Lesson plan

Turing's method

In mathematics, Turing's method is used to verify that for any given Gram point there lie m + 1 zeros of , in the region , where 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 we find a list of Gram points \{g_i \mid 0\leqslant i \leqslant m \} and a complementary list \{h_i \mid 0\leqslant i \leqslant m \} , where is the smallest number such that : (-1)^i Z(g_i + h_i) > 0, where Z(t) is the Hardy Z function. Note that 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} 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 , in the region .

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Article summary

In mathematics, Turing's method is used to verify that for any given Gram point there lie m + 1 zeros of , in the region , where 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 we find a list of Gram points \{g_i \mid 0\leqslant i \leqslant m \} and a complementary list \{h_i \mid 0\leqslant i \leqslant m \} , where is the smallest number such that : (-1)^i Z(g_i + h_i) > 0, where Z(t) is the Hardy Z function. Note that 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} 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 , in the region .

Key claims

  1. publication date: 1953 — source: Wikipedia · confidence 0.95

Discussion questions

  1. Summarise the article in your own words. Which single claim would you verify first, and why?
  2. The record states that the publication date is "1953". What evidence would count against this claim?
  3. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
  4. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

Bibliography

  1. Wikipedia https://en.wikipedia.org/w/index.php?oldid=1268306590

Key claims and bibliography are generated from the same tracked claims and sources as the article page.