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To verify a Riemann zeta zero reliably, work on the critical line with arbitrary-precision arithmetic, evaluate the real Riemann–Siegel function Z(t), bracket each sign change, refine the bracket with a safeguarded solver, and compare your isolated-root count with an independent zero count. Recompute at higher precision and accept digits only when the root interval, residual, and count remain stable.
What you are actually checking
The nontrivial zeros of the Riemann zeta function satisfy ζ(s)=0 in the complex variable s. The Riemann Hypothesis predicts that every nontrivial zero has real part 1/2. A numerical check normally tests a candidate on that line, writing s=1/2+it, where t is real and positive.
Directly evaluating the complex value ζ(1/2+it) is useful for confirmation, but it is not the best function for scanning. Define the real-valued Riemann–Siegel function
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For real t, the phase factor makes Z(t) real (apart from numerical error), and a zero of Z(t) corresponds to a zero of ζ(1/2+it). Sign changes of Z therefore provide convenient root brackets. They are evidence of a crossing, not by themselves a proof that no second zero lies between your samples.
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Why ordinary floating point can produce a plausible wrong answer
A displayed value such as 0.0000000001 does not establish that a zeta value is zero. Several error sources can overlap:
- Rounding: finite-precision numbers round every intermediate result.
- Cancellation: large terms can nearly cancel, leaving a small result whose relative error is much larger than the apparent value.
- Truncation and approximation: infinite series, integrals, or asymptotic formulas are evaluated with finitely many terms or samples.
- Error propagation: an inaccurate phase, height, or earlier function value contaminates later calculations.
Common binary64 arithmetic carries roughly 16 decimal digits, but a root calculation can lose several of them through cancellation or through a poorly conditioned refinement step. A coarse grid can also jump over two closely spaced roots: the sampled signs may be equal even though the function crossed zero twice. Conversely, an unconstrained Newton iteration can converge repeatedly to the same root from different starting points, giving a false impression that a whole interval has been searched.
Keep the height, the real part 1/2, the phase, and every intermediate endpoint in arbitrary precision. Do not create an ordinary Python float first and then convert it to a high-precision number; the lost bits cannot be recovered.
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| Approach | What it is good for | What it cannot establish alone |
|---|---|---|
| Binary floating point | Quick exploratory plots and rough locations | Stable published digits, close-root separation, or reliable completeness |
| Arbitrary precision | Controllable rounding, high-precision values, and reproducible refinement | That no root was missed unless you bracket systematically and perform a count |
| Interval or ball arithmetic | Enclosing values and proving that a sign is positive or negative despite rounding | A complete list without a search strategy and zero-count argument |
For a practical Python workflow, mpmath supplies arbitrary-precision zetazero, nzeros, siegelz, and grampoint functions. Arb provides ball arithmetic, adaptive precision, and Turing-based counting routines. Use arbitrary precision for routine discovery and refinement; use interval or ball enclosures when a sign decision near zero must be certified.
Set precision deliberately
Set the working precision before constructing any inputs. Choose more digits than you plan to publish, with guard digits for the height, phase evaluation, and root refinement. There is no universal height-independent guard-digit number. The defensible rule is empirical: rerun at a substantially higher precision and verify that the reported digits, residual, and bracket agree.
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import mpmath as mp
mp.mp.dps = 80
half = mp.mpf('0.5')
t = mp.mpf('14.13472514173469379045725')
s = mp.mpc(half, t)
Use strings or integer-based constructions for decimal constants. Writing mp.mpf(0.5) is harmless for one-half because it is exactly representable in binary, but decimal heights should be written as strings. Never pass a binary float through a function merely because the final variable has an arbitrary-precision type.
Evaluate the right functions
Use siegelz(t) for real sign scans and for bracket refinement. Evaluate the complex zeta function independently at the final candidate.
z_real = mp.siegelz(t)
z_complex = mp.zeta(s)
print(z_real)
print(z_complex)
On the critical line, Z(t) should be real. A small imaginary component in an independently evaluated quantity is a useful warning that the requested precision is inadequate or that an approximation has lost accuracy. mpmath can select among Borwein, Riemann–Siegel, and Euler–Maclaurin zeta algorithms; the appropriate method depends on the region. Do not assume that one formula is optimal at every height.
Scan for candidate roots and preserve brackets
A scan should produce intervals, not just a list of approximate coordinates. Evaluate adjacent points at the same stated precision and retain both endpoints, their values, and their signs. A simple uniform scan is appropriate for a modest interval:
def sign_of(x):
if x == 0:
return 0
return 1 if x > 0 else -1
def scan_brackets(a_text, b_text, steps):
a = mp.mpf(a_text)
b = mp.mpf(b_text)
step = (b - a) / steps
brackets = []
left = a
f_left = mp.siegelz(left)
for k in range(1, steps + 1):
right = a + k * step
f_right = mp.siegelz(right)
sl = sign_of(f_left)
sr = sign_of(f_right)
if sl == 0:
brackets.append((left, left))
elif sr == 0 or sl != sr:
brackets.append((left, right))
left, f_left = right, f_right
return brackets
The grid spacing must be fine enough for the height range and the behavior of Z. A sign-change scan is not a completeness proof: two roots inside one cell can cancel each other’s sign changes. Gram points provide a traditional, structured set of sampling locations. mpmath exposes grampoint(n); Gram and Rosser blocks are particularly useful for organizing low-index searches. You still need to verify intervals and counts rather than assuming every Gram interval contains exactly one zero.
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Refine each bracket with a safeguarded solver
Start with bisection or a bracketed hybrid method. At every iteration, maintain the invariant that the endpoints have opposite signs, except when an endpoint itself evaluates to zero. A Newton or secant step may accelerate the final digits, but reject it when it leaves the bracket or when its function value is not trustworthy.
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def bisect_z(t_left, t_right, tol):
left = mp.mpf(t_left)
right = mp.mpf(t_right)
f_left = mp.siegelz(left)
f_right = mp.siegelz(right)
if f_left == 0:
return left, left, f_left
if f_right == 0:
return right, right, f_right
if sign_of(f_left) == sign_of(f_right):
raise ValueError('Endpoints do not bracket a sign change')
while right - left > tol:
mid = (left + right) / 2
f_mid = mp.siegelz(mid)
if f_mid == 0:
return mid, mid, f_mid
if sign_of(f_left) != sign_of(f_mid):
right, f_right = mid, f_mid
else:
left, f_left = mid, f_mid
mid = (left + right) / 2
return left, right, mp.siegelz(mid)
Return the final interval, not only its midpoint. The interval width is a direct statement about the location uncertainty. If you use a Newton correction, compute a derivative at the same precision, keep the old bracket, and fall back to bisection whenever the proposed point is outside the interval or does not reduce it.
Validate digits by increasing precision
Suppose a refinement at 80 decimal places returns an interval and midpoint. Repeat the entire evaluation and refinement at a higher precision, rather than merely printing the old midpoint with more digits. Compare three independent quantities:
- Location: the higher-precision interval should lie inside, or closely agree with, the lower-precision interval at the digits you intend to report.
- Residual: evaluate
|Z(t)|at the candidate and also evaluate|ζ(1/2+it)|. Both should decrease consistently as precision and refinement improve. - Method agreement: where practical, change the zeta evaluation method or use a second library. Agreement that survives the change is stronger than repeated calls to one potentially flawed path.
A small residual is necessary but not sufficient. Near a flat crossing, a function can be small over a comparatively wide interval; near a close pair, a coarse search can miss both roots. Publish only the digits supported by the stabilized interval and residual, not every digit emitted by the software.
Check that no zero was missed
Completeness requires an independent count. Let N(T) denote the number of nontrivial zeros up to height T, using the counting routine’s endpoint and multiplicity convention. For an interval (A,B], compare your number of isolated roots with N(B)-N(A). If the counts disagree, stop and investigate the scan, endpoint handling, precision, or the assumption that every candidate lies on the critical line.
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mpmath provides nzeros for zero counting. Its documentation describes Gram/Rosser organization for low-index searches and application of Turing’s method above index 400,000,000. Arb also documents Turing-based counting and increasing precision until the sign of Z(t) is determined. A count is independent of your particular list of sign changes, so it can expose a missed pair or a duplicate root.
As a documentation example, mpmath gives N(10^7)=21,136,125. This is an example from the documentation, not a newly measured statistic. Use the same routine and convention at both interval endpoints when making your own comparison.
lower_count = mp.nzeros(A)
upper_count = mp.nzeros(B)
expected = upper_count - lower_count
found = len(unique_roots)
if found != expected:
raise RuntimeError('Isolated-root count does not match zero count')
Do not count a root twice when neighboring scan cells share an endpoint. Merge overlapping final intervals and deduplicate candidates whose certified intervals intersect.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Cross-check the first zero correctly
The first nontrivial zero is commonly quoted as
1/2 + 14.13472514173469379045725i.
Use it as a software sanity check, not as evidence that your general search is complete. At high precision:
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t0 = mp.mpf('14.13472514173469379045725')
s0 = mp.mpc(mp.mpf('0.5'), t0)
print(mp.siegelz(t0))
print(mp.zeta(s0))
print(mp.zetazero(1))
The independently computed siegelz value and complex zeta value should both be close to zero, and zetazero(1) should agree in its reported digits. Treat the library’s zero finder as a cross-check; for a claim about a searched interval, retain your own brackets and perform the count.
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Common failure modes and fixes
A visually zero value is not stable
Symptom: the displayed digits change when precision is raised. Fix: increase precision before constructing inputs, rerun the complete calculation, and report only common stable digits.
The scan finds fewer roots than expected
Symptom: your sign-change list is shorter than the independent count. Fix: reduce the grid spacing, add Gram-point sampling, inspect intervals containing very small values, and search for a close pair inside every unexplained gap.
The solver returns the same root repeatedly
Symptom: different starting guesses converge to one coordinate. Fix: use disjoint retained brackets and reject any iteration that leaves its bracket. An unconstrained solver is a refinement tool, not an isolation method.
Complex and real evaluations disagree
Symptom: Z(t) appears near zero but ζ(1/2+it) is not, or the latter has a significant imaginary component. Fix: raise precision, verify that t was not first stored as a binary float, and compare evaluation methods.
A zero lies at a scan endpoint
Symptom: adjacent cells appear to have inconsistent signs or the root is counted twice. Fix: treat an endpoint zero as one candidate, merge intervals that touch, and apply the counting routine’s endpoint convention consistently.
The count changes with precision
Symptom: a Turing or sign determination is undecided at the current precision. Fix: increase precision until the sign or interval enclosure is resolved. Ball arithmetic is useful when a tiny value must be certified as positive or negative.
Comparing evaluation and search strategies
| Decision | Option | Practical implication |
|---|---|---|
| Arithmetic | Arbitrary precision | Choose digits explicitly and validate by rerunning higher. |
| Arithmetic | Interval or ball | Carry enclosures and certify signs when ordinary approximations are too close to zero. |
| Zeta formula | Borwein | Available in mpmath; suitability depends on the region and requested accuracy. |
| Zeta formula | Riemann–Siegel | Useful for high-height evaluations when its error control is adequate. |
| Zeta formula | Euler–Maclaurin | Another mpmath option whose cost and accuracy depend on the height. |
| Root isolation | Unbracketed Newton or secant | Fast when close to a known root, but can jump intervals or duplicate a root. |
| Root isolation | Bracketed bisection or hybrid | Slower but preserves an explicit location enclosure. |
| Completeness | Residuals only | Confirms individual candidates but cannot show that none were missed. |
| Completeness | Independent zero count | Compares the number found with a Turing-based or equivalent count. |
A reproducible reporting checklist
- State the arithmetic precision used for discovery, refinement, and the higher-precision rerun.
- Record the exact decimal endpoints of every final bracket and the evaluated endpoint signs.
- Give the midpoint or representative root only to digits supported by the stabilized interval.
- Report the residuals for both
Z(t)andζ(1/2+it). - Identify the zeta evaluation method when the library allows method selection.
- Explain the scan spacing or Gram-point scheme and how overlapping candidates were merged.
- Give the expected count from
N(B)-N(A), the number of isolated roots, and the convention used at endpoints. - State whether an independent implementation or interval/ball calculation confirmed the signs.
The essential distinction is between finding a small value and certifying a zero. A reliable computation combines controlled arithmetic, a retained bracket, precision-stability evidence, and an independent count. That combination addresses rounding, cancellation, approximation error, close roots, duplicate convergence, and missed zeros instead of relying on a convincing-looking decimal.
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