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Proof of the Riemann Hypothesis

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Alvaro H. Salas at National University of Colombia
Alvaro H. Salas
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Abstract

I give a proof of Riemann Hypothesis
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A proof of the Riemann Hypothesis.
Let ali HxLbe the inverse of the log
integral function li HxL.In the paper
https : arxiv.org pdf 1203.5413.pdf
I found this result :
I define :
RHHnLΠpn-aliHnL¤
n log5
2HnL.
We have :
ali HnL<pn<Jn
nN10000 ali HnLfor every n ³1010. On the other hand,
limn®¥
ΠaliHnLKJ n
nN10000 -1O
nlog5
2HnL=limn®¥
ΠnlogHnLKJ n
nN10000 -1O
nlog5
2HnL=0, de donde limn®¥ RHHnL=0.
CONESTOCREO
HABER DEMOSTRADO LA
HIPÒTESISDERIEMANN !
2
ProofRiemann.nb
In[1167]:=
LimitBn Log@nDKJ n
nN10000 -1O
n Log@nD52,n® ¥F
Out[1167]=
0
Mathematica implementation of
ali HxL
ali@x_D:=ModuleB8p, f, Ρ<,
:p=Ix1-10-xHLog@Log@xDD+Log@xD-1LMN;
f@Ξ_D:=LogIntegral@ΞD;
Ρ =
NBRootB-4800 n fH3L@pD3+4800 f@pDfH3L@pD3-4800 p f¢@pDfH3L@pD3+2400 p2f¢¢@pDfH3L@pD3-
800 p3fH3L@pD4+7200 n f¢¢@pDfH3L@pDfH4L@pD-7200 f@pDf¢¢@pDfH3L@pDfH4L@pD+
7200 p f¢@pDf¢¢ @pDfH3L@pDfH4L@pD-3600 p2f¢¢@pD2fH3L@pDfH4L@pD-
1200 n p fH3L@pD2fH4L@pD+1200 p f@pDfH3L@pD2fH4L@pD-1200 p2f¢@pDfH3L@pD2fH4L@pD+
1800 p3f¢¢@pDfH3L@pD2fH4L@pD-1800 n f¢@pDfH4L@pD2+1800 f@pDf¢@pDfH4L@pD2-
1800 p f¢@pD2fH4L@pD2+900 n p f¢¢ @pDfH4L@pD2-900 p f@pDf¢¢@pDfH4L@pD2+
1800 p2f¢@pDf¢¢@pDfH4L@pD2-450 p3f¢¢ @pD2fH4L@pD2-300 n p2fH3L@pDfH4L@pD2+
300 p2f@pDfH3L@pDfH4L@pD2-600 p3f¢@pDfH3L@pDfH4L@pD2-75 n p3fH4L@pD3+
75 p3f@pDfH4L@pD3-2160 n f¢¢@pD2fH5L@pD+2160 f@pDf¢¢@pD2fH5L@pD-
2160 p f¢@pDf¢¢ @pD2fH5L@pD+1080 p2f¢¢@pD3fH5L@pD+1440 n f¢@pDfH3L@pDfH5L@pD-
1440 f@pDf¢@pDfH3L@pDfH5L@pD+1440 p f¢@pD2fH3L@pDfH5L@pD+
720 n p f¢¢ @pDfH3L@pDfH5L@pD-720 p f@pDf¢¢@pDfH3L@pDfH5L@pD-
720 p3f¢¢@pD2fH3L@pDfH5L@pD+240 n p2fH3L@pD2fH5L@pD-240 p2f@pDfH3L@pD2fH5L@pD+
480 p3f¢@pDfH3L@pD2fH5L@pD-360 n p f¢@pDfH4L@pDfH5L@pD+
360 p f@pDf¢@pDfH4L@pDfH5L@pD-360 p2f¢@pD2fH4L@pDfH5L@pD+
180 n p2f¢¢@pDfH4L@pDfH5L@pD-180 p2f@pDf¢¢ @pDfH4L@pDfH5L@pD+
360 p3f¢@pDf¢¢@pDfH4L@pDfH5L@pD+120 n p3fH3L@pDfH4L@pDfH5L@pD-
120 p3f@pDfH3L@pDfH4L@pDfH5L@pD-72 n p2f¢@pDfH5L@pD2+72 p2f@pDf¢@pDfH5L@pD2-
72 p3f¢@pD2fH5L@pD2-36 n p3f¢¢@pDfH5L@pD2+36 p3f@pDf¢¢ @pDfH5L@pD2-
360 n p f¢¢ @pD2fH6L@pD+360 p f@pDf¢¢@pD2fH6L@pD-360 p2f¢@pDf¢¢@pD2fH6L@pD+
180 p3f¢¢@pD3fH6L@pD+240 n p f¢@pDfH3L@pDfH6L@pD-240 p f@pDf¢@pDfH3L@pDfH6L@pD+
240 p2f¢@pD2fH3L@pDfH6L@pD-120 n p2f¢¢@pDfH3L@pDfH6L@pD+
120 p2f@pDf¢¢@pDfH3L@pDfH6L@pD-240 p3f¢@pDf¢¢ @pDfH3L@pDfH6L@pD-
40 n p3fH3L@pD2fH6L@pD+40 p3f@pDfH3L@pD2fH6L@pD+60 n p2f¢@pDfH4L@pDfH6L@pD-
60 p2f@pDf¢@pDfH4L@pDfH6L@pD+60 p3f¢@pD2fH4L@pDfH6L@pD+
- +
ProofRiemann.nb
3
60 p2f@pDf¢@pDf@pDf@pD+60 p3f¢@pD2f@pDf@pD+
30 n p3f¢¢@pDfH4L@pDfH6L@pD-30 p3f@pDf¢¢ @pDfH4L@pDfH6L@pD+
ð13J800 fH3L@pD4-1800 f¢¢@pDfH3L@pD2fH4L@pD+450 f¢¢ @pD2fH4L@pD2+600 f¢@pDfH3L@pD
fH4L@pD2+75 n fH4L@pD3-75 f@pDfH4L@pD3+720 f¢¢@pD2fH3L@pDfH5L@pD-480 f¢@pD
fH3L@pD2fH5L@pD-360 f¢@pDf¢¢@pDfH4L@pDfH5L@pD-120 n fH3L@pDfH4L@pDfH5L@pD+
120 f@pDfH3L@pDfH4L@pDfH5L@pD+72 f¢@pD2fH5L@pD2+36 n f¢¢@pDfH5L@pD2-
36 f@pDf¢¢@pDfH5L@pD2-180 f¢¢ @pD3fH6L@pD+240 f¢@pDf¢¢@pDfH3L@pDfH6L@pD+
40 n fH3L@pD2fH6L@pD-40 f@pDfH3L@pD2fH6L@pD-60 f¢@pD2fH4L@pDfH6L@pD-
30 n f¢¢@pDfH4L@pDfH6L@pD+30 f@pDf¢¢ @pDfH4L@pDfH6L@pDN+
ð12J2400 f¢¢@pDfH3L@pD3-2400 p fH3L@pD4-3600 f¢¢@pD2fH3L@pDfH4L@pD-
1200 f¢@pDfH3L@pD2fH4L@pD+5400 p f¢¢@pDfH3L@pD2fH4L@pD+
1800 f¢@pDf¢¢@pDfH4L@pD2-1350 p f¢¢@pD2fH4L@pD2-300 n fH3L@pDfH4L@pD2+
300 f@pDfH3L@pDfH4L@pD2-1800 p f¢@pDfH3L@pDfH4L@pD2-225 n p fH4L@pD3+
225 p f@pDfH4L@pD3+1080 f¢¢ @pD3fH5L@pD-2160 p f¢¢@pD2fH3L@pDfH5L@pD+
240 n fH3L@pD2fH5L@pD-240 f@pDfH3L@pD2fH5L@pD+1440 p f¢@pDfH3L@pD2fH5L@pD-
360 f¢@pD2fH4L@pDfH5L@pD+180 n f¢¢@pDfH4L@pDfH5L@pD-180 f@pDf¢¢ @pDfH4L@pD
fH5L@pD+1080 p f¢@pDf¢¢ @pDfH4L@pDfH5L@pD+360 n p fH3L@pDfH4L@pDfH5L@pD-
360 p f@pDfH3L@pDfH4L@pDfH5L@pD-72 n f¢@pDfH5L@pD2+72 f@pDf¢@pDfH5L@pD2-
216 p f¢@pD2fH5L@pD2-108 n p f¢¢ @pDfH5L@pD2+108 p f@pDf¢¢@pDfH5L@pD2-
360 f¢@pDf¢¢@pD2fH6L@pD+540 p f¢¢@pD3fH6L@pD+240 f¢@pD2fH3L@pDfH6L@pD-
120 n f¢¢@pDfH3L@pDfH6L@pD+120 f@pDf¢¢ @pDfH3L@pDfH6L@pD-720 p f¢@pD
f¢¢@pDfH3L@pDfH6L@pD-120 n p fH3L@pD2fH6L@pD+120 p f@pDfH3L@pD2fH6L@pD+
60 n f¢@pDfH4L@pDfH6L@pD-60 f@pDf¢@pDfH4L@pDfH6L@pD+180 p f¢@pD2
fH4L@pDfH6L@pD+90 n p f¢¢@pDfH4L@pDfH6L@pD-90 p f@pDf¢¢ @pDfH4L@pDfH6L@pDN+
ð1J4800 f¢@pDfH3L@pD3-4800 p f¢¢ @pDfH3L@pD3+2400 p2fH3L@pD4-
7200 f¢@pDf¢¢@pDfH3L@pDfH4L@pD+7200 p f¢¢@pD2fH3L@pDfH4L@pD+
1200 n fH3L@pD2fH4L@pD-1200 f@pDfH3L@pD2fH4L@pD+2400 p f¢@pDfH3L@pD2fH4L@pD-
5400 p2f¢¢@pDfH3L@pD2fH4L@pD+1800 f¢@pD2fH4L@pD2-900 n f¢¢ @pDfH4L@pD2+
900 f@pDf¢¢@pDfH4L@pD2-3600 p f¢@pDf¢¢@pDfH4L@pD2+1350 p2f¢¢@pD2fH4L@pD2+
600 n p fH3L@pDfH4L@pD2-600 p f@pDfH3L@pDfH4L@pD2+1800 p2f¢@pDfH3L@pDfH4L@pD2+
225 n p2fH4L@pD3-225 p2f@pDfH4L@pD3+2160 f¢@pDf¢¢@pD2fH5L@pD-
2160 p f¢¢ @pD3fH5L@pD-1440 f¢@pD2fH3L@pDfH5L@pD-720 n f¢¢@pDfH3L@pDfH5L@pD+
720 f@pDf¢¢@pDfH3L@pDfH5L@pD+2160 p2f¢¢ @pD2fH3L@pDfH5L@pD-
480 n p fH3L@pD2fH5L@pD+480 p f@pDfH3L@pD2fH5L@pD-1440 p2f¢@pDfH3L@pD2fH5L@pD+
360 n f¢@pDfH4L@pDfH5L@pD-360 f@pDf¢@pDfH4L@pDfH5L@pD+720 p f¢@pD2fH4L@pD
fH5L@pD-360 n p f¢¢@pDfH4L@pDfH5L@pD+360 p f@pDf¢¢@pDfH4L@pDfH5L@pD-
1080 p2f¢@pDf¢¢@pDfH4L@pDfH5L@pD-360 n p2fH3L@pDfH4L@pDfH5L@pD+
360 p2f@pDfH3L@pDfH4L@pDfH5L@pD+144 n p f¢@pDfH5L@pD2-
144 p f@pDf¢@pDfH5L@pD2+216 p2f¢@pD2fH5L@pD2+108 n p2f¢¢ @pDfH5L@pD2-
+ - +
4
ProofRiemann.nb
144 p f@pDf¢@pDf@pD2+216 p2f¢@pD2f@pD2+108 n p2f¢¢ @pDf@pD2-
108 p2f@pDf¢¢@pDfH5L@pD2+360 n f¢¢ @pD2fH6L@pD-360 f@pDf¢¢@pD2fH6L@pD+
720 p f¢@pDf¢¢ @pD2fH6L@pD-540 p2f¢¢@pD3fH6L@pD-240 n f¢@pDfH3L@pDfH6L@pD+
240 f@pDf¢@pDfH3L@pDfH6L@pD-480 p f¢@pD2fH3L@pDfH6L@pD+240 n p f¢¢ @pDfH3L@pD
fH6L@pD-240 p f@pDf¢¢@pDfH3L@pDfH6L@pD+720 p2f¢@pDf¢¢ @pDfH3L@pDfH6L@pD+
120 n p2fH3L@pD2fH6L@pD-120 p2f@pDfH3L@pD2fH6L@pD-120 n p f¢@pDfH4L@pD
fH6L@pD+120 p f@pDf¢@pDfH4L@pDfH6L@pD-180 p2f¢@pD2fH4L@pDfH6L@pD-
90 n p2f¢¢@pDfH4L@pDfH6L@pD+90 p2f@pDf¢¢ @pDfH4L@pDfH6L@pDN&, 3FF.n®x>;
N@ΡDF;
In[1168]:=
TableBSetPrecision@8x, ali@LogIntegral@xDD<, 20D,:x, 11, 25, 1
2>F
Out[1168]=
8811.000000000000000000 , 11.000000000336104478 <,
811.707106781186547524 , 11.707106781231447101 <,
812.414213562373095049 , 12.414213562377547362 <,
813.121320343559642573 , 13.121320343559711219 <,
813.828427124746190098 , 13.828427124746021093 <,
814.535533905932737622 , 14.535533905931703913 <,
815.242640687119285146 , 15.242640687119122234 <,
815.949747468305832671 , 15.949747468305110587 <,
816.656854249492380195 , 16.656854249491829023 <,
817.363961030678927720 , 17.363961030679043063 <,
818.071067811865475244 , 18.071067811865241026 <,
818.778174593052022768 , 18.778174593050405150 <,
819.485281374238570293 , 19.485281374235661644 <,
820.192388155425117817 , 20.192388155419376261 <,
820.899494936611665342 , 20.899494936601971773 <,
821.606601717798212866 , 21.606601717782329075 <,
822.313708498984760390 , 22.313708498960398430 <,
823.020815280171307915 , 23.020815280136748271 <,
823.727922061357855439 , 23.727922061310771085 <,
824.435028842544402964 , 24.435028842481617772 <<
In[1169]:=
TableBSetPrecision@8x, LogIntegral@ali@xDD<, 20D,:x, 11, 25, 1
2>F
Out[1169]=
8811.000000000000000000 , 10.999999999987243981 <,
811.707106781186547524 , 11.707106781158293174 <,
812.414213562373095049 , 12.414213562324073692 <,
813.121320343559642573 , 13.121320343486175375 <,
813.828427124746190098 , 13.828427124646802682 <,
814.535533905932737622 , 14.535533905808316391 <,
815.242640687119285146 , 15.242640686971977715 <,
815.949747468305832671 , 15.949747468138236073 <,
816.656854249492380195 , 16.656854249307752269 <,
817.363961030678927720 , 17.363961030481267045 <,
818.071067811865475244 , 18.071067811657719915 <,
818.778174593052022768 , 18.778174592837444834 <,
819.485281374238570293 , 19.485281374019361778 <,
820.192388155425117817 , 20.192388155203943256 <,
820.899494936611665342 , 20.899494936390741628 <,
821.606601717798212866 , 21.606601717579074773 <,
822.313708498984760390 , 22.313708498768349386 <,
823.020815280171307915 , 23.020815279958750210 <,
823.727922061357855439 , 23.727922061150099608 <,
824.435028842544402964 , 24.435028842342202182 <<
ProofRiemann.nb
5
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