Integrating the Riemann-Siegel Zeta function and ets

The Riemann-Siegel Zeta function Z(t) is defined here:

http://mathworld.wolfram.com/Riemann-SiegelFunctions.html

For our purposes we want to change scales, setting t = 2 pi x / ln(2),
and use Z(x) instead. I integrated |Z(x)| between successive zeros, for
zeros up to 100.409754. Below I list every inteval between zeros where
the integral is greater than one.

The point of this business is to give what you might call a generic
goodness measure for ets; meaning one not attached to any particular
prime limit. The result seems better than what we get for maximal
values of |Z(x)|, and much better than what we can glean from gaps
between the zeros.

If we look at successively larger values, we get 2, 5, 7, 12, 19, 31,
41, 53, 72 ..., and this makes a lot of sense to me. The so-called
"Omega theorems", about the rate of growth of the high values of
|Z(x)|, do not seem strong enough to show this is an infinte list,
though it starts out looking as if it is planning on being one. I
think I'll write to some people more expert than I am and inquire.

If we take the values over one by decades, we get:

1-10: 2, 3, 5, 7, 10
11-20: 12, 15, 17, 19
21-30: 22, 24, 26, 27, 29
31-40: 31, 34, 36
41-50: 41, 43, 46, 50
51-50: 53, 58, 60
61-70: 63, 65, 68
71-80: 72, 77, 80
81-90: 84, 87, 89
91-100: 94, 96, 99

It seems the density may be falling off slowly.

[1.559311781, 2.319105165] 1.103823
[2.759142784, 3.356405400] 1.044063
[4.779747405, 5.295822634] 1.131648
[6.710827976, 7.183072612] 1.162332
[9.797225769, 10.20350285] 1.082282
[11.82260542, 12.24853409] 1.269599
[14.86604170, 15.23665791] 1.104057
[16.88134757, 17.22203271] 1.032175
[18.74431544, 19.13037920] 1.313799
[21.84461333, 22.20308465] 1.258178
[23.84734791, 24.16705528] 1.092055
[25.78054223, 26.09283267] 1.031155
[26.92536457, 27.26360905] 1.185939
[28.77144315, 29.07689211] 1.000619
[30.80395665, 31.16093004] 1.403777
[33.89177893, 34.21059373] 1.241437
[35.83815669, 36.12289081] 1.028887
[40.82320329, 41.15537120] 1.423937
[42.89664942, 43.18457394] 1.035628
[45.83210532, 46.15561125] 1.356067
[49.79781990, 50.08281814] 1.111229
[52.83584779, 53.15446302] 1.486620
[57.92538202, 58.23716835] 1.358357
[59.77541720, 60.04861404] 1.131000
[62.88678487, 63.14811332] 1.049023
[64.88227375, 65.16035560] 1.269821
[67.90486013, 68.18884771] 1.254592
[71.78033774, 72.10918271] 1.625363
[76.85025545, 77.12671468] 1.311364
[79.93215353, 80.20726288] 1.247325
[83.87267811, 84.13938972] 1.241945
[86.87178850, 87.15758094] 1.439474
[88.90088275, 89.15353029] 1.124501
[93.84133446, 94.11907762] 1.394050
[95.82981785, 96.06991440] 1.045052
[98.91449741, 99.20014010] 1.510412

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> If we look at successively larger values, we get 2, 5, 7, 12, 19, 31,
> 41, 53, 72 ..., and this makes a lot of sense to me. The so-called
> "Omega theorems", about the rate of growth of the high values of
> |Z(x)|, do not seem strong enough to show this is an infinte list...

Never mind. They are strong enough...

this is really hot, and i wish i understood it . . . maybe if manfred
schroeder wrote a book on it . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> The Riemann-Siegel Zeta function Z(t) is defined here:
>
> http://mathworld.wolfram.com/Riemann-SiegelFunctions.html
>
> For our purposes we want to change scales, setting t = 2 pi x / ln
(2),
> and use Z(x) instead. I integrated |Z(x)| between successive zeros,
for
> zeros up to 100.409754. Below I list every inteval between zeros
where
> the integral is greater than one.
>
> The point of this business is to give what you might call a generic
> goodness measure for ets; meaning one not attached to any particular
> prime limit. The result seems better than what we get for maximal
> values of |Z(x)|, and much better than what we can glean from gaps
> between the zeros.
>
> If we look at successively larger values, we get 2, 5, 7, 12, 19,
31,
> 41, 53, 72 ..., and this makes a lot of sense to me. The so-called
> "Omega theorems", about the rate of growth of the high values of
> |Z(x)|, do not seem strong enough to show this is an infinte list,
> though it starts out looking as if it is planning on being one. I
> think I'll write to some people more expert than I am and inquire.
>
> If we take the values over one by decades, we get:
>
> 1-10: 2, 3, 5, 7, 10
> 11-20: 12, 15, 17, 19
> 21-30: 22, 24, 26, 27, 29
> 31-40: 31, 34, 36
> 41-50: 41, 43, 46, 50
> 51-50: 53, 58, 60
> 61-70: 63, 65, 68
> 71-80: 72, 77, 80
> 81-90: 84, 87, 89
> 91-100: 94, 96, 99
>
> It seems the density may be falling off slowly.
>
>
> [1.559311781, 2.319105165] 1.103823
> [2.759142784, 3.356405400] 1.044063
> [4.779747405, 5.295822634] 1.131648
> [6.710827976, 7.183072612] 1.162332
> [9.797225769, 10.20350285] 1.082282
> [11.82260542, 12.24853409] 1.269599
> [14.86604170, 15.23665791] 1.104057
> [16.88134757, 17.22203271] 1.032175
> [18.74431544, 19.13037920] 1.313799
> [21.84461333, 22.20308465] 1.258178
> [23.84734791, 24.16705528] 1.092055
> [25.78054223, 26.09283267] 1.031155
> [26.92536457, 27.26360905] 1.185939
> [28.77144315, 29.07689211] 1.000619
> [30.80395665, 31.16093004] 1.403777
> [33.89177893, 34.21059373] 1.241437
> [35.83815669, 36.12289081] 1.028887
> [40.82320329, 41.15537120] 1.423937
> [42.89664942, 43.18457394] 1.035628
> [45.83210532, 46.15561125] 1.356067
> [49.79781990, 50.08281814] 1.111229
> [52.83584779, 53.15446302] 1.486620
> [57.92538202, 58.23716835] 1.358357
> [59.77541720, 60.04861404] 1.131000
> [62.88678487, 63.14811332] 1.049023
> [64.88227375, 65.16035560] 1.269821
> [67.90486013, 68.18884771] 1.254592
> [71.78033774, 72.10918271] 1.625363
> [76.85025545, 77.12671468] 1.311364
> [79.93215353, 80.20726288] 1.247325
> [83.87267811, 84.13938972] 1.241945
> [86.87178850, 87.15758094] 1.439474
> [88.90088275, 89.15353029] 1.124501
> [93.84133446, 94.11907762] 1.394050
> [95.82981785, 96.06991440] 1.045052
> [98.91449741, 99.20014010] 1.510412

Roger that! I was going to post something to this effect, but I
still haven't studied the mathworld entry.

-Carl

>this is really hot, and i wish i understood it . . . maybe if manfred
>schroeder wrote a book on it . . .
>
//
>>
>> The point of this business is to give what you might call a generic
>> goodness measure for ets; meaning one not attached to any particular
>> prime limit. The result seems better than what we get for maximal
>> values of |Z(x)|, and much better than what we can glean from gaps
>> between the zeros.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> If we look at successively larger values, we get 2, 5, 7, 12, 19, 31,
> 41, 53, 72 ..., and this makes a lot of sense to me.

This continues 72, 130, 171 ...

> If we take the values over one by decades, we get:
>
> 1-10: 2, 3, 5, 7, 10
> 11-20: 12, 15, 17, 19
> 21-30: 22, 24, 26, 27, 29
> 31-40: 31, 34, 36
> 41-50: 41, 43, 46, 50
> 51-50: 53, 58, 60
> 61-70: 63, 65, 68
> 71-80: 72, 77, 80
> 81-90: 84, 87, 89
> 91-100: 94, 96, 99
>
> It seems the density may be falling off slowly.

This pattern also continues. The density is falling off, but very
slowly. However the really significant ets are marked by major spikes,
which grow ever more major.

101-110: 103, 106, 109
111-120: 111, 113, 118
121-130: 121, 125, 130
131-140: 135, 137, 140
141-150: 144, 145, 149
151-160: 152, 159
161-170: 161, 164, 167
171-180: 171
181-190: 183, 190
191-200: 193, 198

Here is what we have for 101-200; over 1.3 we have 103, 111, 118, 121,
130, 140, 152, 159, 171, 183, 190, 193 and 198.

[102.7999443, 103.0793314] 1.340775
[105.8454247, 106.0892716] 1.181299
[108.8567520, 109.1034903] 1.025274
[110.9293379, 111.2010820] 1.394739
[112.8424527, 113.1050591] 1.261675
[117.8183072, 118.0990707] 1.544280
[120.9526475, 121.2217233] 1.316426
[124.8240486, 125.0682415] 1.239477
[129.8557778, 130.1420437] 1.634018
[134.9134275, 135.1413271] 1.054602
[136.8838124, 137.1336386] 1.209806
[139.8355968, 140.1167252] 1.548424
[143.7742322, 144.0090406] 1.101443
[144.9633882, 145.2016246] 1.161664
[148.8281106, 149.0720246] 1.280257
[151.9272068, 152.2036245] 1.593855
[158.8160194, 159.0739639] 1.436994
[160.8867930, 161.1399165] 1.272837
[163.9511536, 164.1657206] 1.046830
[166.9758931, 167.2096100] 1.086573
[170.8680796, 171.1331178] 1.652856
[182.8591559, 183.1297521] 1.643410
[189.8015621, 190.0604329] 1.520966
[192.8977322, 193.1469215] 1.453742
[197.9298229, 198.1820821] 1.464293