Turing's method

For the code-breaking method, see Turingery.

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. It was discovered by Alan Turing and published in 1953, although that proof contained errors and a correction was published in 1970 by R. Sherman Lehman.

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

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

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

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

and

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

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).