# Turing's method

In mathematics, Turing's method is used to verify that for any given Gram point gm there lie m + 1 zeros of ζ(s), in the region 0 < Im(s) < Im(gm), 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 gi 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 gi 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(gm).

## References

1. ^ Edwards, H. M. (1974). Riemann's zeta function. Pure and Applied Mathematics. 58. New York-London: Academic Press. ISBN 0-12-232750-0. Zbl 0315.10035.
2. ^ 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.
3. ^ Lehman, R. S. (1970). "On the Distribution of Zeros of the Riemann Zeta‐Function". Proceedings of the London Mathematical Society. s3-20 (2): 303–320. doi:10.1112/plms/s3-20.2.303.