Turing's method

In mathematics, Turing's method is used to verify that for any given Gram point g_m there lie m + 1 zeros of ζ(s), in the region 0 < Im(s) < Im(g_m), where ζ(s) is the Riemann zeta function.^[1] It was discovered by Alan Turing and published in 1953,^[2] although that proof contained errors and a correction was published in 1970 by R. Sherman Lehman.^[3]

For every integer i with i < n we find a list of Gram points ${\displaystyle \{g_{i}\mid 0\leqslant i\leqslant m\}}$ and a complementary list ${\displaystyle \{h_{i}\mid 0\leqslant i\leqslant m\}}$, where g_i is the smallest number such that

${\displaystyle (-1)^{i}Z(g_{i}+h_{i})>0,}$

where Z(t) is the Hardy Z function. Note that g_i may be negative or zero. Assuming that ${\displaystyle h_{m}=0}$ and there exists some integer k such that ${\displaystyle h_{k}=0}$, then if

${\displaystyle 1+{\frac {1.91+0.114\log(g_{m+k}/2\pi )+\sum _{j=m+1}^{m+k-1}h_{j}}{g_{m+k}-g_{m}}}<2,}$

and

${\displaystyle -1-{\frac {1.91+0.114\log(g_{m}/2\pi )+\sum _{j=1}^{k-1}h_{m-j}}{g_{m}-g_{m-k}}}>-2,}$

Then the bound is achieved and we have that there are exactly m + 1 zeros of ζ(s), in the region 0 < Im(s) < Im(g_m).

References