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.

13999525, 30783329, 30930927, 37592215, 40870156, 43628107, 46082042, 46875667, 49624541, 50799238, 55221454, 56948780, 60515663, 61331766, 69784844, 75052114, 79545241, 79652248, 83088043, 83689523, 85348958, 86513820, 87947597

Offset: 1

Author

Charles R Greathouse IV, Sep 17 2012

Keywords

• nonn
• nice

Comments

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.

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).

Trudgian showed that this sequence is of positive (lower) density.

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

References

• R. S. Lehman, On the distribution of zeros of the Riemann zeta-function, Proc. London Math. Soc., (3), v. 20 (1970), pp. 303-320.
• 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.

Links

• R. P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), pp. 1361-1372.
• R. P. Brent, J. van de Lune, H. J. J. te Riele and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. II, Math. Comp. 39 (1982), pp. 681-688.
• X. Gourdon, The 10^13 first zeros of the Riemann zeta-function and zeros computation at very large height (2004).
• E. C. Titchmarsh, On van der Corput's Method and the zeta-function of Riemann IV, Quarterly Journal of Mathematics os-5 (1934), pp. 98-105.
• Timothy Trudgian, On the success and failure of Gram's Law and the Rosser Rule, Acta Arithmetica, 2011 | 148 | 3 | 225-256.

Crossrefs

Cf. A002505, A114856.