M(1024) = 7,189,337,839
The browser does not compute this point. Hurst reports it in arXiv:2607.07566, submitted 8 July 2026. The normalized value below is calculated from the imported integer when this page runs: .
Two finite truths and the wall beyond them
Everything You Can Count Says Yes
Count the Möbius function and the primes as far as this browser can reach. Every visible point supports two seductive inequalities, then two published zero-sum certificates show why finite agreement is not proof and certify failure far beyond the plotted horizon.
For more than a century, computation kept agreeing with stories that were globally false. This page does not ask you to admire that irony. It gives you the counting machine, the raw arrays, and the certificate ledgers.
Four different statements live here. A checked interval is not an isolated value. An existence bound is not a located witness. The distinction is the whole instrument.
Layer one · count
Run to the edge of your machine
The worker factors consecutive blocks. It computes every Möbius value, every cumulative Mertens sum, every prime flag, and every prime count in the chosen interval. The logarithmic integral uses the principal-value convention li(x) = Ei(log x).
Ready.
selected x
waiting
M(x)
waiting
|M(x)| / sqrt(x)
waiting
pi(x) - li(x)
waiting
|M(x)| / sqrt(x), false conjecture line at 1
Run the count to draw the exact integer data.
pi(x) - li(x), sign-change line at 0
Run the count to draw the exact prime counts and numerical integral.
M(1025) = -258,560,632,948
This is also isolated, not a check of every smaller integer. The live normalized value is . Hurst reports validation checks, but the preprint does not report a fully independent computation at this scale.
Your plot did not find a failure. That sentence contains no evidence that a failure is absent beyond the plot. Now exchange scale for a certificate.
Layer two · certify
Put pressure on the error budget
Each bench imports a finite-sum result and every stated allowance from an accessible peer-reviewed paper. The browser recomputes the final inequalities. It does not pretend to re-sum the tens of thousands or hundreds of millions of zeta zeros that are not shipped in this single file.
Mertens certificate, 2014 ledger
Saouter and te Riele report K2 ≈ -1.008867 from zeros below 74,000 and bound the omitted zero contribution through 300,000 by 5.69 × 10-8. Stress that published remainder until the conclusion breaks.
calculating
calculating
Prime sign certificate, 2015 ledger
Saouter, Trudgian, and Demichel used 525 million zeros. Their optimized ledger starts from the published mean J(omega0), subtracts five analytic allowances, roundoff, and the interval-shrinking allowance. Stress every subtractive term together.
calculating
calculating
The logic ledger
No row is allowed to impersonate another.
| kind | statement | what it licenses | what it does not license |
|---|---|---|---|
| checked interval | Every M(x) through 1016, published in Hurst 2018 | The least Mertens failure is greater than 1016 | No conclusion at 1016 + 1 or beyond |
| isolated evaluation | Published M(1024) and M(1025) values | Those two values only, subject to the preprint's validation | No contiguous extension from 1016 |
| existence certificate | A Mertens failure exists before approximately exp(1.96 × 1019), reported by Kim and Nguyen in 2025 | A rigorous upper bound if the manuscript's certified computation is accepted | No explicit integer counterexample and no known sign of the first one |
| existence certificate | More than 7.17 × 10152 consecutive integers with pi(x) > li(x) occur in the 2015 published interval | A positive run somewhere inside that interval | Not positivity throughout the interval and not the first crossing |
| explicit witness | None known for either least failure | Nothing to inspect yet | An existence proof must not be drawn as a located point |
The check
Recomputed from raw browser arrays
Waiting for the first segmented count.
| quantity | live value |
|---|---|
| status | waiting |
Conventions and free choices
- Indexing: sums include every integer n with 1 ≤ n ≤ x. The prime count includes primes p ≤ x.
- li convention: principal-value li(x) = Ei(log x), not the integral beginning at 2. The page evaluates the NIST DLMF power series with Kahan summation.
- Numerical interval: the displayed li interval bounds the positive series tail under real arithmetic. Binary64 rounding is not outward-rounded, so it is a numerical error estimate, not a rigorous interval proof.
- Free choices: the endpoint, fixed segment size of 32,768, selected scrubber position, and certificate stress factors are reader choices. Stress factors other than 1 are counterfactual diagnostics, not published results.
- Timing: elapsed time is device-specific. Integer sums and counts are exact within JavaScript's safe-integer range used here.
What remains open, and what was narrowed
The exact least integer violating |M(x)| < sqrt(x) is unknown, and so is its sign. The first x with pi(x) > li(x) is unknown. The Riemann hypothesis remains open. Its Mertens equivalent is the weaker family M(x) = O(x^(1/2 + epsilon)) for every positive epsilon, not the disproved sharp inequality.
The supplied build specification called exp(1.004 × 10^33) the current Mertens upper bound. Primary-source checking found later bounds, including the 2025 Kim and Nguyen preprint's approximately exp(1.96 × 10^19), so the older number is retained only as the 2014 bench's historical output.
Revers's peer-reviewed 2026 article and DOI were verified, and its abstract confirms improved Lehman error terms. The exact numerical table required for the supplied 2-million-zero reproduction was not accessible in primary full text during the 29 July 2026 check. Those exact 2026 numbers are not used by this page. The fully accessible 2015 peer-reviewed certificate is operated instead.