The MRB constant fulfilling the Riemann Hypothesis
The MRB constant fulfilling the Riemann Hypothesis
Riemann Hypothesis: 'By a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge.' It is a little ,more obvious,that since Limit (-1)^n (n^(1/n)-1) ==0, we can use a remainder of terms from the conditionally convergent series to equal any number, S, close to 0 to add to a sufficiently large sequence of partial sums to finish getting any desired result So starting with partial series like in the following, since both positive and negative sums diverge and can be made arbitrarily large, or small, the partial sums can be made to oscillate around S and eventually converge to it.
(*Toget1*)-7++++2++--+-++-+--+----++++//N
5/32
2
1/4
2
3/8
2
2
+1/8
2
1/24
3
1/6
2
1/12
3
1/18
2
1/9
3
2/9
3
1/3
3
1/10
2
1/20
5
1/5
5
1/6
6
1/14
2
1/28
7
1/7
7
1/10
10
1/11
11
1/13
13
1/14
14
1/15
15
1/17
17
1/19
19
1/21
21
1/22
22
1/26
26
1/30
30
1/34
34
Out[]=
1.00723
(*TogetGoldenRatio=1.618.Noticesimilarityofthesumfor"1."*)-11++++2++--+-+++-+--+-++++//N
5/32
2
1/4
2
3/8
2
2
+1/8
2
1/24
3
1/6
2
1/12
3
1/18
2
1/9
3
2/9
3
1/3
3
1/10
2
1/20
5
1/5
5
1/18
6
1/6
6
1/14
2
1/28
7
1/7
7
1/10
10
1/11
11
1/13
13
1/14
14
1/15
15
1/22
22
1/26
26
1/30
30
1/34
34
Out[]=
1.61689
(*Togetsqrt(2)*)-13++++2++-+--+++--+++-++-+-+----+-++++++//N
5/32
2
1/4
2
3/8
2
2
+1/12
2
1/48
3
1/8
2
1/24
3
1/6
2
1/12
3
1/9
3
1/18
2
1/9
3
2/9
3
1/3
3
3/40
2
1/40
5
1/50
2
1/25
5
1/10
2
1/20
5
2/25
5
1/5
5
1/18
6
1/6
6
1/14
2
1/28
7
1/7
7
1/10
10
1/22
2
1/44
11
1/11
11
1/26
2
1/52
13
1/13
13
1/14
14
1/15
15
1/17
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1/19
19
1/21
21
1/22
22
1/23
23
1/26
26
1/30
30
1/34
34
1/38
38
1/42
42
1/46
46
Out[]=
1.40966
(*Toget)*)-28+++++2+++++--+++-++++-++-+-++-+--++++++++++++//N
3/32
2
5/32
2
1/4
2
3/8
2
2
+1/12
2
1/48
3
1/24
2
1/36
3
1/8
2
1/24
3
1/54
2
1/18
3
1/6
2
1/12
3
1/18
2
1/9
3
2/9
3
1/3
3
3/40
2
1/40
5
1/50
2
1/25
5
1/10
2
1/20
5
1/5
5
1/18
6
1/6
6
3/56
2
1/56
7
1/14
2
1/28
7
1/7
7
1/10
10
1/22
2
1/44
11
1/11
11
1/26
2
1/52
13
1/13
13
1/14
14
1/30
2
1/60
15
1/15
15
1/34
2
1/68
17
1/17
17
1/19
19
1/22
22
1/26
26
1/30
30
1/34
34
1/38
38
1/42
42
1/46
46
1/58
58
1/62
62
1/66
66
1/70
70
1/74
74
Out[]=
2.72249
(*Togetpi)*)-16++++2++-+++++++++//N
5/32
2
1/4
2
3/8
2
2
+1/8
2
1/24
3
1/6
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1/12
3
1/18
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1/9
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1/3
3
1/10
2
1/20
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1/6
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1/14
2
1/28
7
1/10
10
1/14
14
1/22
22
1/26
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1/30
30
1/34
34
Out[]=
{3.14849}
(*Toget1/2*)-2++2---+-+-+//N
3/8
2
2
+1/6
2
1/12
3
2/9
3
1/3
3
1/5
5
1/6
6
1/7
7
1/10
10
1/11
11
1/14
14
Out[]=
0.507174
Code from the Notebook Assistant:
(*Definethetermsoftheseries*)terms=Table[(-1)^n(n^(1/n)-1),{n,1,100}];(*Cumulativesumfunctionthatalternatesbetweenpositiveandnegativetermstoreachatargetsum*)RearrangeToTarget[terms_,target_]:=Module[{sum=0,pos=Select[terms,#>0&],neg=Select[terms,#<0&],result={}},While[Abs[sum-target]>0.01&&(Length[pos]>0||Length[neg]>0),If[sum<target,If[Length[pos]>0,sum+=First[pos];AppendTo[result,First[pos]];pos=Rest[pos]],If[Length[neg]>0,sum+=First[neg];AppendTo[result,First[neg]];neg=Rest[neg]]];];{sum,result}](*Calculatetherearrangedsumtoreachthetarget*)rearrangedSum=RearrangeToTarget[terms,0.5]