions like the Fejér kernel are exponentially sensitive to off-line zeros (growing as exp(2b*delta) off the real line by the Paley-Wiener theorem). Detector power is a property of the test-function family, not the Weil criterion itself. This is computationally demonstrated in scripts/24_detector_power.py.

8. Speiser's theorem — the one that actually worked

THEOREM (Speiser; the paper is in Mathematische Annalen 110, cited variously as 1934 and 1935).

    RH   <=>   zeta'(s) has no zeros in the strip  0 < Re(s) < 1/2

Why, roughly. The functional equation makes zeta in the left half-strip a reflected copy of zeta in the right half-strip times an explicit analytic factor. Differentiating and running an argument-principle / Rolle-type count, a zero of zeta strictly right of the critical line forces a zero of zeta' strictly left of it. Levinson and Montgomery (1974) proved the quantitative form: up to height T, the number of zeros of zeta' in the left half-strip 0 < Re s < 1/2 and the number of zeros of zeta in that same half-strip agree to within O(log T). (By the functional equation, zeros of zeta off the line come in pairs mirrored across it, so the left half-strip holds exactly half of the off-line zeros; both counts are zero precisely under RH.)

Numerically, the two zeros of zeta' of smallest positive imaginary part:

    zeta'(s) = 0   at   s = 2.4631619 + 23.29832 i
                   and  s = 1.2864968 + 31.70825 i

both comfortably right of Re s = 1/2, as RH requires. (zeta' also has real zeros in the left half-plane, interleaved with the trivial zeros of zeta: s = -2.7172628..., s = -4.9367621....)

Productive? YES — uniquely so on this list. Speiser's theorem is the engine of Levinson's method: zeros of zeta' are easier to count than zeros of zeta, and Levinson (1974) used this to prove that more than one third of the non-trivial zeros lie on the critical line — later improved by Conrey (1989) to more than 2/5, and further since. See docs/08-why-it-is-hard.md §1.3 for the current record and, importantly, for why a positive proportion is progress on a different question than Theta = 1/2. Still: no other criterion in this document has yielded an unconditional quantitative theorem about the zeros. If you want a worked example of an equivalence that paid, this is it.

9. De Bruijn–Newman — RH with no margin

Developed in full in docs/05-de-bruijn-newman.md; the statement belongs in this catalogue.

Deform xi by a heat flow in a time parameter t, producing a family H_t with H_0 equal, up to normalisation, to Xi.

THEOREM (de Bruijn, 1950; Newman, 1976). There is a finite real constant Lambda_dBN such that H_t has only real zeros exactly when t >= Lambda_dBN. Hence RH <=> Lambda_dBN <= 0.

THEOREM (de Bruijn, 1950). Lambda_dBN <= 1/2; since improved — Polymath15 (2018-19) proved <= 0.22, and docs/05 notes a commonly-cited further refinement to <= 0.2.

THEOREM (Rodgers and Tao; arXiv 2018, journal version around 2020). Lambda_dBN >= 0 — Newman's conjecture, whose motivating slogan was that RH, if true, is only barely so.

Together: RH <=> Lambda_dBN = 0 exactly. This is the most philosophically loaded equivalence here. It says RH is not merely true-or-false but marginally true if true at all — so any argument carrying slack, any inequality with a constant to spare, is provably incapable of proving RH. It also explains, retroactively, the thin margins you keep meeting elsewhere in this document: Robin's 0.56% at n = 5040, lambda_1 = 0.023, the 1/log N decay in Báez-Duarte.

Productive? Yes, in the specific sense that Rodgers–Tao is a hard unconditional theorem obtained by working on this side of the equivalence. Code: zeta/heatflow.py.

10. Others, briefly (and hedged)

11. Why so many equivalences, and still no proof

The general obstructions — the parity problem, Davenport–Heilbronn, the limits of zero-density methods, the graveyard of failed attacks — are docs/08-why-it-is-hard.md's subject and I will not duplicate them. What belongs here is the narrower question: why does the sheer abundance of exact equivalences not constitute progress?

Difficulty is conserved, and you can watch it move. RH is one statement about the zeros of one function. A criterion re-encodes exactly that information in a different language, so the hardness must reappear somewhere — and it always does, visibly:

    criterion              where the difficulty went
    ---------------------  ----------------------------------------------------
    Mertens (§2)           cancellation in sum mu(n); no handle on it
    Riesz / H-L (§3)       45 digits of cancellation at x = 100, growing in x
    Li (§4)                computing lambda_n needs the zeros you wanted to find
    Nyman-Beurling (§5)    d_N^2 ~ C/log N: unreachably slow, ill-conditioned
    Robin / Lagarias (§6)  extremal sigma on superabundant n, which is RH again
    Weil (§7)              the arithmetic surface that would prove positivity
                           does not exist

The correct reaction to a beautiful new equivalence is therefore "where did the hard part go?" — and there is always an answer. Nothing has been gained; a label has been changed.

There is no margin to work with. Rodgers–Tao turned a feeling into a theorem: Lambda_dBN >= 0 is proved and Lambda_dBN = 0 is RH. Whatever proves RH must be exactly tight. That rules out an entire style of argument — the kind where you bound something by something else with room to spare — and it is not a soft observation but a consequence of a published theorem.

A practical filter. An equivalence is productive if and only if it lets you prove something new without proving RH. By that test only three items in this catalogue have paid: the error-term version (§1 — an entire industry of conditional theorems), Speiser (§8 — Levinson's method, and a positive proportion of zeros unconditionally on the line), and de Bruijn–Newman (§9 — Rodgers–Tao). Robin, Lagarias, Li, Nyman–Beurling, Riesz and Mertens are exactly true, fully rigorous, genuinely beautiful, and have to date produced no partial progress whatsoever. Saying so is not pessimism; it is the only way to tell the two categories apart.

And a note on what the abundance means. It is often reported as "we're closing in". Read it the other way: RH sits at a junction where many independent parts of mathematics meet, which is strong evidence that it is true and deep, and correspondingly weak evidence that it is nearly proved.

Where to go next