黎曼猜想不是等式
The Riemann hypothesis is not equality
Wu he fa
Freedom mathematical researchers Shandong Heze Shanxian shandong heze zhongxing
town wu ji village , China Han
Mobile phone [1**********] qq1508960914
Manuscript submitted April 12, 2016 , accepted April 25,2016
Abstract : The Riemann hypothesis is the most important in today's international maths, expect
to solve the most famous incomplete mathematics problem, it is incomplete in the classical
mathematical analysis form the product of paradox. As long as the classical mathematical analysis
system does not get rid of their incomplete defects, Riemann hypothesis can not completely solve,
even the most basic also failed to set up or not. But what is of the Riemann hypothesis? Why is
it important? Still not had a relatively complete system of certificate for introduction, explanation
and elaboration, only the reference of existing irrelevant answer of so-called "mathematics
specialized books, documents or ramble", rather than the law of trust in science. So even all
mathematicians not to mention ordinary people will only know "the Riemann hypothesis" the
question. Riemann hypothesis is not completely equation solved this difficult problem. Key words : Euler product, Frequency even the product of law, The Riemann hypothesis is not
equal
Personal profile Wu he fa, 1972. 10. 2, the number and change rule of science of founder,
shandong heze zhongxing town wu ji village, zip code 274329. in 2015 published the
number and change rule Science>>[1], including the research field: the scientific nature of the
Euler's constant, Riemann conjecture, infinite series , Godel's incomplete theorem and so
on[2]-[3]-[4]-[5]-[6].
1. Euler's proof of Euler's constnta[7]-[8]-[9] Euler in his paper some review of the Infinite Series(Various Observations about Infinite Series)in
the product of Euler formula"to"prove Riemann zeta function,and published in 1737by the
academy of sciences.Riemann zeta function can be written in the form of Euler's product
ς(s)=∑1111=1++s +s +....... , and the left is equal to the Riemann zeta f unction, s s n 234
the right product is extended to all prime number p 111111=. . . ... ... ∏-s -s -s -s -s -s 1-21-31-51-71-p p prime 1-p (1) Prove process only to form simple and omit the infinite, It is the incomplete method that the
Euler once used. From(2) less (1 ),all the even items: type, we get
of ζ(s ) =1+rid 11111111+++... , ζ(s ) =++s +s +... (2) s s s s s 234224s 68
1111111(1-) ς=+++++... (3) (s ) s s s s s s s 3239152127
(4)
The cut all multiple item 2 and 3. Multiples items can be visible to the right
, to repeat the above steps are sieve to available:
(5)
At last, Euler got ς(S ) =1 (6) ∏-s 1-p P , Pr ime
To prove more closely, we need to notice when R (s) > 1, sieved the right item to 1, and follow the
convergence of dirichlet series. Is derived from the above all can zeta the interesting but the results of the paradox....(1-11111)(1-)(1-)(1-)(1-) ς(1)=1 117532
It can be written as ......(10/11)(6/7)(4/5)(2/3)(1/2) ζ(1)=1,
1+1/2+1/3+1/4+1/5+...=2·3·5·7·11·13·...../(1·2·4·6·8·...) (7)
Euler proved to be incorrect, derived 1+ 1111+++...... +=ln(n +1) +r 234n
2. Euler's proof is not complete, the following to correct
2.1. The Euler's constant r (Euler - Maxieluoni constant) are two different irrational Numbers [1].
The source of the Euler's constant r : Euler's constant r is objective existence, it is
indisputable fact that the number of frequency - Euler's constant r is already fully respect Euler
found that with the inevitable number frequency conditions, the frequency from several science,
so it is the series of classical mathematics to several scientific frequency of a leap, number and
frequency law make us believe that the objective of this explanation is reasonable. In 1734 , Euler
using Newton's achievements, won the first harmonic finite terms the value of the sum of the
series . 1+1/2+1/3+…+1/n=ln(n+1)+r, he process is as follows: By
ln(1+x)=x-x2/2+x3/3-x 4/4+…, ﹙－1＜x ≤1), x=1,2,3,4,...,1/n,
ln(1+1/n)=ln[(n+1)/n]=1/n-1/2·1/n2+1/3·1/n3-1/4·1/n4+…;
The above all kinds of additive, and tidy: 1+1/2+1/3+1/4+…+1/n=ln(1+n)+r ,
r=1/2(1/22+1/32+1/42+…+1/n2)-1/3(1/23+1/33+1/43+…)
+1/4(1/244+1/344+1/444+…)-1/5(1/255+1/355+1/45+…)+…
=1/2∑1/n2-1/3∑1/n3+1/4∑1/n4-1/5∑1/n5+…+[(-1)n-1]/n∑1/nn . (n=2,∞) ①
This is a youth mathematician Euler make great contributions to harmonic progression and
analysis, it is the modern theory of modern classic series is the important measure to avoid.
2.2. The reciprocal law of the reciprocal of natural numbers
1/n=1/(1+n)+1/(1+n)2+1/(1+n)3+...+1/(1+n)n -1+1/n·1/(1+n)n .
n=1, 1=1/2+1/22+1/23+1/24+…=1/2(1-1/2n )/(1-1/2)+1·(1/2) n =1;
n=2, 1/2=1/3+1/32+1/33+1/34+…=1/3(1-1/3n )/(1-1/3)+1/2·(1/3n );
n=3, 1/3=1/4+1/42+1/43+1/44+…=1/4(1-1/4n )/(1-1/4)+1/3·(1/4n );
… … … …
The law of the number and frequency of scientific significance is that it's identity, is not imprecise
approximation theory of classical mathematics, regardless of any natural number n is set up.The
above all together, the left=1+1/2+1/3+1/4+…+1/n
The right=[1/2+1/3+1/4+…+1/(n+1)]+[1/22+1/32+1/42+…+1/(n+1)2]
+[1/23+1/33+1/43+…+1/(n+1)3]+[1/24+1/34+1/44+…+1/(n+1)4]+…
On the left on the right side of the equal, so
1-1/(n+1)=[1/22+1/32+1/42+…]+[1/23+1/33+1/43+…]+… ② ∴n →∞ so
1=[1/22+1/32+1/42+…]+[1/23+1/33+1/43+…]+[1/24+1/34+1/44…]+… ,
1/2=1/2[1/22+1/32+1/42+…]+1/2[1/23+1/33+1/43+…]+…
＞1/2[1/22+1/32+1/42+…]-1/3[1/23+1/33+…]+1/4[1/24+1/34+…]-…= r＞0 ,
The 1/2 > r > 0. This is the number of frequency - Euler's constant r preliminary conclusions.
Prove that 0
Euler and others: 0＜r ＜1/3, n →∞,There is
ζ(2)=1+1/22+1/32+1/42+…=(π2)/6 ; ζ(3)=1+1/23+1/33+1/43+…=1.202056903…Ap éry's constant
ζ(4)=1+1/24+1/34+1/44+…=(π4)/90=1.0823…;
ζ(5)=1+1/25+1/35+1/45+…=1.03692775…,
ζ(6)=1+1/26+1/36+1/46+…=(π6)/945=1.002476…; ……
ζ(8)=1+1/2+1/3+1/4+…=(π8)/9450=1.0039827…; …… [6]
Clearly ζ(2)＞ζ(3)＞ζ(4)＞ζ(5)＞ζ(6)＞ζ(7)＞ζ(8)＞……
Let R=1/2ζ(2)-1/3ζ(3)+1/4ζ(4)-1/5ζ(5)+…+(-1)(n-1)(ζn )/n
=1/2(1+1/22+1/32+1/42+…)-1/3(1+1/23+1/33+1/43+…)+1/4(1/24+1/34+1/44+…)-…
+(-1)n-1(1+1/2n +1/3n +1/4n +…)/n
=(1/2-1/3+1/4-1/5+…)+[1/2∑1/n2-1/3∑1/n3+1/4∑1/n4-1/5∑1/n5+…],
=(1-ln2)+r ,(n=2, ∞) ③
Because R is alternating series number of diminishing the deformation of culvert, according to
the alternating series on the folding method, always ς(7) ς(2) 1111111have R
ς(8) ς(2) ς(3) ς(4) ς(5) ς(6) ς(7) ς(8) -+-+-+=0. 6284.... ∴instead it still exists R
R=1-ln2+r＜0.6284…, r ＜0.6284…+ln2 -1=0.321520…＜1/3.
note : ln2=1-1/2+1/3-1/4+1/5-1/6+…+[(-1)n-1]/n . Euler's constant r since have no one to prove
whether it is rational, now to prove - Euler number frequency constant r is irrational. In (3) type R
= 1 - ln2 + r
numbers .Finished.
3. The Riemann zeta function principle: s for arbitrary number; S - up, p
for prime Numbers. ζ
(sn )=1+1/2s +1/3s +1/4s +......+1/ns >1>∏（1-p -s ).
prove: ⑴.let ζ(s1)=1+1/2s , (1/2s ) ζ(s1)=1/2s +1/4s ,
Namely (1-1/2s ) ζ(s1)=ζ(s1)-1/2s ζ(s1)=1-1/4s ;
⑵.let ζ(s2)=1+1/2s +1/3s , so ζ(s2)/2s =1/2s +1/4s +1/6s ,
ζ(s2)/(1-1/2s )=1/2s +1/4s +1/6s , 1/3s (1-1/2s ) ζ(s2)=1/3s +1/9s -1/12s -1/18s ,
(1-1/3s )(1-1/2s ) ζ(s2)=1-1/4s -1/6s -1/9s +1/12s +1/18s +1/24s .
⑶. let ζ(s3)=1+1/2s s+1/3s s+1/4s s s; ζ(s3)/2s =1/2s s+1/4s +1/6s s+1/8s
1/3S ζ(s3)=1/3s +1/9s -1/18s -1/24s ,
( 1-1/3S )(1-1/2S ) ζ(s3)=1-1/6S -1/8S -1/9S +1/18S +1/24S .
(1-1/4S )(1-1/3S )(1-1/2S ) ζ(s3)=1-1/4S -1/6S -1/8S -1/9S +1/18S (1-1/4S )+1/24S (1-1/4S );
⑷. ........................ ⑸. letζ(sn )=1+1/2S +1/3S +1/4S ......+1/nS ,..........,
In turn have1/2s ζ(sn )=1/2S +1/4S +1/6S +1/8S +...+1/(2n)S ,........
Above 5 step derivation shows that Euler first conclusion namely elimination all even term is not
set up; The second conclusion elimination all multiples of 2 and 3; The third repeat of the last
conclusion is equal to 1. Because the one on the right: increase the negative even term and are
not even equal to and less than 1; The last important: if the increase of negative even term is
even with the increase of the sum and approximate as 0, so the original type zeta don't (s),
available
S = up, 1 + 1/2s + / 3s + 1/ns +.. = 1. The paradox arises when s = 1, therefore Euler's turn to the right
of the conclusion as 1 to deal with, only his personal subjective not objective facts. Summarize
the above argument, to prove that the number and frequency that the laws of Riemann zeta
function. The end.
4.The Riemann hypothesis is not equality
Under the classical mathematical theory system of real number of incomplete, all
conclusions are incomplete, especially product of Euler's expression, doubly interesting -- -- it
decided the Riemann hypothesis success or failure, a scientific sense. Euler primes the reciprocal
of the sum of the "divergence" incomplete prove one: namely Euler incomplete to prove:[11]
1111111ln (∑) =ln(∏p ) =+(+++......) ∑∑-122n 1-p P P 23P 4P n =1P 11111
Wry chronicling the so-called proof, in fact what also have no proof that -- -- because this is a
virtual equation. Even his white push prove correct, its conclusion is not established.
Discussion : Riemann hypothesis is not equality, it does not is single, guess,
approximate mathematical problem but a has been shown, gradually clear and establish a
scientific system, bringing the update of a series of scientific data and theory. This is the
true value of the Riemann h ypothesis.
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2016-01-2] Additional :