To verify a nontrivial Riemann zeta zero reliably, work on the critical line with the real Riemann–Siegel function Z(t), evaluated at arbitrary precision. Bracket every candidate sign change, refine the bracket with a safeguarded solver, rerun at higher precision until the digits and residual stop changing, and compare your isolated-root count with an independent zero count such as Turing’s method. A small computed value of |ζ(1/2+it)| by itself is not proof that you found a zero or that you found them all.
The workflow below is designed to expose the two ways ordinary floating-point calculations most often produce convincing but wrong answers: a value can be inaccurate because of rounding, cancellation, truncation or propagated error, and a scan can jump over two nearby roots. Keep the height, phase and intermediate results in arbitrary precision, preserve brackets, and report the precision and interval used for every zero.
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What a numerical check must establish
The Riemann zeta function is ζ(s). The nontrivial zeros are expected in the strip 0 < Re(s) < 1; the Riemann hypothesis predicts that they all have real part 1/2. A numerical check normally asks whether a particular height t gives a zero on that line, where s = 1/2 + it.
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Z(t) = eiθ(t) ζ(1/2 + it).
For real t, a zero of Z(t) is a zero of ζ(1/2 + it). Using Z for a scan is important: sign changes can be located with real arithmetic, while the complex zeta value remains an independent check.
A convincing result has three parts
- Isolation: an interval [a, b] whose endpoint evaluations have opposite signs (or an interval enclosure proving a root).
- Refinement: a much narrower interval and a reported approximation whose residual is small at the stated working precision.
- Completeness: an independent count showing that the number of roots you isolated in the height range equals the number expected there.
Without the third part, a list of accurate-looking roots can still omit a pair that your sampling grid skipped.
Why ordinary floating point can give a plausible wrong answer
Binary floating point, commonly binary64, stores only a fixed number of significant bits. The error budget is broader than the last displayed decimal place. The mpmath technical guide identifies rounding and cancellation from finite precision, plus truncation or approximation when infinite sequences or continuous functions are replaced by finite calculations. Error can also propagate from an earlier inaccurate value into every later operation.
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Cancellation near a zero
Several large terms in a zeta or Riemann–Siegel evaluation may nearly cancel. The final result can be tiny even when the absolute error in those terms is much larger than the tiny answer. Printing more decimal places does not recover information that was lost during the calculation.
A grid can jump over two roots
If two zeros lie between consecutive sample points, the sign at both endpoints may be the same. A sign-scan that reports only endpoint changes will miss both. Sampling the complex magnitude does not solve this: a coarse grid can pass over a narrow pair, and a local minimum of |ζ| need not be a zero.
An unconstrained solver can revisit one root
Newton or secant iterations started from different guesses can converge to the same zero. They can also leave the region you intended to search. A converged iteration and a small residual therefore do not prove that the root is isolated or that neighboring roots were found.
Height and phase are part of the calculation
Do not convert an arbitrary-precision height to a binary64 value before evaluating Z(t). The phase in the factor eiθ(t) and the argument 1/2 + it must be formed at the same deliberate precision as the function evaluation. Keep decimal inputs as arbitrary-precision numbers rather than ordinary language-level floats.
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Choose an arithmetic model and evaluation method
The following distinctions matter more than the choice of a particular root-finding syntax.
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| Decision | Option | What it provides | What it does not prove |
|---|---|---|---|
| Arithmetic | Binary floating point | Fast exploratory calculations. | Stable digits near cancellation, or a reliable enclosure. |
| Arithmetic | Arbitrary precision | Adjustable precision and repeatable precision-increase tests. | Automatic completeness; an arbitrary-precision result can still be based on an inadequate scan. |
| Arithmetic | Interval or ball arithmetic | Enclosures that account for rounding and can certify signs when intervals exclude zero. | A certificate if the interval remains too wide to determine the sign. |
| Zeta evaluation | Borwein, Riemann–Siegel or Euler–Maclaurin formulas | Different algorithms suited to different regions; mpmath supports all three families. | Identical speed or error behavior at every height. |
| Root isolation | Unbracketed Newton or secant | Fast local convergence when a good starting point is already known. | Isolation, containment or protection against converging to a duplicate. |
| Root isolation | Bracketed bisection or a safeguarded hybrid | Preserves an interval containing a sign-changing root. | Completeness outside the intervals you searched. |
| Completeness | Residual and visual inspection | A useful local sanity check. | Evidence that no zero was skipped. |
| Completeness | Independent zero count, such as Turing’s method | A global check on how many zeros should occur in a height range. | A substitute for accurate local brackets. |
Arbitrary precision is a process, not a magic digit count
There is no height-independent rule such as “add 20 guard digits.” Select a working precision above the digits you intend to publish, include extra digits for the height, phase and refinement, then validate the choice by repeating the entire calculation at higher precision. Keep increasing precision until the root interval, residual and zero count are unchanged at the digits you report.
Interval and ball arithmetic
Arb is an arbitrary-precision ball-arithmetic library that documents adaptive precision and Turing-based counting. When an evaluation returns an interval that lies wholly above or below zero, its sign is determined at that precision. If the interval overlaps zero, increase precision or subdivide the interval; do not treat an indeterminate sign as a sign change.
Reproducible workflow
1. Define the height range and the output requirement
Write down whether you need one named zero, all zeros in [T0, T1], or the first n zeros. Decide how many decimal digits you will publish. The required precision depends on the height, the separation of nearby zeros, the evaluation method and the requested tolerance, so the final answer must be validated rather than guessed.
2. Set precision before creating numbers
In mpmath, set mp.dps before constructing the height and any constants. Use strings such as mp.mpf('14.0'), not binary floating-point literals, when a decimal value is intended. Keep guard digits internally and round only when formatting the final result.
3. Evaluate the real Z-function for the scan
Use mpmath’s siegelz(t) for real-line values. Its zeta implementation can select Borwein, Riemann–Siegel or Euler–Maclaurin algorithms according to the region. Record the method and precision if your software allows explicit selection; changing the method is a useful independent check.
4. Locate and preserve brackets
Sample a sufficiently fine grid and retain every pair of adjacent points with opposite signs. Store each endpoint, its sign and the precision used. Gram points, available through mpmath’s grampoint, provide a structured set of sample locations, but Gram’s law is not an unconditional completeness guarantee. A missed sign change or an even number of roots in one interval still requires a count-based check.
When the function value is so close to zero that its sign is uncertain, increase precision or use an interval evaluation. Never coerce an uncertain value to a sign merely because its printed decimal representation is positive or negative.
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Bisection is slow but retains the sign-changing interval. A bracketed hybrid method can use secant or Newton steps for speed, provided it rejects any step outside [a, b] and falls back to bisection. Stop only when the interval width is below your requested uncertainty and the function evaluations are accurate enough to support that decision.
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6. Check both Z(t) and ζ(1/2+it)
At the refined height, evaluate s = mp.mpf('0.5') + mp.j*t, then evaluate both mp.siegelz(t) and mp.zeta(s). The two checks test different parts of the computation: Z is the real scan function, while ζ is the original complex function. Agreement should be tested at the working precision, not inferred from a short printed value.
The first nontrivial zero is a useful format check. The mpmath documentation gives it as 1/2 + 14.13472514173469379045725i. Reproducing those digits is not, by itself, a proof of your implementation’s completeness or of additional digits beyond those you have validated.
7. Compare with an independent zero count
Use a zero-counting routine for the same height interval. mpmath’s nzeros provides counts up to a height; for an interval, compare the counts at its upper and lower boundaries, taking care with the endpoint convention. Its documented search machinery uses Gram/Rosser blocks at low indices and applies Turing’s method above index 400,000,000. Arb also documents Turing counting with precision increased until the sign of Z(t) is determined.
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N(b) − N(a) = number of isolated zeros in (a,b],
with any zero exactly on a boundary handled according to the counting routine’s convention. A mismatch means the search is incomplete, a bracket is duplicated, an endpoint was counted differently, or the count itself needs higher precision. Investigate the mismatch instead of adjusting the list to make the totals agree.
8. Repeat independently
Raise the working precision and rerun the scan, refinement and count. Where practical, change the evaluation method or use a second library. Accept a digit only when the refined interval, residual and independent count remain stable. A second run at higher precision is a practical verification technique recommended in mpmath’s numerical-analysis guidance, but it is a test of stability rather than a formal proof in every situation.
Minimal mpmath example with a retained bracket
The following example illustrates the mechanics for the first zero. The sample endpoints are deliberately broad; for a production scan, generate and retain many brackets over the complete range of interest.
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mp.dps = 80
def Z(t):
return mp.siegelz(t)
a = mp.mpf('14')
b = mp.mpf('15')
fa = Z(a)
fb = Z(b)
if fa == 0:
b, fb = a, fa
elif fa * fb > 0:
raise ValueError('The endpoints do not bracket a sign change')
for _ in range(300):
m = (a + b) / 2
fm = Z(m)
if fm == 0:
a = b = m
fa = fb = fm
break
if fa * fm <= 0:
b, fb = m, fm
else:
a, fa = m, fm
if b - a < mp.mpf('1e-70'):
break
root = (a + b) / 2
s = mp.mpf('0.5') + mp.j * root
print(mp.nstr(root, 70))
print(mp.nstr(Z(root), 20))
print(mp.nstr(mp.zeta(s), 20))This code proves only that the retained interval contains a sign-changing point at the evaluation precision. It does not show that no second root lies inside the interval, nor that no root was skipped before a or after b. Those claims require a finer isolation strategy and a count.
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Scanning strategies and their limits
Uniform sampling
A uniform grid is easy to audit: record the start, end and step as arbitrary-precision values, evaluate every point, and retain all sign changes. Its weakness is that no fixed step is a universal guarantee at every height. Closely spaced roots or a pair with no net sign change between grid points can defeat it. Reduce the step adaptively where values are small, where the phase changes rapidly, or where a count indicates missing roots.
Gram-point scanning
Gram points are natural sampling locations for Z(t) and can organize a search into Gram or Rosser blocks. They are efficient for indexing low zeros, but Gram’s law has exceptions. Treat the points as a search framework, not as a proof that every interval contains exactly one root.
Adaptive subdivision
Start with broad intervals, then subdivide whenever an endpoint sign is uncertain, a function value is unusually small, or the interval contains more roots according to the independent count than your brackets account for. Subdivision also helps separate a close pair before refinement.
How to validate published digits
Suppose the midpoint of your final bracket is t*. Report more than the rounded value:
- the bracket endpoints and its width;
- the working precision and the higher precision used for confirmation;
- the value of Z(t*) and, separately, the complex residual ζ(1/2 + it*);
- the evaluation method or library version when it is relevant to reproducibility;
- the height range and the zero-counting result used for completeness.
Round only after the higher-precision run. If the first run gives 40 matching decimals but the 60-digit rerun changes the 25th decimal, publish no more than the stable digits. A residual written as “zero” because it underflowed or was rounded to zero is not evidence; print enough significant digits to show its scale.
Interpreting a zero count
The count is a global constraint, not a substitute for local numerical work. If your isolated-root list has fewer entries than the count predicts, inspect unsampled intervals and regions where the sign was indeterminate. If it has more entries, check for duplicated brackets, endpoint double-counting and repeated convergence to one root.
As a scale check, the mpmath zeta-functions documentation gives N(107) = 21,136,125. This is a documentation example, not a newly measured statistic. It illustrates why a count should be treated as a separate computation: at high heights, a visually inspected list is not a practical completeness argument.
Troubleshooting common failure modes
There is no sign change at the expected zero
Check that you evaluated Z(t), not the complex value ζ directly, and that both endpoints were formed at arbitrary precision. Increase precision if either endpoint is close to zero. Then subdivide the interval; a pair of roots can produce equal endpoint signs.
The displayed digits change when precision increases
The original precision was insufficient, or the evaluation suffered cancellation. Recreate all numbers after increasing mp.dps; do not merely reprint values that were already rounded. Continue until the interval, residual and count stabilize.
Two starting guesses return the same root
Use a retained bracket for each candidate and compare the intervals, not just the rounded roots. If the intervals overlap, rescan and subdivide. An unbracketed solver is allowed to converge repeatedly to the same attractor.
Z(t) is small but ζ is not comparably small
Verify the argument and phase: use 1/2 plus mp.j*t, not a binary-float imaginary component. Check that the same precision and consistent value of t were used. A method-selection or cancellation problem is also possible; rerun with higher precision and, where available, another zeta algorithm.
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The isolated count disagrees with nzeros or Turing’s method
First check endpoint conventions and whether a root lies exactly at a boundary. Then search for skipped sign changes, unresolved signs and duplicate brackets. If the count routine reports an indeterminate sign or interval, increase precision before interpreting the result.
Increasing precision makes the calculation very slow
Use a staged workflow: scan at a moderate precision, retain candidate brackets, then spend high precision only on refinement and verification. Choose an evaluation formula suited to the height, and avoid printing or converting every intermediate value. Speed is useful, but never replace a count or bracket with a faster unverified guess.
A compact reproducibility record
For each run, keep a machine-readable record containing:
- the software and library versions;
- the decimal precision and any interval or ball radius;
- the evaluation formula and options;
- the scan range, grid or Gram-point rule, and every retained bracket;
- the root interval, midpoint, residuals and stopping tolerance;
- the independent count, its height boundaries and endpoint convention;
- the higher-precision rerun and the digits that remained unchanged.
This record lets another reader distinguish a genuinely isolated, counted zero from a short decimal produced by an unconstrained solver.
Bottom line
Use arbitrary precision to evaluate the real Riemann–Siegel function, bracket before refining, keep every interval and sign at its stated precision, verify with the complex zeta function, and certify completeness with an independent count such as Turing’s method. Ordinary floating point and a tiny residual are suitable for exploration, not for claiming that a zero is correct to published digits or that no neighboring zero was missed.
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