Fig 1: The functions eta -, mu -, zeta - and xi -- absolute value in red, angle in green proximity to 1 blue - showing the infinite pole at s = 1, trivial zeros on x = 0, y < 0, and the non-trivial zeros on x = ½. The trivial zeros coming from the sin function in the analytic continuation are at the angle shifts at even integers on the negative real axis. The corresponding zeros of show in the central foci of angle shift with the absolute value and angle reflecting the function's symmetry between s and 1 - s.
Taming Riemann's Tiger: Has a young Indian i-Phone Programmer
Solved the Most Challenging Enigma in Mathematics?
Chris King dhushara.com Genotype 1.0.7 16-6-2015
For critical fault see >>>
The Riemann hypothesis is the most challenging outstanding unsolved problem in mathematics, with a million dollar prize from the Clay Institute awaiting anyone who can demonstrate a proof. Many of the most famous minds in mathematics have tried and failed to solve this problem. Editors of some journals are besieged with proofs and many mathematicians refuse to even look at a purported proof from a mixture of skepticism that the effort will be worthwhile, combined with a trace of jealousy towards anyone who would dare to try. Accomplished world-class mathematicians have thrown just about every sophistry one could imagine into attempted proofs, including using the diverse L-functions of number fields, elliptic curves and modular forms, looking for a natural operator whose real zeros coincide with those of zeta, using the random Hermitian matrices of quantum mechanics, and even string theory, exploding the enigma into cosmological dimensions.
The following quotes, running from Riemann to Littlewood illustrate the dilemma of RH:
'It is very probable that all roots [of rotated xi] are real [i.e. lie on the critical line]. Certainly one would wish for a stricter proof here;
I have meanwhile temporarily put aside the search for this after some fleeting futile attempts.' - Bernhard Riemann
'There have probably been very few attempts at proving the Riemann hypothesis, because,
simply, no one has ever had any really good idea for how to go about it!' - Atle Selberg
'The truth has defeated not only all the evidence of the facts and of common sense,
but even a mathematical imagination as powerful as that of Gauss.' - Godfrey Hardy.
'There is a growing body of evidence that there is a conspiracy among L-functions
- a conspiracy that is preventing us from solving RH.' - Brian Conrey
'There is no evidence whatever for it (unless one counts that it is always nice when any function has only real roots).
One should not believe things for which there is no evidence.' - John Littlewood
The fact that the zeros of zeta do appear to lie on the critical line x = 1/2 provides a tantalizing hint of a deep symmetry, which becomes even more profound when viewed in terms of Riemann's xi function (above), which is symmetric about the critical line.
The intrinsic difficulty of the Riemann hypothesis is that the non-trivial zeros all lying on the critical line x = ½ is equivalent to the accumulated primes being distributed in limit order = ½ , as Albert Ingham proved where , and . Thus we need either to be able to prove every zero is on the critical line, or prove the prime distribution is as above, but the two are on opposite sides of the Euler product equivalence which holds only for x > 1.
The difficulty is that some irregularities grow extremely slowly, so that unimaginably high values of the zeros, or primes, might have to be explored without coming to a definitive answer. John Littlewood proved that there exists a value where without determining what these values would be. Skewes then produced an unimaginably huge upper bound. The bound came down more recently thanks to te Riele and later Hudson and Bayes to , but the problem remains beyond computational verification.
Neither do abstract methods, such as looking for patterns in abstract L-functions help, because the primes of number fields and the higher degree Euler products of elliptic curves are generated ultimately through the founding natural number primes themselves, invoking a hall of mirrors effect, as noted in Brian Conrey's comment above.
One way around the lack of explicit prime information in the constant 1's of zeta is to exploit the Mobius function:
(see appendix B) since , would guarantee the Mobius function would converge for x > ½, and show there were no infinities (and hence no zeta zeros).
Roupam's paper, which is compact and direct and immediately readable, without relying on other research, takes this idea deeper by using the Mobius function, which, unlike zeta, explicitly carries encoded prime information in its Dirichlet coefficients and splicing it with a fractional part integral formula for eta , closely related to zeta (see appendix A), resulting in the provable result in the lemma, which establishes that every integer has residue -1 (figs 2,3) when the Mobius coefficients are continuously convolved with the inverse fractional part function .
Fig 2: Consecutive cases from 2 to 12 showing prime equanimity towards the integers - the harmonic and prime factorization irregularities
convolving to residue -1 in each integer, regardless of their prime factorizations.
Because this lemma is provable for all n to infinity, it sets itself apart from the computational and theoretical ambiguities of trying to prove results about the zeros of , or the primes, which, no matter how far we go, might have anomalies at inconceivably greater values, because of ultra-slow sub-logarithmic growth of key irregularities (see discussion at the end).
Fig 3: Showing the functional trends in Roupam Ghosh's argument. Graphs for n=1:500, of ,
, ,, ,
and . Right: Plot of with contributing values in each row 1(cyan) and -1 (magenta).
This suggests that despite appearing to be simply an inverse harmonic function includes prime information decrypting .
This result is intriguing and novel as far as I know. It reveals a deep symmetry between the primes and the whole numbers, implying the primes are as evenly distributed in a non-mode-locked manner as they can be. The proof then uses two-sided bounds on the exponential growth of the convolved function, which is more tractable to integral analysis than the primes themselves, to show the Dirichlet series of is convergent forand hence that here, and so there are no zeta zeros, and so RH is true. The Mobius function proves to be key to RH because the zeta zeros are determined by the divergences of and these are a direct result of the graininess of the prime-determined coefficients, which appear explicitly in the quasi-alternating Dirichlet series of (see fig 5).
This leads us back to formulations of the Riemann hypothesis, in terms of the Farey sequences of successive fractions between 0 and 1, up to a given denominator, appearing in conservative chaos and dynamical mode-locking, involving Fibonacci and then golden numbers as non-mode locked limits. Two equivalents of RH state:
This is saying that the Farey fractions are as evenly distributed as they can be (to order n1/2) given that they are by definition not evenly distributed, but determined by fractions with all (prime) common factors removed.
Summary of the Riemann Hypothesis Proof by Roupam Ghosh: http://arxiv.org/pdf/1009.1092
2: Dirichlet eta function η(s) and ν(x)
Where is the fractional part of x
See appendix C for a comparable formula for
3: The criterion for the Riemann hypothesis
Rearranging variables from (1),
Using Fubini-Tonelli finiteness in the bound*, we swap integration and summation
Now since , ,
but is constant on any [k,k+1), so
So, using uniqueness of Dirichlet coefficients .
Note: While this result seems highly surprising and suggests something very novel about prime equanimity, as noted above, it may actually be an expression of the fact that , despite appearing to be a simple harmonic inverse, also has prime encoded information, as shown in the right of fig 3, which deconvolves the prime encoding in , as the above result is effectively guaranteed by the fact that , because, if f has the stated values then .
Now let and
Note that , since
, again using Fubini-Tonelli.
Hence if the LHS converges to zero as for , i.e.
and so is convergent for , where it equals (3)
Comment: Although (2) might appear to be a null result because = and the whole LHS appears to effectively cancel in the limit, , potentially allowing zeros of eta to coincide with divergences of , the LHS tending to 0 does imply has to be convergent, as the only zeros of on would give LHS =, which has no zeros in this region, but the RHS tends to 0, by the assumption on the order of fn and inequality (18) holds true throughout, confirming the two series display equivalent convergence, as illustrated in fig 5.
Fig 5: Left: Dirichlet series computations for 2 million terms for LHS above and below for 0.4+15.13i (left) and 0.6+15.13i (right), confirming they both converge for . Upper Right: Raw Mobius Dirichlet series for 1000 terms, again showing convergence for (compare fig 1), with the zeta zeros showing as divergence tongues. The key to the location of the zeta zeros is the convergence of the Mobius Dirichlet series. Lower Right: Raw Dirichlet series depictions of zeta using eta, mu and zeta itself. While the strictly positive unit coefficients of the zeta function Dirichlet series result in divergence for x < 1, the strictly alternating eta function Dirichlet series is convergent down to x = 0. The Mobius series is a grainy alternating series, with the coefficients being determined by the prime distribution. Clumping of the 1 or -1 values of the coefficients leads to periods of divergence. The zeta zeros thus lie on x = 1/2 because the prime-determined Mobius coefficients are as evenly distributed among the integers as they can be given that they can't be - prime equanimity - leading to convergence of the Dirichlet series for x > ½. Differing forms of L-function, from elliptic curves and Modular forms to number fields also have critical roots because they apply algebraic symmetries to the Dirichlet coefficients, which, despite the complexity of their higher degree Euler products, ultimately manifest the equanimity of the natural primes in relation to the natural integers. Far Right: The Mobius series also encodes the imaginary values of zeta zeros. Sequences on either side of the first zeta zero (here a pole tongue) at x = 0.4, 0.5 and 0.6 + 14.13i show linear accumulation, due to a vanishing change in the argument. To the right of 1/2 the fluctuations decrease and to the left increase, with a constant size trend on 1/2 .
4: The proof of the Riemann hypothesis
We have the following bounds
Let us assume the following: (5)
with the integrals from 0 to 1 all zero since
Fig 6: Numerical confirmation of inequalities (2), (4) and (6) defining the
two sided constraints between fn(x) andat s = 0.65 for n = 1 to 100.
From (5) (7)
Now as (7) 0 for all .
If , then its Mellin transform can be expressed as a Dirichlet series for , since both and are step functions and the transform converges.
Since , by uniqueness of Dirichlet series.
Hence from (3) converges for all
But this contradicts assumption (5), that converges only for , implying the LHS in (2) is convergent only for and hence by the contrapositive of (3).
Hence and convergent for .
Furthermore convergent for , so , thus confirming RH.
In retrospect, the most difficult part of proving RH in terms of the Mobius function is establishing a bound on the order of the Mertens function . By convolving with the integral formula for , Roupam produces derived functions f and g. Determining the order in theorem 4 appears to be an equivalent problem to that of M(n).
Mertens conjectured that M(n)< n1/2 , but Odlyzko and te Riele proved M(n)/n1/2 > 1.06 for some large n and have conjectured that sup(M(n)/n1/2)= , implying that M(n) is not O(n1/2) (Odlyzko A., te Riele H. 1985 Disproof of the Mertens conjecture http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf ).
Now this is exactly what will give convergence for any point to the right of x=1/2, but not on x=1/2 and it is what one wants to guarantee convergence of to the right of 1/2, but not on it, which is what corresponds to the zeros of being on x=1/2, but none to the right of 1/2.
Now Titchmarsh proved that RH is equivalent to (as noted in Odlzyko's paper), so this is the key result that needs to be proven for M, or for RoupamÕs f. Theorem 3 proves the equivalent of implies RH, but theorem 4 is then required to complete a proof of RH.
It has been conjectured based on RH and properties of the zeta zeros that, displaying the sub logarithmic growth mentioned earlier (http://www.cs.uleth.ca/~nathanng/RESEARCH/mobius2b.pdf), following the proof that the related function(Ingham A.1932 The Distribution of Prime Numbers, Camb. Univ. Pr. p 100 Th 34).
An intriguing property of is that the probability , which appears to be have unchanging estimates for an interval of 1000 values from any given k when values of k are investigated up into the billions. The distribution thus appears to be steady at the asymptotic probability, by contrast with the distribution of the primes. Note also that , where Fn is the Farey sequence of order n, again linking and M(n) with .
If we are investigating bounds on the growth of M(n), it is pertinent to note that one can never have more than 3 consecutive non-zero entries of because any four successive whole numbers have one which is a multiple of 4 and hence has =0. The probability P can be equated to what is left after sieving out all the multiples of 22=4, 32=9, 52=25 etc. which remove all the integers containing squared or higher powers of prime factors and may place limits on the order of M(n).
There thus may still be further evidence that comes to light in the Mobius function that could clinch what seems one of the most promising strategies to date, particularly given the direct encoding of prime information into the Dirichlet series coefficients determining convergence or otherwise evident in fig 5.
Appendix A: Standard Analytic Continuation and Euler Product Formulae
Dirichlet sum and Euler product:
Mellin transform of zeta:
This leads to an analytic continuation using xi, where :
From above we get
Appendix B: Derivation of the Mobius Dirichlet series coefficients
Notice this involves strict integer divisibility g(n\d) to combine the inverse powers of ns whereas the lemma convolution involves real number division of .
Appendix C: Derivation of the fractional parts integral formula for Zeta
This provides an analytic continuation of zeta as far as x = 0, which is simpler and provides a completely different description from Riemann's own traditional analytic continuation above.
Appendix D: Integral formula for Mobius, Mertens function and the Riemann Hypothesis
, and integrating by parts,, where M is the Mellin transform. Now by Mellin inversion, valid forand for, assuming RH. Hence the Mellin transform integral is convergent and, which is thus equivalent to RH, since the argument is invertible.