Support run `e6241336`: one Epstein value, checked

The answer: yes. At dps=60,

abs(epstein_completed(0.8+85.7i, (2,1,3))) = 1.617452632377971461454064e-58

which is 1.0109 x the claimed 1.6e-58. That is 1.1% away, so it agrees to within an order of magnitude by a wide margin: log10 of the ratio is 0.0047, against the 1.0 an order of magnitude would allow.

The full complex value at dps=60 is

-1.566243383527633990474456e-58 - 4.037755336103688597427175e-59 i

Reproduce with python hunts/support_e6241336/measure.py (about 80 s). Data: results.json.

How much of that number is real

dps=60 does not buy 60 digits here. epstein_completed splits the Mellin transform of the form's theta series at t = 1; at s = 0.8 + 85.7i the pole terms alone are of size 1e-3 while the answer is of size 1e-58, so about 55 digits cancel. The working precision is dps + 10, which leaves roughly 15.

Measured, rather than argued:

dps=40 is still the roundoff floor and is 5 orders too large. dps=50 has about 5 correct digits, dps=60 has 16.0 (relative error 1.021e-16 against the dps=120 value), and 80 and 120 agree to all 25 digits printed. So the dps=60 return answers the question asked and is good to 16 digits, but a caller who prints 25 of them is printing 9 digits of noise.

This reproduces hunts/dps_cap (Hunt #12), which measured the same quantity for a different reason: 80 and 120 here match its recorded converged value 1.617452632377971296327251e-58 digit for digit.

The three checks, none of which is "run it again"

The ladder above only says the routine converges. It does not say it converges to the right thing, so three checks were run against routes that do not share its cancellation.

A. Normalisation, at a point where the Dirichlet series converges. Lambda_Q(s) / ((sqrt(d)/pi)^s Gamma(s)) must equal sum_{(m,k) != (0,0)} Q(m,k)^{-s}. At s = 4, summed over a 801 x 801 lattice, the relative difference is 1.9e-17 for (1,1,6) and 3.7e-17 for (2,1,3) and (2,-1,3). So the completion the routine applies is the one the docstring claims, with d = |D|/4 = 5.75.

B. The Dedekind identity, at the actual point. Discriminant -23 has class number 3 with reduced forms (1,1,6), (2,1,3), (2,-1,3), and

sum_Q Lambda_Q(s) = 2 (sqrt(23)/2pi)^s Gamma(s) zeta(s) L(s, chi_-23)

the 2 being the number of units. The right-hand side was built from mpmath's zeta and Hurwitz zeta, with chi_-23 the Legendre symbol (n/23). That route reaches a quantity of size 1e-58 with no cancellation at all: the smallness comes from Gamma(0.8+85.7i) directly, which mpmath evaluates to full relative accuracy. The two sides agree to 4.0e-17 at dps=60 and 6.3e-76 at dps=120.

This is the check that makes the number quotable. A defect in the lattice sums would have to conspire across three forms to leave that identity standing, and the identity is the one place where a route with no cancellation meets a route with 55 digits of it.

C. The functional equation across the inverse class. Lambda_(2,1,3)(s) against Lambda_(2,-1,3)(1-s): relative difference 4.4e-16 at dps=60 and 1.5e-75 at dps=120.

One defect found, in this run's own instrument

The first pass built s = mpc("0.8","85.7") at module scope, where mp.dps is mpmath's default 15. That rounds the argument to 53 bits. Since d/ds log Lambda is about log|s|, roughly 4.5, at this point, a half-ulp error in 85.7 moves the magnitude in its 15th digit: the low-precision argument returns 1.617452632377967865948261e-58, which is 2.1e-15 off. The first pass therefore disagreed with Hunt #12 at the 16th digit and, briefly, took its own value for the reference one.

epstein_completed cannot repair this. It calls mpmathify on whatever it is handed, inside workdps(dps + _GUARD), but an mpf that arrives short of digits stays short. Passing a string or building s inside a high-precision block both work; passing a Python complex, or an mpc built at the ambient default, does not. The same trap silently capped this run's first version of check C at 2.5e-15, because 1 - s was formed at the ambient precision.

Recorded as a thread in HANDBACK.json, not acted on: touching zeta/ is outside this run's scope.

Scope

One point, one form, two routines. Nothing here is evidence for or against the Riemann hypothesis (docs/08-why-it-is-hard.md). Nothing here says anything about where the Davenport-Heilbronn off-line zero sits, even though 0.8 + 85.7i is near it: epstein_completed on the form (2,1,3) and the Davenport-Heilbronn function are different objects, and this run evaluated only the first. The word this run is entitled to is measured, at the first rung of the ladder for the value itself and at the second for the agreement between independent routes.