"One cannot inquire into the foundations and nature of mathematics without delving into the question of the operations by which the mathematical activity of the mind is conducted. If one failed to take that into account, then one would be left studying only the language in which mathematics is represented rather than the essence of mathematics."
-- Luitzen Egbertus Jan Brouwer (https://en.wikipedia.org/wiki/L._E._J._Brouwer).
Relevant Reference Link:
https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true2
http://www.dtc.umn.edu/~odlyZko/doc/zeta.html
http://www-personal.umich.edu/~hlm/paircor1.pdf
https://en.wikipedia.org/wiki/Riemann_hypothesis
https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis
Hmm. The short answer to our question is a resounding definitive yes! Wow! The great Riemann and the great Hardy (https://en.m.wikipedia.org/wiki/G.H.Hardy) saw it all.
Here's the important idea while accepting the truth of the Riemann Hypothesis.
Σ(1/ks from k = 1 to k = ∞)
= 0
= Σ[ 1 / ( pn * l )sn from l = 1 to l = ∞ ]
+ Σ(1 / msn over all positive integers, m, not divisible by pn)
= 1 / (pn)sn * Σ(1 / lsn from l = 1 to l = ∞)
+ Σ(1 / msn over all positive integers, m, not divisible by pn);
But clearly, Σ(1 / lsn from l = 1 to l = ∞) = 0.
Therefore,
Σ(1 / msn over all positive integers, m, not divisible by pn)
= Σ[1 / ( pn * l )sn from l = 1 to l = ∞] = 0 (q.e.d.)
where s is any complex nontrivial simple zero of the Riemann Zeta Function (RZF) and where sn is the nth complex nontrivial simple zero of RZF and where pn is the nth prime number.
Moreover , we conclude:
pn exists if and only if sn exists.
(Note: In addition, we can prove the Generalized Riemann Hypothesis (GRH) is true as a result of our work here.)
"The art of doing mathematics consists in finding that special case which contains all the germs of generality." -- David Hilbert.
*****
Furthermore,
Σ[ 1 / ( pn * l )sn from l = 1 to l = ∞ ] = Σ[ ( pn * l )-sn from l = 1 to l = ∞]
= 0
= Σ[ ( pn * l )-sn from l = 1 to l = pn - 1]
+ Σ[ ( pn * l )-sn from l = pn to l = ∞].
Therefore, we have the following important equation:
1. Σ[ ( pn * l )-sn from l = pn to l = ∞]
=
− Σ[ ( pn * l )-sn from l = 1 to l = pn - 1]
where π(l) = n − 1 when l = pn − 1 and π(l) = n when l = pn
and where π(l) is the essential Prime-Counting Function.
We observe four obvious properties of equation one:
I. pn is the greatest prime number that divides pn * l when l < pn.
II. pn is the most common prime number that divides pn * l when l ≥ pn.
III. However, properties, I and II, do not imply Re(sn) = 1/2 (the Riemann Hypothesis). We have already accepted the truth of the Riemann Hypothesis.
IV. − Σ[ ( pn * l )-sn from l = 1 to l = pn − 1] is a complex number.
_____________________________________________________________
Essential Data Analysis using Wolfram Source Code (http://develop.wolframcloud.com/):
We let sn = N[ZetaZero[n],100];
S = Σ[ ( pn * l )-sn from l = 1 to l = pn − 1]
= Sum[(l*pn)^(-1*N[ZetaZero[1],100]),{l,1, pn − 1}];
Two Examples:
{n =68,
pn =337,
sn = 0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 +
178.3774077760999772858309354141844261831323614612725037014890408037488465646405879270706711681797739i,
S = 0.0653855572161115382311773533705347751885983095125623782395149343075206594352102178974266808615407 +
0.0267775235346093395162641450603589366761091439610849330981463305347426474394999736288889468384325i};
{n =949,
pn = 7489,
sn = 0.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 +
1360.393144762191152312089751910287813454655741604841462385320842334125119375400268288336773124123036i,
S = 0.0555643071881259461735964460926832809903221209682727919440948480061202123690626853472898204429053 +
0.0437180092288374001335409906388001928687233184040356180082684817339775439389815272654560780043058i}.
Notes:
MatrixForm[Sort[Table[{n =RandomInteger[{1,10000}],p = Prime[n],s =N[ZetaZero[n],100],Sum[(l*p)^(-1*s),{l,1,p-1}]},100]]]
is the code to generate randomly one hundred sets of data for values, n, between 1 and 10000, inclusively.
_____________________________________________________________
Excellent Reference Links:
'On the Number of Prime Numbers less than a
Given Quantity.(Ueber die Anzahl der Primzahlen unter einer
gegebenen Grosse.) ',
http://www.claymath.org/sites/default/files/ezeta.pdf;
'THE PAIR CORRELATION OF ZEROS OF THE ZETA FUNCTION'
by Prof. H. L. M.,
http://www-personal.umich.edu/~hlm/paircor1.pdf;
'Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics...';
http://www.dtc.umn.edu/~odlyZko/doc/zeta.html;
http://www.dtc.umn.edu/~odlyZko/zeta_tables/index.html;
'Tabulating values of the Riemann-Siegel Z function along the critical line',
http://web.mit.edu/kenta/www/six/parallel/2-Final-Report.html.
*****
https://www.quora.com/What-is-a-step-by-step-solution-to-find-one-of-the-non-trivial-zeroes-of-the-Riemann-Z%C3%A8ta-function
@ David,
Thanks for the link to the Riemann paper!
With respect to your claims, it is not clear what is the relation between sn and pn. Seemingly, the claims remain true for every pair of (s,p), where s is any root of the Rzf and p is any prime number. However: How can one omit the divergence on both sides of your equations?
Regards, Joachim
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