Calculation of approximated values of the zeta

Calculation of approximated values of zeta

Leonhard Euler (1707-1783)

in preparation

Round down

+ and - mean differences between the values in the textbook and 000.
? means a large difference.

zeta(2k)(1-1/2^{2k})
02 1.23370055013616982735431 137
04 1.01467803160419205454625 345
06 1.00144707664094212190647 857
08 1.00015517902529611930298 723
10 1.00001704136304482548818 389 ???
12 1.00000188584858311957590 883
14 1.00000020924051921150010 636
16 1.00000002323715737915670 767
18 1.00000000258143755665977 284
20 1.00000000028680769745558 199
22 1.00000000003186677514044 360
24 1.00000000000354072294392 050
26 1.00000000000039341246691 444
28 1.00000000000004371244859 229
30 1.00000000000000485693682 340
32 1.00000000000000053965957 068
34 1.00000000000000005996217 146
36 1.00000000000000000666246 337
38 1.00000000000000000074027 370
40 1.00000000000000000008225 263
42 1.00000000000000000000913 918
44 1.00000000000000000000101 546
46 1.00000000000000000000011 282
48 1.00000000000000000000001 253

(the number of error is one)

zeta(2k) (Euler might have the following list)
02 1.64493406684822643647241 516
04 1.08232323371113819151600 368
06 1.01734306198444913971451 791
08 1.00407735619794433937868 522
10 1.00099457512781808533714 594 ?
12 1.00024608655330804829863 799 ++++
14 1.00006124813505870482925 854
16 1.00001528225940865187173 257 ???
18 1.00000381729326499983985 646
20 1.00000095396203387279611 315
22 1.00000023845050272773299 000 ??
24 1.00000005960818905125947 961
26 1.00000001490155482836504 123
28 1.00000000372533402478845 705
30 1.00000000093132743241966 818
32 1.00000000023283118336765 054 -
34 1.00000000005820772087902 700
36 1.00000000001455192189104 198
38 1.00000000000363797954737 865
40 1.00000000000090949478402 638
42 1.00000000000022737368458 246 -----
44 1.00000000000005684341987 627
46 1.00000000000001421085482 803
48 1.00000000000000355271369 133 --

zeta(2k)/2^{2k} (From the above two lists)
02 0.41123351671205660911810
04 0.06764520210694613696975
06 0.01589598534350701780804
08 0.00392217717264822007570
10 0.00097753376477325984896 ?
12 0.00024420070472492872273 +
14 0.00006103889453949332915
16 0.00001525902225127271503 ???
18 0.00000381471182744318008
20 0.00000095367522617534053
22 0.00000023841863595259255 ??
24 0.00000005960464832831555
26 0.00000001490116141589813
28 0.00000000372529031233986
30 0.00000000093132257548284
32 0.00000000023283064370808 -
34 0.00000000005820766091685
36 0.00000000001455191522858
38 0.00000000000363797880710
40 0.00000000000090949470177
42 0.00000000000022737367545 -
44 0.00000000000005684341886
46 0.00000000000001421085471
48 0.00000000000000355271368 -

Round off

+ and - mean some differences between the values in the textbook and 500.
? means a large difference.

zeta(2k)(1-1/2^{2k})
02 1.23370055013616982735431 137
04 1.01467803160419205454625 345
06 1.00144707664094212190647 857 -------
08 1.00015517902529611930298 723 ----
10 1.00001704136304482548818 389
12 1.00000188584858311957590 883 ---------
14 1.00000020924051921150010 636 --
16 1.00000002323715737915670 767 -----
18 1.00000000258143755665977 284
20 1.00000000028680769745558 199
22 1.00000000003186677514044 360
24 1.00000000000354072294392 050
26 1.00000000000039341246691 444
28 1.00000000000004371244859 229
30 1.00000000000000485693682 340
32 1.00000000000000053965957 068
34 1.00000000000000005996217 146
36 1.00000000000000000666246 337
38 1.00000000000000000074027 370
40 1.00000000000000000008225 263
42 1.00000000000000000000913 918 --------
44 1.00000000000000000000101 546
46 1.00000000000000000000011 282
48 1.00000000000000000000001 253

(Euler might not round off.)

zeta(2k)/2^{2k}
02 0.41123351671205660911810 379
04 0.06764520210694613696975 023
06 0.01589598534350701780803 934
08 0.00392217717264822007569 798
10 0.00097753376477325984896 205 ?
12 0.00024420070472492872272 915 +
14 0.00006103889453949332915 217
16 0.00001525902225127271502 489 ???
18 0.00000381471182744318008 361
20 0.00000095367522617534053 115
22 0.00000023841863595259254 639 ??
24 0.00000005960464832831555 910 --
26 0.00000001490116141589812 678
28 0.00000000372529031233986 476
30 0.00000000093132257548284 477
32 0.00000000023283064370807 986 -
34 0.00000000005820766091685 554 -
36 0.00000000001455191522857 861
38 0.00000000000363797880710 494
40 0.00000000000090949470177 375
42 0.00000000000022737367544 328
44 0.00000000000005684341886 081
46 0.00000000000001421085471 520 -
48 0.00000000000000355271367 880 -

We used Euler-Maclaurin with a=10 and B_0-B_60.

The sum of the power of prime numbers

Values of high degree -> Values of low degree.
Euler changed methods of computation.
So we must understand where he changed them.

The following values are correct.

02 0.452247420041065 50068 +156.50
04 0.076993139764246 84494 +5.16
06 0.017070086850636 51295 +2.5
08 0.004061405366517 83056 -2.8
10 0.000993603574436 98021 -803.98
12 0.000246026470034 54567 -1.5
14 0.000061244396725 46447
16 0.000015282026219 33934
18 0.000003817278703 17499 -
20 0.000000953961124 10362 -
22 0.000000238450445 87670 +
24 0.000000059608185 49833 -
26 0.000000014901554 60631 +
28 0.000000003725334 01091 -
30 0.000000000931327 43155 -4.43
32 0.000000000232831 18331 -
34 0.000000000058207 72087
36 0.000000000014551 92189
38 0.000000000003637 97880
40 0.000000000000909 49470
42 0.000000000000227 37367
44 0.000000000000056 84341
46 0.000000000000014 21085
48 0.000000000000003 55271
(---,+++ mean difference from the values in the textbook.)

s=36, 34, 32
[1/2^s]

02 0.25
04 0.0625
06 0.015625
08 0.00390625
10 0.0009765625
12 0.000244140625
14 0.00006103515625
16 0.000015258789062 5
18 0.000003814697265 625
20 0.000000953674316 40625
22 0.000000238418579 1015625
24 0.000000059604644 775390625
26 0.000000014901161 19384765625
28 0.000000003725290 2984619140625
30 0.000000000931322 574615478515625
32 0.000000000232830 64365386962890625
34 0.000000000058207 6609134674072265625
36 0.000000000014551 915228366851806640625
38 0.000000000003637 97880709171295166015625
40 0.000000000000909 4947017729282379150390625
42 0.000000000000227 373675443232059478759765625
44 0.000000000000056 84341886080801486968994140625
46 0.000000000000014 2108547152020037174224853515625
48 0.000000000000003 552713678800500929355621337890625

s=32, 30, 28, 26, 24, 22, 20, 18, 16
[p.237]
A(2)+A(3)-A(6)-B(2)-B(3)+B(6)+1/6^s-(1/25^s+1/35^s+1/49^s)
A(a)=zeta(s)(1-1/a^s), B(a)=1-1/a^s
Round down (15)
Round down (20)

02 0.453891391902471 71689 ???
04 0.076993205698789 58289 ???
06 0.017070086856372 24580 ???
08 0.004061405366518 10701 ---
10 0.000993603574436 13435 ???
12 0.000246026470033 98293
14 0.000061244396725 99998 (-)
16 0.000015282026219
18 0.000003817278702
20 0.000000953961124 -1
22 0.000000238450446
24 0.000000059608184
26 0.000000014901555
28 0.000000003725333
30 0.000000000931326 -3
32 0.000000000232830
We can get close values by s=16.

s=14, 12, 10, 8
[p.234 middle]
Values of high degree -> Values of low degree.

d=20 (middle)
02 0.452247420041065 +157
04 0.076993139764247 +5
06 0.017070086850637 +2
08 0.004061405366518 -3
10 0.000993603574437 -804
12 0.000246026470035 -2
14 0.000061244396725

d=15 (middle)
02 0.452247420041065 +157
04 0.076993139764247 +5
06 0.017070086850637 +2
08 0.004061405366517 -2
10 0.000993603574437 -804
12 0.000246026470034 -1
14 0.000061244396725

s=10, 8, 6, 4,2
[p.234 low]
Values of high degree -> Values of low degree.

d=20 (low)
02 0.452247420041065 +157
04 0.076993139764246 +6
06 0.017070086850637 +2
08 0.004061405366518 -3
10 0.000993603574437 -804

d=15 (low)
02 0.452247420041065 +157
04 0.076993139764247 +5
06 0.017070086850637 +2
08 0.004061405366518 -3
10 0.000993603574437 -804

Euler might have the following list.

02 0.452247420041065 +157
04 0.076993139764246 +6
06 0.017070086850637 +2
08 0.004061405366518 -3
10 0.000993603574437 -804
12 0.000246026470035 -2
14 0.000061244396725
16 0.000015282026219
18 0.000003817278702
20 0.000000953961124 -1
22 0.000000238450446
24 0.000000059608184
26 0.000000014901555
28 0.000000003725333
30 0.000000000931326 -3
32 0.000000000232830
34 0.000000000058207
36 0.000000000014551
38 0.000000000003637
40 0.000000000000909
42 0.000000000000227
44 0.000000000000056
46 0.000000000000014
48 0.000000000000003

He might use "1/2^s" for s=32 and 34,
the method at p.237 from 16 to 30,
the method at p.234 (middle) from 8 (or 12) to 14,
the method at p.234 (low) from 2 to 6 (or 10).
Then we can express the indices 62 and 110=20*1+30*3.

p.234 (middle)

02 0.452247420041224

Euler's value

02 0.452247420041222

The difference =-0.000000000000002
So many 2s!! 2 is the irregularity index of p=157.

Reg: 2005, Jul 9
Title: Calculation of approximated values of the zeta
Method:webpage
Content:
https://math0.pm.tokushima-u.ac.jp/~hiroki/major/eulercal-e.html
EP: 55621

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