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 1953¹, 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 .

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 .

References

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