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 and a complementary list , where gi is the smallest number such that
where Z(t) is the Hardy Z function. Note that gi may be negative or zero. Assuming that and there exists some integer k such that , then if
and
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
- ^ 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.
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