E091: Partial Euler products on the critical line

E091 hero

E091: Partial Euler products on the critical line

Tags: analysis, quantitative-exploration, visualization, riemann-zeta, dirichlet-series, numerics

Highlights

  • Focused numeric experiment with a single main figure.

  • Parameters saved to params.json for reproducibility.

  • Defaults are chosen, so the experiment remains feasible for the CI “slow” suite.

What is computed

  • Compare (\zeta(1/2+it)) to partial Euler products (\prod_{p\le P} (1-p^{-s})^{-1}) at a fixed t while increasing the prime cutoff P.

  • Plot the mismatch to illustrate non-convergence on the critical line.

Notes

  • The Euler product is only convergent for (\Re(s)>1); this experiment visualizes the failure mode at (\Re(s)=1/2).

Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

We compare zeta(1/2 + i t) (t=10.0) to partial Euler products truncated at primes <= p_max.

The Euler product does not converge on the critical line, so the partial products do not stabilize as p_max grows (in contrast to the Re(s) > 1 case).

params.json (snapshot)
{
  "mp_dps": 70,
  "prime_cutoffs": [
    10,
    30,
    100,
    300,
    1000,
    3000,
    10000
  ],
  "t": 10.0
}

References

  • See the zeta / Dirichlet-series references in refs.bib.