A117538 Locations of the increasing peak values of the integral of the absolute value of the Riemann zeta function between successive zeros on the critical line. This can also be defined in terms of the Z function; if t and s are successive zeros of a renormalized Z function, z(x) = Z(2 Pi x/log(2)), then take the integral between t and s of |z(x)|. For each successively higher value of this integral, the corresponding term of the integer sequence is r = (t+s)/2 rounded to the nearest integer.

2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973

Offset: 0

Author

Gene Ward Smith, Mar 27 2006

Keywords

• hard
• more
• nonn

Comments

The fractional parts of the numbers r = (t+s)/2 above are very unevenly distributed. For all of the values in the table, the integers are in fact the unique integers contained in the interval of zeros [t, s] of z(x). An interesting challenge to anyone wishing to do computations related to the zeta function would be to find the first counterexample, where in fact the peak value interval did not contain the corresponding integer. Perhaps even more than the peak values of the zeta function themselves, these integrals are extremely closely related to relatively good equal divisions of the octave in music theory.

References

• Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
• Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986
• Paris, R. B. and Kaminski, D., Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001

Links

• Andrew Odlyzko, The first 100,000 zeros of the Riemann zeta function, accurate to within 3*10^(-9)
• Wikipedia, Z function
• Index entries for zeta function.

Crossrefs

Cf. A117536, A117537, A117539, A054540.

Extensions

Extended by T. D. Noe, Apr 21 2010