Turing's method explained

In mathematics, Turing's method is used to verify that for any given Gram point there lie m + 1 zeros of, in the region, where 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 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 is the smallest number such that ( -1 ) ^{i} Z(g _{i} + h _{i)} > 0 ,

where Z(t) is the Hardy Z function. Note that may be negative or zero. Assuming that

h _{m} = 0 and there exists some integer k such that h _{k} = 0 , then if1 + (1 . 91 + 0 . 114log (g _{m+k} / 2 \pi) + \sum ^{m + k -1} _{j = m +1} h _{j})/(g _{m+k} - g _{m}) < 2 ,

and

-1 - (1 . 91 + 0 . 114log (g _{m/2\pi)} + \sum ^{k -1} _{j =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, in the region .

Notes and References

1. Book: Edwards, H. M. . Harold Edwards (mathematician)

   . Harold Edwards (mathematician) . Riemann's zeta function . Pure and Applied Mathematics . 58 . New York-London . . 1974 . 0-12-232750-0 . 0315.10035.

2. Turing. A. M. . Alan Turing . Some Calculations of the Riemann Zeta‐Function . 10.1112/plms/s3-3.1.99 . 1953. . s3-3. 1. 99–117 .
3. Lehman. R. S. . On the Distribution of Zeros of the Riemann Zeta‐Function . 10.1112/plms/s3-20.2.303. 1970. Proceedings of the London Mathematical Society . s3-20. 2 . 303–320 .