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.

We may define the n-limit Pepper ambiguity, for any odd number n greater than one, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest.

We may define the n-limit Pepper ambiguity, for any odd number greater than one n, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest. In the 7-limit, that means we look at the ratios of the errors for the nearest approximations to 3/2, 5/4, 5/3, 7/4, 7/5 and 7/6 to the next nearest.

We may define the n-limit Pepper ambiguity, for any odd number n greater than one, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest. In the 9-limit, that means we look at the ratios of the errors for the nearest approximations to 3/2, 5/4, 5/3, 7/4, 7/5, 7/6, 9/8, 9/7 and 9/5 to the next nearest.

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.

The reason for the choice of 12 as a starting point is from musical practice; 12 is the standard equal division of the octave of Western music. The subsequent values where this integral is greater than it is for 12 are also equal divisions. While all the values tabulated are such that the integer of the integer sequence is actually contained in the interval between two successive zeros, it must eventually happen that a counterexample would be found. Another interesting question is the density of this sequence; it is not clear if it is increasing in density or not.

We may define the n-limit Pepper ambiguity, for any odd number n greater than one, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest.