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Turing's method

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For the code-breaking method, see Turingery.

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. It was discovered by Alan Turing and published in 1953, although that 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 ⩽ i ⩽ m } {\displaystyle \{g_{i}\mid 0\leqslant i\leqslant m\}} and a complementary list { h i ∣ 0 ⩽ i ⩽ m } {\displaystyle \{h_{i}\mid 0\leqslant i\leqslant m\}} , where gi is the smallest number such that

( − 1 ) i Z ( g i + h i ) > 0 , {\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 h m = 0 {\displaystyle h_{m}=0} and there exists some integer k such that h k = 0 {\displaystyle h_{k}=0} , then if

1 + 1.91 + 0.114 log ⁡ ( g m + k / 2 π ) + ∑ j = m + 1 m + k − 1 h j g m + k − g m < 2 , {\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

− 1 − 1.91 + 0.114 log ⁡ ( g m / 2 π ) + ∑ j = 1 k − 1 h m − j g m − g m − k > − 2 , {\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. Vol. 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.
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