Classical zero-free region constant

Upper bound, last-listed row

| $4.896$ | [BTY2026] | Bellotti–Trudgian–Yang: $\zeta(\sigma+it)\ne 0$ for $t\ge 3$ and $\sigma > 1 - 1/(4.896\log t)$; extends to $\lvert t\rvert \ge 2$ using verified RH up to $H = 3\cdot 10^{12}$. A companion Theorem 2 sharpens the constant further to $4.8594$ conditionally on unpublished improvements to prior inputs. |tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $2/\log \gamma\_{1} \approx 0.755106$ | - | Unconditional: $\gamma\_{1} = 14.1347\dots$ is the ordinate of the lowest zero of $\zeta$, and that zero has $\sigma = 1/2$, so any admissible $R$ must satisfy $1/2 \leq 1 - 1/(R\log\gamma\_{1})$. Optimal assuming RH. |bound cell edited 2026-09-05

Checked against material for its cited source

• Another row, not one shown above: bound row $9.64591$ line 13 citing [S1970] — the claim we checked: that Stechkin's cited 1970 paper gives the upper bound this row attributes to it, 9.64591. What we read could not settle it for the edition the entry cites: the Russian original's Theorem 2 gives R = 9.65 (with T = 12), and we found no finer value of R in it; the body of the English translation could not be read. Rosser and Schoenfeld's 1975 Theorem 1 gives R = 9.645908801, which rounds to 9.64591, and they present it as improving Stechkin's Theorem 2 with a smaller R. This verdict does not cover whether the bound itself holds. source link (read in the Russian original (Matematicheskie Zametki 8(4), 1970, pp. 419–429), all eleven pages, from Math-Net.ru's scan; the English translation our entry cites (Mathematical Notes 8(4), pp. 706–711) could not be reached, because Springer answered with a bot challenge and no archived copy was found. Rosser and Schoenfeld 1975 (Mathematics of Computation 29(129), pp. 243–269) was read at p. 250 from an archived copy of the AMS file). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

1 bound row(s) here have been checked against material for their cited source (1 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. A row not listed here has NOT been checked.