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 [math]\displaystyle{ \{g_i \mid 0\leqslant i \leqslant m \} }[/math] and a complementary list [math]\displaystyle{ \{h_i \mid 0\leqslant i \leqslant m \} }[/math], where gi is the smallest number such that
- [math]\displaystyle{ (-1)^i Z(g_i + h_i) \gt 0, }[/math]
where Z(t) is the Hardy Z function. Note that gi may be negative or zero. Assuming that [math]\displaystyle{ h_m = 0 }[/math] and there exists some integer k such that [math]\displaystyle{ h_k = 0 }[/math], then if
- [math]\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} \lt 2, }[/math]
and
- [math]\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}} \gt -2, }[/math]
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. 58. New York-London: Academic Press. ISBN 0-12-232750-0.
- ↑ 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. 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.
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