The Riemann zeta function and L-functions¶

This page covers the analytic core of Numerisect: the definition and Euler product of \(\zeta(s)\), its continuation and functional equation, the trivial and nontrivial zeros, the Riemann hypothesis and what it is equivalent to, the computational machinery of Hardy's \(Z\) function, Gram points, \(N(T)\), \(S(T)\) and the Turing method, the Riemann–Siegel formula, the explicit formula linking zeros to \(\psi(x)\) and \(\pi(x)\), the statistics of zero spacings and the GUE connection, Dirichlet characters and \(L\)-functions with the generalised Riemann hypothesis, Dedekind zeta functions, and finally the ball arithmetic that lets Numerisect say certified about some of these results and only exploratory about others. The tools are documented in Riemann zeta tools and the zeta lab.

Certified and exploratory are not synonyms

Numerisect uses those two words in a technical sense, and this page uses them the same way. Certified means a rigorous Arb ball enclosure, or a decision that could only be reached because an enclosure excluded zero. Exploratory means a number that is probably right and proves nothing — a truncated sum, a histogram bin, a plot sample taken from enclosure midpoints, or any PARI floating-point output. The last section explains why the distinction is real and not decorative.

Definition and Euler product¶

For \(\mathrm{Re}\,s > 1\),

\[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \left(1 - \frac{1}{p^s}\right)^{-1}. \]

The identity is Euler's, and expanding each factor as a geometric series and multiplying out is exactly the statement that every integer factors into primes in one way. That is the whole reason a function of a complex variable has anything to say about primes: the Euler product is unique factorization written analytically.

Taking logarithms of the product and differentiating gives

\[ -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}, \]

which is the bridge used in every proof of the prime number theorem and the source of the explicit formula below.

In Numerisect. Euler-product comparison (POST /api/zeta/euler-product) evaluates the truncated product over primes from FLINT's n_primes_t sieve with acb_pow, compares it against acb_dirichlet_zeta, and reports the deviation and a rigorous truncation bound — the half-plane condition \(\mathrm{Re}\,s > 1\) is checked as a ball comparison, so a request that cannot be decided is refused rather than answered. Every partial value, deviation and bound on that page is certified.

Continuation, the functional equation, and the zeros¶

\(\zeta\) continues to a meromorphic function on all of \(\mathbb{C}\), analytic except for a simple pole at \(s = 1\) with residue 1. The continuation is governed by the completed function

\[ \xi(s) = \tfrac{1}{2}\, s (s - 1)\, \pi^{-s/2}\, \Gamma\!\left(\tfrac{s}{2}\right) \zeta(s), \]

which is entire and satisfies the functional equation

\[ \xi(s) = \xi(1 - s). \]

Unwinding gives the asymmetric form

\[ \zeta(s) = 2^s \pi^{s-1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma(1 - s)\, \zeta(1-s). \]

Two families of zeros follow.

Trivial zeros at \(s = -2, -4, -6, \ldots\), forced by the poles of \(\Gamma(s/2)\) in the completed function; equivalently by the sine factor.
Nontrivial zeros, all lying in the critical strip \(0 < \mathrm{Re}\,s < 1\). The functional equation makes them symmetric about \(s \mapsto 1 - s\), and \(\overline{\zeta(\bar s)} = \zeta(s)\) makes them symmetric about the real axis, so they come in quadruples \(\rho, \bar\rho, 1-\rho, 1-\bar\rho\) unless \(\mathrm{Re}\,\rho = 1/2\), when the quadruple degenerates to a pair.

The line \(\mathrm{Re}\,s = 1/2\) is the critical line, the axis of that symmetry. The first few nontrivial zeros have ordinates \(14.134725141\ldots\), \(21.022039639\ldots\), \(25.010857580\ldots\).

In Numerisect. Evaluate \(\zeta(s)\) (POST /api/zeta/evaluate, acb_zeta) returns rigorous enclosures for real part, imaginary part, magnitude and argument, and rejects \(s = 1\). Completed xi and Dirichlet eta (POST /api/zeta/xi-eta) use acb_dirichlet_xi and acb_dirichlet_eta. Functional equation (POST /api/zeta/functional-equation) evaluates both sides independently in FLINT/Arb and reports the equation as verified only when the two complex balls overlap — which is a genuine check, not a restatement. Stieltjes constants (POST /api/zeta/stieltjes, acb_dirichlet_stieltjes) give rigorous enclosures for the \(\gamma_n\) in the Laurent expansion at \(s = 1\).

The Riemann hypothesis¶

Conjecture (Riemann, 1859). Every nontrivial zero of \(\zeta\) satisfies \(\mathrm{Re}\,\rho = \tfrac{1}{2}\).

This is a conjecture. Numerisect never states it, or anything derived from it, as a theorem.

What is proved: infinitely many zeros lie on the critical line (Hardy, 1914); a positive proportion do (Selberg), and at least two fifths do (Conrey, 1989); the first \(3 \times 10^{12}\) zeros by ordinate do, verified rigorously by Platt and Trudgian (2021). None of that is evidence about the rest, in the strict sense that none of it constrains the zeros above the computed height.

What RH is equivalent to¶

The reason RH is the problem in prime number theory is that it is exactly a statement about the error term in the prime number theorem. The following are equivalent to RH:

\[ \psi(x) = x + O\!\left(\sqrt{x}\,\log^2 x\right), \qquad \pi(x) = \mathrm{li}(x) + O\!\left(\sqrt{x}\,\log x\right). \]

Schoenfeld made the second explicit: RH is equivalent to \(|\pi(x) - \mathrm{li}(x)| < \dfrac{\sqrt x \log x}{8\pi}\) for all \(x \ge 2657\).

Unconditionally, all that is known is the de la Vallée Poussin form, coming from a zero-free region just left of \(\mathrm{Re}\,s = 1\): \(\psi(x) = x + O\!\left(x\,e^{-c\sqrt{\log x}}\right)\), which is very much weaker. The general principle is that a zero at \(\mathrm{Re}\,\rho = \sigma\) contributes a term of size \(x^{\sigma}\) to the error, so the supremum of the real parts of the zeros is the exponent in the error term. RH says that supremum is \(1/2\), the smallest it can be.

Two equivalents outside analysis are worth knowing because Numerisect computes both sides of them: \(M(x) = O(x^{1/2 + \varepsilon})\) for the Mertens function, and the same for the summatory Liouville function — see the modular page, which also records that the stronger Mertens conjecture \(|M(x)| < \sqrt x\) is false.

Hardy's \(Z\) function and the geometry of the critical line¶

To find zeros on the critical line one wants a real function whose sign changes are the zeros. That is Hardy's \(Z\):

\[ Z(t) = e^{i\theta(t)}\, \zeta\!\left(\tfrac{1}{2} + it\right), \qquad \theta(t) = \arg \Gamma\!\left(\tfrac{1}{4} + \tfrac{it}{2}\right) - \tfrac{t}{2}\log \pi . \]

The Riemann–Siegel theta function \(\theta\) is chosen precisely so that the phase cancels: \(Z(t)\) is real for real \(t\), and \(|Z(t)| = |\zeta(1/2 + it)|\). A sign change of \(Z\) between two points is therefore a zero of \(\zeta\) on the critical line, of odd order, in that interval.

Gram points and Gram's law¶

\(\theta\) is increasing for \(t\) large, so \(\theta(t) = n\pi\) has a unique solution \(g_n\), the \(n\)-th Gram point; \(g_0 = 17.8455995404\ldots\). Since the leading behaviour of \(Z\) is dominated by its first term \(2\cos\theta(t)\), one expects \(Z(g_n)\) to alternate in sign with \(n\):

Gram's law (an empirical rule, not a theorem). \((-1)^n Z(g_n) > 0\).

It usually holds, and it fails infinitely often. Its first failure is at \(n = 126\), where \(g_{126} = 282.4547208234621746\ldots\) and \(Z(g_{126}) = -0.0276294988571999\ldots\) although 126 is even. When Gram's law holds on a stretch, each Gram interval \([g_n, g_{n+1})\) contains exactly one zero and counting is trivial; the failures are exactly where a rigorous count needs more than the rule. A Gram block is a maximal run \([g_n, g_{n+k})\) with good endpoints and \(k-1\) bad interior points, and it is the unit in which exceptions are organised.

In Numerisect. Gram point (POST /api/zeta/gram, acb_dirichlet_gram_point) gives one \(g_n\) as a rigorous enclosure. Gram blocks and exceptions (POST /api/zeta/gram-blocks) scans an index range with acb_dirichlet_gram_point and acb_dirichlet_hardy_z. Every verdict is certified: a Gram point is called good or bad only when the enclosure of \((-1)^n Z(g_n)\) strictly excludes zero, and otherwise it is reported as inconclusive. Scanning indices 120–131 reports exactly one exception and one Gram block, \(n = 125\) of length 2 with pattern gbg.

Counting zeros: \(N(T)\), \(S(T)\), and Turing's method¶

Let \(N(T)\) be the number of zeros with \(0 < \mathrm{Im}\,\rho \le T\).

Theorem (Riemann–von Mangoldt).

\[ N(T) = \frac{\theta(T)}{\pi} + 1 + S(T), \qquad S(T) = \frac{1}{\pi}\arg \zeta\!\left(\tfrac{1}{2} + iT\right), \]

where the argument is by continuous variation from \(2\) to \(2 + iT\) to \(1/2 + iT\). Expanding \(\theta\),

\[ N(T) = \frac{T}{2\pi}\log\frac{T}{2\pi e} + \frac{7}{8} + S(T) + O(1/T). \]

So the smooth part \(\theta(T)/\pi + 1\) is known exactly and everything difficult is in \(S(T)\). Unconditionally \(S(T) = O(\log T)\), and in practice \(S(T)\) is usually small — it stays between \(-1\) and \(1\) for a very long way — but it is known to be unbounded, and a single unnoticed excursion would mean a missed zero.

Why the smooth term is not a count

\(\theta(T)/\pi + 1\) is the main term of Riemann–von Mangoldt, and rounding it is a guess. Numerisect labels the smooth count exploratory wherever it appears — including the \(\theta(T,\chi)/\pi\) count on the \(L\)-zeros page — and never prints it as the number of zeros.

Turing's method turns this into a rigorous count. The key is not a bound on \(S(T)\) pointwise but on its average: explicit bounds on \(\int_{T_1}^{T_2} S(t)\,dt\) constrain how many zeros can hide, and combining that with the number of sign changes actually observed pins \(N(T)\) to a unique integer. FLINT implements this in acb_dirichlet_zeta_nzeros.

In Numerisect. Count zeros through \(T\) (POST /api/zeta/count) succeeds only when FLINT isolates a unique integer; otherwise it reports inconclusive and asks for higher precision or a different endpoint, rather than rounding. This is certified. Zero-counting remainder \(S(T)\) (POST /api/zeta/backlund-s) reports acb_dirichlet_backlund_s, acb_dirichlet_backlund_s_bound, acb_dirichlet_zeta_nzeros and acb_dirichlet_hardy_theta; the values are certified, but the plotted \(S(T)\) trace is exploratory, because a plot shows enclosure midpoints. Consecutive zeros (POST /api/zeta/zeros, acb_dirichlet_hardy_z_zeros) returns rigorous intervals for consecutive zeros, each printed with its radius. N(100) = 29 is one of the pinned test values.

A sign-change count is a lower bound

Certified sign changes on a finite grid prove that at least that many zeros of odd order lie in the range. A coarse grid can step over a closely spaced pair. Raising the resolution raises the bound; nothing claims completeness.

The Riemann–Siegel formula¶

Evaluating \(\zeta(1/2 + it)\) by the defining series is impossible (it does not converge) and by Euler–Maclaurin is expensive at large \(t\). The Riemann–Siegel formula, found by Siegel in Riemann's unpublished notes and published in 1932, gives \(Z(t)\) in \(O(\sqrt t)\) terms:

\[ Z(t) = 2 \sum_{n=1}^{\nu} \frac{\cos\bigl(\theta(t) - t \log n\bigr)}{\sqrt n} \; + \; R(t), \qquad \nu = \left\lfloor \sqrt{\tfrac{t}{2\pi}} \right\rfloor . \]

The first part is the main sum: a truncated Dirichlet series with the phase folded in, of length \(\nu \approx \sqrt{t/2\pi}\). The remainder has an asymptotic expansion in correction terms

\[ R(t) \sim (-1)^{\nu - 1} \left(\frac{t}{2\pi}\right)^{-1/4} \sum_{k \ge 0} C_k(p)\left(\frac{t}{2\pi}\right)^{-k/2}, \qquad p = \sqrt{\tfrac{t}{2\pi}} - \nu , \]

where each \(C_k\) is an explicit combination of derivatives of a fixed function of the fractional part \(p\). \(C_0\) alone already gives several digits; the series is asymptotic, so adding terms helps up to a point and then stops helping.

An asymptotic expansion with the remainder dropped is not a certification

Riemann zeta tools records that the imported research prototype contained an incomplete custom approximation described as Riemann–Siegel, and that Numerisect does not expose that claim. Dropping the omitted remainder of an asymptotic formula is not a proof. What Numerisect exposes instead is acb_dirichlet_zeta_rs, which folds a rigorous remainder bound into the returned ball, together with acb_dirichlet_zeta_rs_bound and an independent acb_dirichlet_zeta evaluation. Every Riemann–Siegel row is therefore certified.

In Numerisect. Riemann–Siegel remainder (POST /api/zeta/riemann-siegel), with \(t \ge 10\) and \(0 \le K \le 20\) correction terms.

The explicit formula¶

This is the theorem that makes zeta a statement about primes rather than a curiosity.

Theorem (von Mangoldt's explicit formula). For \(x > 1\) not a prime power,

\[ \psi(x) = x \;-\; \sum_{\rho} \frac{x^{\rho}}{\rho} \;-\; \log 2\pi \;-\; \tfrac{1}{2}\log\!\left(1 - x^{-2}\right), \]

the sum being over nontrivial zeros, taken symmetrically in \(\rho\) and \(\bar\rho\).

Read the terms. \(x\) is the main term — this alone is the prime number theorem. The final two are small and explicit (they come from the pole and the trivial zeros). The sum over zeros is the entire fluctuation of the primes about their average: each zero \(\rho = \beta + i\gamma\) contributes an oscillation of amplitude \(x^{\beta}/|\rho|\) and frequency \(\gamma\) in \(\log x\). The zeros are the frequencies of the primes.

Riemann's original version is for the counting function. With \(J(x) = \sum_{k \ge 1} \pi(x^{1/k})/k\),

\[ J(x) = \mathrm{li}(x) - \sum_{\rho} \mathrm{li}\!\left(x^{\rho}\right) - \log 2 + \int_x^{\infty} \frac{dt}{t(t^2 - 1)\log t}, \qquad \pi(x) = \sum_{n \ge 1} \frac{\mu(n)}{n} J\!\left(x^{1/n}\right). \]

The Möbius coefficients strip out the contribution of prime squares, cubes and higher powers — the same inversion that produces Riemann's \(R(x)\) on the distribution page.

What "prime reconstruction from zeros" means¶

Truncating the sum at the first \(N\) zeros gives an approximation to \(\psi(x)\) or \(\pi(x)\) that oscillates around the true step function and sharpens as \(N\) grows; the jumps at prime powers emerge from the interference of the terms. It is a striking demonstration and it is not a computation of \(\pi(x)\): the discarded tail is not bounded, so the result is an estimate, always.

π(100) from 200 zeros

curl -sX POST 127.0.0.1:8765/api/zeta/explicit-prime-count \
  -H 'X-Numerisect-Token: <token>' -H 'content-type: application/json' \
  -d '{"bound":"100","zeros":200,"precision":30,"threads":8}'

The response reports the exact \(\pi(100) = 25\) from primecount alongside a convergence table at \(N = 1, 2, 4, 8, \ldots\) zeros. The estimate walks toward 25; it never becomes a certification. The one piece that is bounded rigorously is the tail integral, enclosed in \(\left[0,\; \log\!\left(\frac{x^2}{x^2-1}\right) / (2\log x)\right]\).

In Numerisect. Reconstruct \(\pi(x)\) from zeros (POST /api/zeta/explicit-prime-count) uses acb_dirichlet_hardy_z_zeros for the zeros, acb_hypgeom_ei for \(\mathrm{li}(x^\rho)\), arb_hypgeom_li for the principal term, n_moebius_mu for the weights, and primecount for the exact value. Rebuild Chebyshev \(\psi(x)\) (POST /api/zeta/chebyshev-psi) sums \(x^\rho/\rho\) with acb_exp and acb_div and computes the exact \(\psi(x)\) from FLINT's n_primes_t sieve with arb_log_ui; \(\psi(10) = 3\log 2 + 2\log 3 + \log 5 + \log 7 = 7.8320141805054689907\ldots\). Both estimates are exploratory, by construction. Limits: 5000 zeros in a sum, \(x \le 10^7\) for exact \(\psi\), \(x \le 10^{12}\) for \(\pi(x)\).

Zero statistics and the GUE connection¶

The zeros thin out logarithmically: near height \(T\) the average gap is \(2\pi/\log(T/2\pi)\). To compare spacings at different heights they are unfolded — rescaled to unit mean — using \(\theta\): \(\tilde\gamma_n = \theta(\gamma_n)/\pi\), so that consecutive differences \(\delta_n = \tilde\gamma_{n+1} - \tilde\gamma_n\) have mean 1.

Conjecture (Montgomery's pair correlation, 1973). For the unfolded zeros,

\[ \frac{1}{N}\#\{(n, m) : \alpha \le \tilde\gamma_n - \tilde\gamma_m \le \beta \} \;\longrightarrow\; \int_{\alpha}^{\beta}\left(1 - \left(\frac{\sin \pi u}{\pi u}\right)^2\right) du \;+\; \delta(\alpha, \beta). \]

Montgomery proved this for test functions of restricted support, assuming RH; the general statement is conjectural. The famous part is the recognition, in conversation with Freeman Dyson, that \(1 - (\sin \pi u / \pi u)^2\) is exactly the pair correlation of eigenvalues of large random Hermitian matrices from the Gaussian Unitary Ensemble. The zeros repel each other like eigenvalues of a random matrix, which is the main evidence for the Hilbert–Pólya idea that the zeros are the spectrum of some self-adjoint operator.

Odlyzko's computations — at heights up to the \(10^{20}\)-th zero and beyond — match the GUE predictions to remarkable accuracy, and also expose the slow convergence: lower-height statistics deviate visibly, and the agreement improves with height.

Histogram bins are exploratory

Zero ordinates are certified; binned statistics computed from them are not. A histogram is a finite sample of a limiting distribution, and Numerisect labels every bin on the spacing and pair-correlation pages exploratory.

In Numerisect. Zero-spacing histogram (POST /api/zeta/zero-spacing) uses acb_dirichlet_hardy_z_zeros, unfolds with acb_dirichlet_hardy_theta, and draws the GUE surmise with arb_hypgeom_erf. Pair correlation (POST /api/zeta/pair-correlation) computes the same unfolding and the \(1 - (\sin \pi u/\pi u)^2\) curve with arb_sin. Up to 20 000 zeros per run.

Dirichlet characters and L-functions¶

A Dirichlet character modulo \(q\) is a homomorphism \(\chi : (\mathbb{Z}/q\mathbb{Z})^\times \to \mathbb{C}^\times\), extended to \(\mathbb{Z}\) by \(\chi(n) = 0\) when \(\gcd(n, q) > 1\). There are \(\varphi(q)\) of them, forming a group isomorphic to \((\mathbb{Z}/q\mathbb{Z})^\times\) itself. A character is primitive when it does not factor through a smaller modulus; the least such modulus is its conductor. \(\chi\) is even or odd according as \(\chi(-1) = +1\) or \(-1\), and real when it takes only values \(0, \pm 1\) — the real primitive characters are exactly the Kronecker symbols \(\left(\frac{D}{\cdot}\right)\) of fundamental discriminants, which is where quadratic fields re-enter.

The associated Dirichlet \(L\)-function is

\[ L(s, \chi) = \sum_{n \ge 1} \frac{\chi(n)}{n^s} = \prod_p \left(1 - \frac{\chi(p)}{p^s}\right)^{-1}, \qquad \mathrm{Re}\,s > 1, \]

continuing to an entire function for \(\chi\) non-principal. For primitive \(\chi\) of conductor \(q\) there is a functional equation relating \(L(s,\chi)\) to \(L(1-s, \bar\chi)\), with a root number \(\varepsilon(\chi)\) built from the Gauss sum \(\tau(\chi) = \sum_{a} \chi(a) e^{2\pi i a/q}\) and satisfying \(|\varepsilon(\chi)| = 1\).

Dirichlet's theorem on primes in progressions is exactly the statement \(L(1, \chi) \ne 0\) for non-principal \(\chi\), which is why the theorem on the distribution page belongs to this circle of ideas.

Conjecture (generalised Riemann hypothesis, GRH). Every nontrivial zero of every Dirichlet \(L\)-function lies on \(\mathrm{Re}\,s = 1/2\).

GRH is a conjecture. It matters operationally: PARI's quadclassunit and bnfinit assume it for their generator bounds, and Numerisect therefore marks class numbers that could not be certified with bnfcertify as conditional on GRH — see the algebra laboratory.

In Numerisect. Dirichlet character table (POST /api/zeta/characters) uses dirichlet_group_init, dirichlet_char_next, dirichlet_conductor_char, dirichlet_parity_char and dirichlet_order_char; modulus up to 100 000. Evaluate \(L(s,\chi)\) (POST /api/zeta/l-function) computes with acb_dirichlet_l, cross-checked against an independent acb_dirichlet_l_hurwitz evaluation, and reports the root number and Gauss sum; every value is certified. \(L(1, \chi_{-4}) = \pi/4 = 0.78539816339744830961566\ldots\).

Critical-line zeros of \(L(s,\chi)\) (POST /api/zeta/l-zeros) searches a grid for sign changes of \(Z(t, \chi)\), and here the certification is precise: a bracket counts only when the enclosures of \(Z\) at both endpoints are strictly of opposite sign, which proves a critical-line zero of odd order inside. For \(\chi_4(3, \cdot)\), the real primitive odd character of conductor 4, searching \(0.5 \le t \le 30\) on a 400-point grid certifies ten sign changes against a smooth \(\theta(T,\chi)/\pi\) count of 9.497; the first zero is at \(t = 6.020948904697596654\ldots\).

Complex characters have no real \(Z\)

Hardy's \(Z(t, \chi)\) is real-valued only for real characters. For a complex character such as \(\chi_5(2, \cdot)\) the page switches to MODE: magnitude-minima and reports minima of \(|L(\frac12 + it, \chi)|\), which are explicitly exploratory indicators and not certified zeros.

Dedekind zeta functions¶

For a number field \(K\),

\[ \zeta_K(s) = \sum_{\mathfrak{a}} \frac{1}{N(\mathfrak{a})^s} = \prod_{\mathfrak{p}} \left(1 - \frac{1}{N(\mathfrak{p})^s}\right)^{-1}, \]

summed over nonzero integral ideals. It generalises \(\zeta\) (which is \(\zeta_{\mathbb{Q}}\)) and has a simple pole at \(s = 1\) whose residue is given by the analytic class number formula

\[ \mathop{\mathrm{Res}}_{s=1} \zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2}\, h\, R}{w \sqrt{|d_K|}}, \]

with \(r_1, r_2\) the real and complex places, \(h\) the class number, \(R\) the regulator, \(w\) the number of roots of unity, \(d_K\) the discriminant. The extended Riemann hypothesis is the corresponding conjecture for \(\zeta_K\).

For abelian \(K\), \(\zeta_K\) factors into Dirichlet \(L\)-functions. The cleanest case is \(K = \mathbb{Q}(i)\), where \(\zeta_K(s) = \zeta(s)\, L(s, \chi_{-4})\), so the zero set of \(\zeta_K\) is the union of the zeta ordinates and the \(\chi_{-4}\) ordinates: \(6.0209\ldots, 10.2437\ldots, 12.9880\ldots, 14.1347\ldots\) — the fourth of which is the first zeta zero.

The Dedekind page is entirely exploratory

FLINT/Arb has no Dedekind zeta implementation, so this is the one zeta operation that leaves the C helper and runs in PARI/GP (zeta_fields.gp). PARI computes at a requested realprecision and returns floating-point numbers with no rigorous error bound. These are high-precision numerics, not Arb balls, and every value on that page is labelled exploratory — including the zero ordinates.

In Numerisect. Dedekind zeta (POST /api/zeta/dedekind) uses PARI nfinit, polcyclo, lfuncreate, lfuncheckfeq, lfun, lfunrootres, lfunzeros and bnfinit. For \(\mathbb{Q}(i)\) it reports degree 2, signature \((0,1)\), discriminant \(-4\), \(\zeta_K(2) = 1.50670300992298503088\ldots\) and residue \(\pi/4\), and with class_data enabled compares the residue against the class number formula. lfuncheckfeq returns the base-2 logarithm of the functional-equation error, so a large negative value means the \(L\)-function object is internally consistent. Bounds: degree \(\le 8\), \(|d_K| \le 10^{12}\), lfunzeros height \(\le 200\), realprecision 20–200 digits.

Ball arithmetic: what "certified" buys you¶

Floating-point arithmetic gives a number with no attached claim. Round-off accumulates, cancellation destroys digits silently, and a computed value near zero may be zero, may be tiny, or may be pure error — nothing in the result distinguishes those cases. For deciding whether a function changes sign, which is the entire business of counting zeros, that is fatal.

Ball arithmetic (also called midpoint–radius interval arithmetic) carries a radius through every operation. A value is represented as \([m \pm r]\), meaning a rigorous enclosure: the true value is guaranteed to lie in that interval. Each operation returns an enclosure of the true result of that operation applied to anything in the input enclosures. Radii grow, sometimes fast — that is honest, and the remedy is to raise the working precision and recompute — but they never lie. This is what FLINT/Arb implements (Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Computers 66, 2017).

The practical consequences in Numerisect:

A sign is certified when the enclosure excludes zero. \([0.3 \pm 0.01]\) is positive, certainly. \([0.003 \pm 0.01]\) has an unknown sign, and Numerisect reports inconclusive rather than guessing. Every Gram's-law verdict works this way.
A zero ordinate is an interval, not a number. acb_dirichlet_hardy_z_zeros returns an interval provably containing exactly one zero, printed with its radius.
A count is certified when a unique integer is isolated. If the enclosure of \(N(T)\) straddles two integers, the answer is inconclusive and the tool asks for more precision. It never rounds.
A verification is real when both sides are computed independently. The functional-equation page evaluates each side separately and reports success only if the balls overlap.

Why plot samples are exploratory¶

A chart is a finite set of points. Even when each point is computed in Arb, drawing it discards the radius and shows a midpoint, and nothing is known about the function between two adjacent pixels. So:

certified: zero intervals, \(N(T)\) from the Turing method, \(S(T)\), \(\theta(T)\), Gram points and \(Z(g_n)\), Riemann–Siegel rows, Euler-product values and bounds, \(L(s,\chi)\) values, sign-change brackets;
exploratory: the critical-line plot, the Argand trace, the complex heatmap, the \(S(T)\) trace, every histogram and pair-correlation bin, every truncated explicit-formula estimate, and all PARI output on the Dedekind page.

A curve that looks like it misses zero

A critical-line plot that appears to stay away from the axis between two samples proves nothing about that gap. If you want a statement about an interval, use the zero-isolation or sign-change tools, which return enclosures. The plot endpoints exist to suggest where to look, and their reports say so.

Where this appears in Numerisect¶

Topic	Tool (route)	Engine routine	Status
\(\zeta(s)\), magnitude, argument	Evaluate (`/api/zeta/evaluate`)	`acb_zeta`	certified
Euler product versus \(\zeta\)	Euler product (`/api/zeta/euler-product`)	`acb_pow` over `n_primes_t`, `_acb_dirichlet_euler_product_real_ui`	certified
\(\xi\), \(\eta\), continuation	Xi/eta (`/api/zeta/xi-eta`)	`acb_dirichlet_xi`, `acb_dirichlet_eta`	certified
Functional equation, both sides independently	Functional equation (`/api/zeta/functional-equation`)	two FLINT/Arb evaluations, balls compared	certified
\(\gamma_n\) at \(s=1\)	Stieltjes (`/api/zeta/stieltjes`)	`acb_dirichlet_stieltjes`	certified
\(Z(t)\) on the critical line	Hardy \(Z\) (`/api/zeta/hardy`)	`acb_dirichlet_hardy_z`	certified
Gram points	Gram (`/api/zeta/gram`)	`acb_dirichlet_gram_point`	certified
Gram's law, exceptions, blocks	Gram blocks (`/api/zeta/gram-blocks`)	`acb_dirichlet_gram_point`, `acb_dirichlet_hardy_z`	certified, else inconclusive
Consecutive zero intervals	Zeros (`/api/zeta/zeros`)	`acb_dirichlet_hardy_z_zeros`	certified
\(N(T)\) by the Turing method	Count (`/api/zeta/count`)	`acb_dirichlet_zeta_nzeros`	certified, else inconclusive
\(S(T)\) and its bound	Backlund \(S\) (`/api/zeta/backlund-s`)	`acb_dirichlet_backlund_s`, `..._bound`, `acb_dirichlet_hardy_theta`	values certified, trace exploratory
Riemann–Siegel main sum and corrections	Riemann–Siegel (`/api/zeta/riemann-siegel`)	`acb_dirichlet_zeta_rs`, `acb_dirichlet_zeta_rs_bound`, `acb_dirichlet_zeta`	certified
\(\pi(x)\) from the explicit formula	Explicit prime count (`/api/zeta/explicit-prime-count`)	`acb_dirichlet_hardy_z_zeros`, `acb_hypgeom_ei`, `arb_hypgeom_li`, `n_moebius_mu`, `primecount`	exploratory
\(\psi(x)\) from zeros	Chebyshev \(\psi\) (`/api/zeta/chebyshev-psi`)	`acb_exp`, `acb_div`, `n_primes_t`, `arb_log_ui`	exploratory
Unfolded spacings against GUE	Zero spacing (`/api/zeta/zero-spacing`)	`acb_dirichlet_hardy_theta`, `arb_hypgeom_erf`	exploratory
Montgomery pair correlation	Pair correlation (`/api/zeta/pair-correlation`)	`acb_dirichlet_hardy_theta`, `arb_sin`	exploratory
Character tables, conductor, parity, order	Characters (`/api/zeta/characters`)	`dirichlet_group_init`, `dirichlet_char_next`, `dirichlet_conductor_char`, `dirichlet_parity_char`, `dirichlet_order_char`	certified
\(L(s,\chi)\), root number, Gauss sum	\(L\)-function (`/api/zeta/l-function`)	`acb_dirichlet_l`, `acb_dirichlet_l_hurwitz`, `acb_dirichlet_root_number`, `acb_dirichlet_gauss_sum`	certified
Critical-line zeros of \(L(s,\chi)\)	\(L\)-zeros (`/api/zeta/l-zeros`)	`acb_dirichlet_hardy_z`, `acb_dirichlet_hardy_theta`, `acb_dirichlet_l`	brackets certified; smooth count and magnitude minima exploratory
\(\zeta_K\), residue, class number formula	Dedekind (`/api/zeta/dedekind`)	PARI `nfinit`, `lfuncreate`, `lfuncheckfeq`, `lfun`, `lfunrootres`, `lfunzeros`, `bnfinit`	exploratory (PARI floating point)
Critical-line, Argand and heatmap plots	`/api/zeta/line`, `/api/zeta/heatmap`	parallel `acb_zeta` samples	exploratory (midpoints)

References¶

H. M. Edwards, Riemann's Zeta Function, Academic Press (1974; Dover reprint) — the explicit formula, the Riemann–Siegel formula (Ch. 7), Gram points and the computational history. The best first book on this page's subject.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2^nd ed. revised by D. R. Heath-Brown, Oxford (1986) — the standard analytic reference: functional equation, \(N(T)\) and \(S(T)\), zero-density and mean-value theorems.
H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Colloquium Publications 53 — Chapters 5 and 10 on general \(L\)-functions, characters, and GRH.
H. L. Montgomery, "The pair correlation of zeros of the zeta function", Proc. Sympos. Pure Math. 24 (1973).
A. M. Odlyzko, "On the distribution of spacings between zeros of the zeta function", Math. Comp. 48 (1987), and the subsequent large-height computations.
C. L. Siegel, "Über Riemanns Nachlaß zur analytischen Zahlentheorie" (1932) — the Riemann–Siegel formula as recovered from Riemann's notes.
A. M. Turing, "Some calculations of the Riemann zeta-function", Proc. London Math. Soc. (1953) — the method behind rigorous zero counts.
D. J. Platt and T. S. Trudgian, "The Riemann hypothesis is true up to \(3 \cdot 10^{12}\)", Bull. London Math. Soc. 53 (2021).
L. Schoenfeld, "Sharper bounds for the Chebyshev functions \(\theta(x)\) and \(\psi(x)\), II", Math. Comp. 30 (1976) — the explicit RH-equivalent bound.
F. Johansson, "Arb: efficient arbitrary-precision midpoint-radius interval arithmetic", IEEE Trans. Computers 66 (2017) — the ball arithmetic every certified value on this page rests on.
FLINT/Arb acb_dirichlet documentation.
Numerisect: Riemann zeta tools, zeta lab, verification.