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 this 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 $\{g_{i}|0\leqslant i\leqslant m\}$ and a complimentary list $\{h_{i}|0\leqslant i\leqslant m\}$ , 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 $0 < Im(s) < Im(g m)$ .

References

Turing, A. M (1953), "Some Calculations of the Riemann Zeta‐Function", Proceedings of the London Mathematical Society, s3-3 (1): 99–117, doi:10.1112/plms/s3-3.1.99

Lehman, R. Sherman (1970), "On the Distribution of Zeros of the Riemann Zeta‐Function", Proceedings of the London Mathematical Society, s3-20: 303–320, doi:10.1112/plms/s3-20.2.303

Edwards, H.M. (1974), Riemann's zeta function, Pure and Applied Mathematics, vol. 58, New York-London: Academic Press, ISBN 0-12-232750-0, Zbl 0315.10035