Gaps between Zeta function zeros and ets

I took a list of the first 10000 zeros of the Riemann zeta function,
rescaled so that they read in terms of equal divisions of the octave.
The top zero is then 1089.695, so we are looking at ets from 1 to 1089.
The gaps between these zeros are on average 1/log2(n) in the vicinity
of n, so I took the gaps between successive zeros and multiplied by
log2 of the average of the two successive zeros. As I expected, for
the largest gaps these are centered around equal divisions of the
octave. The first fifty of these, in order, are as follows:

954, 1012, 311, 764, 422, 581, 270, 814, 742, 935, 718, 494, 882,
1041, 525, 908, 571, 1065, 342, 836, 653, 851, 1084, 460, 354, 1075,
692, 1029, 684, 566, 624, 472, 711, 400, 863, 639, 988, 243, 997, 441,
643, 597, 373, 1046, 795, 449, 224, 513, 328, 966

Some tendency for the ets in question to be the kind approximating a
lot of primes (311) instead of a relatively few (411) seems to be in
evidence, but it isn't really clear what is going on.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I took a list of the first 10000 zeros of the Riemann zeta function,
> rescaled so that they read in terms of equal divisions of the
octave.
> The top zero is then 1089.695, so we are looking at ets from 1 to
1089.
> The gaps between these zeros are on average 1/log2(n) in the
vicinity
> of n, so I took the gaps between successive zeros and multiplied by
> log2 of the average of the two successive zeros. As I expected, for
> the largest gaps these are centered around equal divisions of the
> octave. The first fifty of these, in order, are as follows:
>
> 954, 1012, 311, 764, 422, 581, 270, 814, 742, 935, 718, 494, 882,
> 1041, 525, 908, 571, 1065, 342, 836, 653, 851, 1084, 460, 354, 1075,
> 692, 1029, 684, 566, 624, 472, 711, 400, 863, 639, 988, 243, 997,
441,
> 643, 597, 373, 1046, 795, 449, 224, 513, 328, 966

Are these in order of the largest gap to the smallest for the first
10000 zeros, or some other ordering. How do you derive 954, for
example. Thanx
>
> Some tendency for the ets in question to be the kind approximating a
> lot of primes (311) instead of a relatively few (411) seems to be in
> evidence, but it isn't really clear what is going on.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

> Are these in order of the largest gap to the smallest for the first
> 10000 zeros, or some other ordering. How do you derive 954, for
> example. Thanx

954 is the largest in the sense that the size of the gap, times
log2(954), is the largest up through 1089. If I'd cut things off at
900, 311 would have been the largest, which is kind of cool. I'm
going to check and see what happens if I integrate Z(x) between the
two zeros of the gap.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I took a list of the first 10000 zeros of the Riemann zeta function,
> rescaled so that they read in terms of equal divisions of the octave.
> The top zero is then 1089.695, so we are looking at ets from 1 to 1089.
> The gaps between these zeros are on average 1/log2(n) in the vicinity
> of n, so I took the gaps between successive zeros and multiplied by
> log2 of the average of the two successive zeros. As I expected, for
> the largest gaps these are centered around equal divisions of the
> octave. The first fifty of these, in order, are as follows:
>
> 954, 1012, 311, 764, 422, 581, 270, 814, 742, 935, 718, 494, 882,
> 1041, 525, 908, 571, 1065, 342, 836, 653, 851, 1084, 460, 354, 1075,
> 692, 1029, 684, 566, 624, 472, 711, 400, 863, 639, 988, 243, 997, 441,
> 643, 597, 373, 1046, 795, 449, 224, 513, 328, 966
>
> Some tendency for the ets in question to be the kind approximating a
> lot of primes (311) instead of a relatively few (411) seems to be in
> evidence, but it isn't really clear what is going on.

I agree.