_dfloat_platt_smk, _dfloat_platt_smk_dd

_dfloat_platt_smk_dd omits the common factor $c = e^{-it_0\log\sqrt{\pi}}$ and writes compact balls of five doubles per entry (the real and imaginary midpoints as double-words and a common radius): S5[5 (k N + m) + 0..4] holds Re hi, Re lo, Im hi, Im lo and the radius, with $N = AB$. M bounds the table of terms (0 for the default $\min(J/2, 5 \cdot 10^6)$).

The rest of the multi-evaluation amplifies an error in the moment $k$ by about $2^{24 + 8.6k}$, while $|S_k|$ decreases like $2^{-13k}$. So the terms $j^{-s}$ and $b_j$ are d3b balls (with phases in d3b or d4b), $S_0$ is summed in d3b balls, and the moments $1 \le k < 14$ are computed in plain double-double arithmetic and $k \ge 14$ in double arithmetic, vectorised over the $j$ of a block, with an a priori error bound added per bucket. The terms are obtained by sieving: $f(j) = f(d) f(j/d)$ and $\log j = \log d + \log(j/d)$ for the smallest prime factor $d$, from a table for $j \le M$ and by splitting $j = xy$ with $x, y \le M$ beyond. The work is split into chunks computed in parallel and summed in a fixed order, so the result does not depend on the number of threads. At the height of the $10^{15}$-th zero this takes about 0.42 microseconds per $j$ on one thread, against 7.7 for the arb_t version at 228 bits.

@misc{flint-doc-dfloat-dfloat-platt-smk,
  author       = {{The FLINT team}},
  title        = {{\_dfloat\_platt\_smk, \_dfloat\_platt\_smk\_dd (dfloat.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/dfloat/_dfloat_platt_smk.html},
  note         = {Generated 2026-11-02}
}