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. 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 $0 < Im(s) < Im(g_{m})$.