cp's OEIS Frontend

A216700 Violations of Rosser's rule: numbers n such that the Gram block [ g(n), g(n+k) ) contains fewer than k points t such that Z(t) = 0, where Z(t) is the Riemann-Siegel Z-function.

%I A216700 #33 Aug 06 2024 09:50:46
%S A216700 13999525,30783329,30930927,37592215,40870156,43628107,46082042,
%T A216700 46875667,49624541,50799238,55221454,56948780,60515663,61331766,
%U A216700 69784844,75052114,79545241,79652248,83088043,83689523,85348958,86513820,87947597
%N A216700 Violations of Rosser's rule: numbers n such that the Gram block [ g(n), g(n+k) ) contains fewer than k points t such that Z(t) = 0, where Z(t) is the Riemann-Siegel Z-function.
%C A216700 A Gram block [ g(m), g(m+k) ) is a half-open interval where g(m) and g(m+k) are "good" Gram points and g(m+1), ..., g(m+k-1) "bad" Gram points. A Gram point is "good" if (-1)^n Z(g(n)) > 0 and "bad" otherwise; see A114856.
%C A216700 Lehman showed that this sequence is infinite and conjectured (correctly) that a(1) > 10^7. Brent (1979) found a(1)-a(15). Brent, van de Lune, te Riele, & Winter extended this to a(104). Gourdon extended the calculation through a(320624341).
%C A216700 Trudgian showed that this sequence is of positive (lower) density.
%C A216700 Note: There is a typo in 7.3 of the Trudgian link showing 13999825, rather than 13999525, as the value for a(1). - _Charles R Greathouse IV_, Jan 27 2022
%D A216700 R. S. Lehman, On the distribution of zeros of the Riemann zeta-function, Proc. London Math. Soc., (3), v. 20 (1970), pp. 303-320.
%D A216700 J. Barkley Rosser and J. M. Yohe and Lowell Schoenfeld, Rigorous computation and the zeros of the Riemann zeta-function, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), vol. 1, North-Holland, Amsterdam, 1969, pp. 70-76. Errata: Math. Comp., v. 29, 1975, p. 243.
%H A216700 R. P. Brent, <a href="http://dx.doi.org/10.1090/S0025-5718-1979-0537983-2 ">On the zeros of the Riemann zeta function in the critical strip</a>, Math. Comp. 33 (1979), pp. 1361-1372.
%H A216700 R. P. Brent, J. van de Lune, H. J. J. te Riele and D. T. Winter, <a href="https://doi.org/10.1090/S0025-5718-1982-0669660-1">On the zeros of the Riemann zeta function in the critical strip. II</a>, Math. Comp. 39 (1982), pp. 681-688.
%H A216700 X. Gourdon, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf">The 10^13 first zeros of the Riemann zeta-function and zeros computation at very large height</a> (2004).
%H A216700 E. C. Titchmarsh, <a href="http://qjmath.oxfordjournals.org/content/os-5/1/98.extract">On van der Corput's Method and the zeta-function of Riemann IV</a>, Quarterly Journal of Mathematics os-5 (1934), pp. 98-105.
%H A216700 Timothy Trudgian, <a href="https://citeseerx.ist.psu.edu/pdf/0d399de3304d64e039762fda43d671732b62c090">On the success and failure of Gram's Law and the Rosser Rule</a>, Acta Arithmetica, 2011 | 148 | 3 | 225-256.
%Y A216700 Cf. A002505, A114856.
%K A216700 nonn,nice
%O A216700 1,1
%A A216700 _Charles R Greathouse IV_, Sep 17 2012