Featured publication · Liju James · 2026
The Mirror: verified classical structure of the Riemann zeta function
An expository and computational verification collection organized around the functional equation, finite zero checks, and the limits of what those checks imply.
30 decimal places
First 1,000 zeros
108
RH remains open
The organizing idea
The functional equation Λ(s)=Λ(1−s) supplies the mirror symmetry. Verified zero locations form a finite, load-bearing layer for computations. The collection then shows why symmetry alone does not settle the Riemann Hypothesis or control every finite-scale application.
What was checked
- The functional equation, mirror pairing of the first 50 nontrivial zeros, special values, and the right-edge halving law at 30 decimal places
- A fresh sieve reproduction of Chebyshev's bias through 108, including the first lead change at x=26,861
- A partial Euler-product illustration for L(s,χ4) at s=1/2
- An independent check that the first 1,000 nontrivial zeros lie on the critical line, pairing Hardy Z sign changes with a Backlund/Turing strip count
The paper does not claim a proof or disproof of the Riemann Hypothesis. Its value lies in reproducible finite verification and in making the logical boundary of that evidence explicit.
Reproducibility
The archive includes six figures, scripts using standard Python together with mpmath and NumPy, and independent re-run reports. A related Weil-positivity numerical study is presented as a sensitivity result, not as a route around the open problem.