A117537 Locations of the midpoints of consecutive zeros of the Riemann zeta function on the critical line with increasingly large normalized spacing.

2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190

Offset: 0

Author

Gene Ward Smith, Mar 27 2006

Keywords

• hard
• nonn

Comments

Equivalently, locations of consecutive real zeros of the Z function. If t and s are consecutive zeros of the Z function, we define their normalized spacing as (s-t)*log((s+t)/(4*Pi)). The sequence above is found by taking r = log(2)(s+t)/(4*Pi) and rounding to the nearest integer. These values r have a marked tendency to be close to integer values and all of the terms of the above sequence are actually contained in the intervals [s, t]*log(2)/(2*Pi).

So far as the first 100000 zeros take us, the integers of the above sequence actually fall inside the normalized intervals of zeros of Z; that is, they fall between two zeros of Z(2*Pi*t/log(2)). It would be a worthwhile project to push this computation far enough to find a counterexample. The integers above, while slightly less clearly linked to music than A117536 and A117538, are nevertheless very clearly closely related to equal divisions of the octave. Large gaps between consecutive zeros, in other words, seem to correspond to good scale divisions, though less exactly than peak values or high integrals do.

References

• Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
• A. Ivic (1985). The Riemann Zeta Function, John Wiley & Sons. ISBN 0-471-80634-X.
• Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986

Links

• A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp., 48 (1987), 273-308.
• A. M. Odlyzko, The first 100,000 zeros of the Riemann zeta function, accurate to within 3*10^(-9)
• Wikipedia, Z Function

Crossrefs

Cf. A117536, A117538, A117539.