cp's OEIS Frontend

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.

These correspond to increasing peaks of the absolute value of the Riemann zeta function along the critical line. If Z'(s)=0 is a positive zero of the derivative of Z, then |Z(s)| is the peak value.

The fractional parts of these values are not randomly distributed; r = log(2) * s(n) / (2*Pi) shows a very strong tendency to be near an integer.

It would be interesting to have theorems on the distribution of the fractional part of the "r" above, for which the Riemann hypothesis would surely be needed. It would be particularly interesting to know if the absolute value's fractional part is constrained to be less than some bound, such as 0.25. This computation could be pushed much farther by someone using a better algorithm, for instance the Riemann-Siegel formula and better computing resources. The computations were done using Maple's accurate but very slow zeta function evaluation. They are correct as far as they go, but do not go very far. The terms of the sequence have an interpretation in terms of music theory; the terms which appear in it, 12, 19, 22 and so forth, are equal divisions of the octave which do relatively well approximating intervals given by rational numbers with small numerators and denominators.

This sequence was extended by examining the peaks of |zeta(0.5+xi)| between each the first million zeros of the zeta function. These record peaks occur between zeros that are relatively far apart. The fractional part of r decreases as the magnitude of r increases. - T. D. Noe, Apr 19 2010

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.