Under the previous weblog postal service most the Riemann Hypothesis, Edwin Steiner nicely observed that solely the collection of primes as nosotros know them,\[
(p_1,p_2,p_3,\dots) = (2,3,5,\dots)
\] give us the prissy wiggly business office that suppresses the positive interference. He said that the primes are non merely prime: they are "as prime number as possible". In other words, the primes are the "primest ones". The pick of the primes inward a higher house may minimize a for sure quantity, the departure from the optimal primeness. All other "vaguely similarly distributed" collections of numbers that aren't quite prime number are doing worse inward something.
What's that?
Why tin give the sack the primes endure described as the "true originals"? What's the most natural, shut-up-and-calculate, constructive mode to depict primes? They are the numbers such that all positive integers \(K\) may endure written as products of powers,\[
K = 2^{M_2} 3^{M_3} 5^{M_5}\dots
\] where the exponents \(M_2,M_3,M_5\) are non-negative integers.
Imagine that yous supersede the numbers \(2,3,5,\dots\) past times others, similar \(2.1,2.9,4.8,\dots\). What volition endure "less prime" most such a laid relatively to the "true original" primes? Well, it's simple. If yous hit the laid of products of powers\[
\{K; K = p_1^{M_{p,1}} p_2^{M_{p,2}} p_3^{M_{p,3}}\dots \}
\] where the exponents \(M_{p,1},M_{p,2},\dots\) are i time to a greater extent than going over non-negative integers, yous larn "something less pure" than the laid of all positive integers. You volition obtain closed to other set. The positive integers yous obtain as the products of powers of the "true original" primes are every bit spaced; all the products of powers of the "fake" primes neglect to endure every bit spaced.
(I introduced novel symbols \(M_{p,1}\) etc. where the alphabetic lineament \(p\) solely indicates that the next integer isn't a prime number itself but rather a label distinguishing dissimilar primes – which may endure mistaken primes.)
What's special most the every bit spaced laid of numbers? Well, if at that spot are electrical charges sitting at points \(K\) from the laid inward a higher house in addition to they repel each other (there is a repulsive potential seat out energy \(E\) inward betwixt pairs of them), they volition attempt to repel each other to perish every bit spaced. So define this seat out energy \(E\) inward closed to mode as a business office of all the "candidate primes"\[
E = E(p_1,p_2,p_3,\dots)
\] Maybe the business office contains merely pairwise interactions, perchance it also includes interactions that depend on private variables \(p_i\) only, perchance the charges of \(p_i\) alongside higher values of \(i\) are smaller to brand \(E\) to a greater extent than convergent, in addition to perchance the details don't quite matter.
At whatever rate, it could endure possible to define a natural plenty business office \(E\) of the "candidate primes" \(p_i\). The pick of the "true primes"\[
(p_1,p_2,p_3,\dots) = (2,3,5,\dots)
\] are where the business office \(E\) realizes its global minimum which nosotros laid to \(E=0\) (although a subtraction of the "natural value" may endure needed to switch to this \(E=0\) convention).
I retrieve that the minimization of \(E\) should explicate all the visual effects of the "wiggly function" from the previous weblog postal service as good as other things. So maybe, nosotros sympathise where the "special ability of the truthful master primes" comes from.
Just to endure sure, i may define a generalized zeta business office for whatever laid of mistaken primes (although I don't quite know which of the properties of \(\zeta(s)\) this business office obeys). Just work the Euler product:\[
\zeta(s;p_1,p_2,p_3,\dots) = \prod_{i=1}^\infty \frac{1}{1 - p_i^{-s}}
\] This formula may endure used fifty-fifty if yous substitute arbitrary "fake primes" for the parameters \(p_i\). However, for generic values of \(p_i\), the business office volition remove keep RH-violating zeroes inward the critical strip, fifty-fifty extra poles inward the critical strip, in addition to volition reject to obey the \(s\leftrightarrow 1-s\) functional relation (or symmetry). I am non for sure whether the business office inward a higher house may endure unambiguously continued to the whole complex plane.
Now, assume that this business office may endure extended in addition to written inward price of closed to "fake roots" of the "fake zeta function" above. If that's non the case, therefore it's probable that this endeavour to attempt RH collapses. The Riemann zeta business office may endure written as a production from the nontrivial zeros, upwards to a rather uncomplicated factor:\[
{\zeta(s) = \pi^{s/2} \frac{\prod_j \left(1 - s/\rho_j \right)}{2(s-1)\Gamma(1+s/2)}.}
\] The production over roots \(\rho_j\) has to evaluated as a restrain of a partial production where nosotros add together the factors alongside the growing \(|{\rm Im}(\rho_j)|\), i.e. nosotros remove keep to add together the roots past times pairs. Now, closed to other "Big If" is that the previous zeta function\[
\zeta(s;p_1,p_2,p_3,\dots)
\] of the mistaken primes may endure rewritten (in a for sure vicinity of the truthful primes) as a business office of the mistaken roots\[
\zeta(s;\rho_1,\rho_2,\rho_3,\dots).
\] In other words, the precise values of the "fake primes" may endure translated to the precise values of the "fake roots" of the mistaken zeta function. This map betwixt "the laid of primes" in addition to "the laid of roots" looks similar a map betwixt 2 dual lattices except that the sets aren't lattices. More precisely, nosotros postulate to bring the natural logarithms \(\log p_i\) of the primes which are "complementary" to the roots inward a similar feel as the momentum in addition to winding inward string theory.
(This should endure done past times methods similar to the quasi-Fourier expansion alongside the frequencies \(\log(p_i)\) on the critical axis – or the vertical axis alongside a root – discussed inward the previous weblog post.) Maybe yous postulate to allow the locations of piffling (and non merely nontrivial) zeros (and perchance the singularity at \(s=1\)) to endure moved as good to fully fit the Euler production from the given "fake primes".
I retrieve that nosotros avoid diverse confusions if nosotros include all the roots \(\rho_j\), including the "mirror ones", in addition to process all of them as independent complex variables. For RH-compliant nontrivial zeroes (those on the critical axis) \(s\), they are paired alongside closed to other root \(s^* = 1-s\). For the hypothetical RH-breaking nontrivial zeroes \(s\), at that spot are iv distinct zeros inward a grouping related past times symmetry,\[
s,1-s, s^*, 1-s^*.
\] But those should endure labeled e.g. as \(\rho_1,\rho_2,\rho_3,\rho_4\) in addition to considered as iv independent complex variables. If that's therefore in addition to if the \(p_i\leftrightarrow \rho_j\) lexicon works, therefore the "fake primes" \(p_i\) volition endure locally holomorphic functions of all \(\rho_j\).
If the laid of "fake roots" \(\rho_j\) is invariant nether complex conjugation i.e. if it contains a \(\rho_j^*\) along alongside every \(\rho_j\), therefore the corresponding "fake primes" \(p_i\) volition endure existent (at to the lowest degree inward closed to vicinity of the truthful primes in addition to truthful roots). And for these existent locations of candidate primes \(p_i\), the potential seat out energy \(E\) volition endure real, too.
So the invariance of the "set of candidate roots" \(\rho_j\) nether the complex conjugation may endure assumed almost everywhere – that's what is related to the reality status for the primes. On the other hand, for full general "fake primes", I believe that the mistaken zeta business office won't obey the functional human relationship exchanging \(s\leftrightarrow 1-s\), therefore inward general, the mistaken primes won't endure translated to mistaken roots that are symmetric alongside honor to the critical axis.
How practice yous consummate our proof of the Riemann hypothesis?
You reckon the vicinity of the "true primes" \(p_i\) in addition to the corresponding "true roots" \(\rho_j\) in addition to investigate the min variation of the latter. If the lexicon industrial plant at all, the variation of a exceptional \(\rho_j\) may endure translated to a variation of all primes \(p_i\).
The RH-compliant roots are similar half-D-branes that are stuck on the critical axis. They can't motility away. If the critical axis reduced the unitary estimate grouping of D-branes to the orthogonal one, a unmarried regular root of the zeta business office would ship the \(O(1)=\ZZ_2\) estimate group. (I truly practice retrieve it makes a deep feel to reckon the roots to endure locations of D-branes, or eigenvalues of a random matrix. The distribution of the roots follows the patterns of random matrix theory.)
On the other hand, if at that spot were an RH-breaking root \(\rho_j\) away from the critical axis, it could motility inward the whole plane, similar a total D-brane that carries the \(U(1)\) estimate group. And I retrieve that this extra liberty of the roots that remove keep partners on the other side of the axis has consequences.
I retrieve that if yous varied such a "bad" root \(\rho_j\) past times the complex \(\Delta \rho_j\), the seat out energy \(E\) – which is stationary at that dot (the get-go derivative is zero) – would remove keep to alter to\[
E\to E+\Delta E, \quad \Delta E = a\cdot \Delta \rho_j^2
\] At to the lowest degree I promise that at that spot would endure a mode to demo that inward this leading approximation has to endure holomorphic. You tin give the sack meet that \(\Delta E\) is complex inward full general but if yous vary the complex conjugate root \(\rho_{j+1}\) past times the complex conjugate amount, yous larn closed to other contribution to \(\Delta E\) that is complex conjugate to the contribution inward a higher house in addition to their marrow volition endure real.
(Ten minutes subsequently I completed the weblog post, I realized that this was a rhetorical fob to fool myself in addition to overlook the possible price \(\Delta \rho_j \cdot \Delta \rho_{j+1}\) that withal guarantee the reality of the seat out energy \(E\) if \(\Delta \rho_j\) in addition to \(\Delta \rho_{j+1}\) are complex conjugate to each other. So I acknowledge that this "mixed term" could endure at that spot as good but I tend to believe that a nicely defined \(E\) volition endure to a greater extent than local fifty-fifty inward the \(\rho\) plane in addition to price of the type \(\Delta \rho_j^2\) in addition to \(\Delta \rho_{j+1}^2\) volition dominate over the mixed ones. The seat out energy \(E\) could endure given past times closed to rather specific facial expression as a business office of the roots in addition to this "locality" could endure demonstrable.)
So alongside the extra optimistic supposition suggested inward the previous paragraph, the second-order price inward \(\Delta E\) deport like\[
{\rm Re} (a\cdot \Delta \rho_j^2)
\] in addition to the funny thing is that this business office may remove keep both signs, basically because \(x^2-y^2\) has a saddle at \((x,y)=(0,0)\). If the seat out energy \(E\) becomes higher – as it should – for closed to stage of \(\Delta \rho\), it becomes lower for the orthogonal direction, \(\Delta \rho_j\to i\cdot \Delta \rho_j\).
It agency that the RH-breaking roots are "too loose" in addition to allow yous both to increase or decrease \(E\). In other words, an min variation of such roots would allow yous to divulge
candidate primes that are fifty-fifty primer than the truthful primeswhich is a contradiction. With a fully specified Definition of \(E\), it should non endure difficult to demo that the truthful primes hit the most every bit spaced laid of the products of powers. There can't endure anything better! But the loose primes away from the critical axis give yous closed to room for an improvement, therefore these RH-violating primes cannot exist.
I must warn yous that at that spot is closed to other possible lethal defect of the projection hither because at that spot is closed to feel inward which I need to assume that the laid of roots \(\rho_j\) remains invariant nether \(\rho\to 1-\rho\) or \(\rho\to 1-\rho^*\) at all times. Otherwise I would endure able to divulge the numbers \(p_i\) that are "primer than primes" fifty-fifty out of variations of the RH-compliant zeros. But I remove keep previously stated that I believe that for full general "candidate primes", i breaks the \(s\leftrightarrow 1-s\) symmetry of the zeros.
Well, piece this looks similar a lethal problem, at that spot is a really small-scale but (subjectively) nonzero remove a opportunity that there's something optimistic that I overlook hither – or a slight variation of the sequence of thoughts that industrial plant every bit good – in addition to the fixed declaration makes feel in addition to yous volition endure able to plough this weblog postal service into a rigorous proof.