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Prime Numbers Paralleling Reality: Possible?

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Post recently published in Science Blogs. Thought of posting it here to keep the blog readers current. Indulge in primes!

All non-trivial zeros of the zeta function have real part one-half

stated Bernhard Riemann in 1859, a German mathematician whose contributions to modern mathematics, and theoretical physics, is wide and deep—a commonly known one is in structuring the layout of Einstein’s theory of general relativity (spacetime conforms to gravity).

Riemann zeta function

The relatively simple form of Riemann zeta function (in the above statement),

equation1

is an infinite series converging on its limit—a mathematical articulation worked out utilizing tools of analysis. This function with some clever number juggling, directed by Euler, transforms itself into a product (∏), that is, a series involving multiplication—as opposed to the above summation (the summation symbol ∑ we are familiar with)—over all primes, bringing the quirk of primes in the scope of palpable. Here we have the most significant milestone in connecting the nature of primes to the tapestry of all numbers (recall that at surface we don’t see a clear scheme in the distribution of prime numbers). The magic lies in the relationship of “product (∏)” to “summation (∑),” known as Euler product formula, with prime numbers coming into play. The above zeta function is then also this:

equation2  (p: prime, over all prime numbers)

Conceiving the dynamics of this function would then help grasp the inner nature of prime numbers, which Riemann did by the above hypothesis. Indeed visualizing the dynamic interplay not only involves seeing the structuring of prime product but also seeing it in the light of playing of the summation function, which involves perceiving through scrupulous analytics and advanced calculus.1

Digging deep

Except for 1, the zeta function has values for both positive and negative numbers, and its value for every negative even number is a zero—but a trivial zero. (We will see what the zero of a function implies in a bit.) The availability of non-trivial zeros is the gripping point in the true portrayal of prime numbers, and it emerges from the zeta function only but under the guidance of complex field involving the above exponentiation with complex numbers (“a + bi” is a complex number, with a as real part and bi an imaginary where the standard i is taken to be √–1). The Riemann Hypothesis says that under the navigation of zeta function, the complex plane brings about a steadfast line that sits at a ½ real value, streaked all the way to infinity rendered by all non trivial zeros—known as the critical line (Figure 1). Infinitely many non-trivial zeros satisfy the Riemann hypothesis,2 and the first ten trillion of them are seen to conform to the hypothesis.3

The first few non-trivial zeros (known as Gram’s zeros) start approximately as:

½ + 14.134725i; ½ + 21.022040i; ½ + 25.010856i

See the ½ real in the complex plane with different “i”s. Important is to note that here all “i” comes to be an irrational number, that is expanding limitlessly without any pattern, but that’s another story, off from the point of this post.

Figure1

Seeing the looming “½” takes exceedingly complex renderings like Equation3 and Riemann’s vision. Significant mathematical maneuvering and background would be required to even come close to how the non-trivial zeros align, but there it is. By it we have a hold of a crisp order executed by prime numbers—the very numbers that at the surface hover haphazardly (Figure 2). And this schematic is written in a regular numerical language right in front of our eyes. The root of the natural number landscape comes to be the tenacious halo of primes.

Figure2

Unifying Principles

Lucid as it is, we haven’t seen the apex yet. In this deep-seated scope of a clear scheme the prime numbers take us further. Their fabric is stunningly indicatory one. It is here we see the dovetailing primes portending the coordination of the physical universe at its inmost depths.

To cut a lengthy and exceedingly labyrinthine story short, the mathematics that goes in describing quantum mechanical landscape constructs on advanced dosages of matrices—a group in an array that abides by certain set principles—algebra, and group theory. Mathematical operators, which underlie the rendering of matrices, are utilized to chart out the statistical mechanical territory of quantum landscape. Every matrix is stamped with a signature algebraic equation. An algebraic equation is like a prescription, realizing which one can decipher the nature of the object. At mathematical level this means finding its roots: incorporating what values in the equation do we get a zero. For example, for an expression x2 – 3x – 4 (i. e. equation x2 – 3x – 4 = 0) the roots come to be –1 and 4. Replacing x with either number annuls the expression, or makes it zero. The degree of the polynomial (algebraic) defines the number of zero(s) the polynomial has. Thus the squared ones, like in the above example, will have two zeros, or roots.

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It is in these roots we merge the math and universe. For mathematical operators that go in describing quantum field these algebraic zeros are referred as eigenvalues—rings a bell? Indeed, it points to the eigenvalues of energy in quantum mechanical setup—that only certain values of energy are allowed.4,5

It is here we have the natures unite. Some such specialized operators cast striking resemblance with the Riemann’s zeta function in a way that the operator’s eigenvalues coincide with the zeta function’s non trivial zeros. It is here that not only diverse mathematical branches meld but also mathematical and physical amalgamate (Figure 3), by the sharp correspondence of the quantum energy values (the eigenvalues) and the non-trivial zeros.

Figure3

We now have prime numbers not only casing a universal principle of symmetry but also doing it in the well defined outlay of tactile quantum realm.5 Their symmetry isn’t on the surface but in the dynamical interplay—the aligning of zeta zeros—that the physical world at its roots dons.

The non-trivial zeros themselves fall in a pattern, and squeeze closer and closer, as we climb up the complex ladder of zeta function. The spacing of non-trivial zeros aligns with the spacing of the eigenvalues. The array of quantum eigenvalues constitutes the spectrum that the non-trivial zeros of zeta function bring forth.  Then, the deep-hidden order of primes is the language of quantum depictions.

This was more than expected!

It is even contemplated that the Riemann function itself can directly be prescribed by an operator which would model a physical system, i. e., a potency of seeing a physical system by the weave of Riemann operator—a physical system of semiclassical quantum chaos to be precise.4 Not chaotic chaos, but chaos of chaos theory which sees a crisp complexion in a rendering that at the surface appears completely erratic. The non-trivial zeta zeros of this operator would be eigenvalues of a semiclassical chaotic system.

The Riemann hypothesis not only substantiates the Prime Number Theorem, it exposes a stubborn structural identity to the prime numbers, and piece them in the all-embracing arena of symmetry. Indeed immense approximations are involved for us to see the diagrammatic of the hypothesis, but they are all with acute mathematical precision.

The nuance of the quantum world vindicates the hypothesis. Do we still need a proof!

The hypothesis isn’t proven or disproven yet,6 but it has incited a great deal of novelties and unified large swaths of mathematics and mathematical physics in the interim. The intricate interconnections that play out behind it is mesmerizingly suggestive, and offer deep insights of the natural structure that is both discrete and abstract at the same time.

——————————————————

References:

  1. John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, A Plume Book, 2003
  2. H. Hardy (a British mathematician) in 1914 proved that infinitely many non-trivial zeros satisfy Riemann Hypothesis (or lie on the critical line): Sur Les zeros de la fonction ζ (s) de Riemann. French. In: Comptes Rendus de l’ Académie des Sciences 158 (1914), pp. 1012-14. Issn: 00014036.
  3. Gourdon (2004), The 1013 First Zeros of the Riemann Zeta Function, and Zeros Computation at Very Large Height.

For an overview (4, 5):

  4. Barry Cipra, A Prime Case of Chaos

  5. Germán Sierra, The Riemann zeros as spectrum and the Riemann hypothesis

6. Clay Mathematics Institute Millennium Problems: http://www.claymath.org/millennium-problems/riemann-hypothesis

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Window of Mathematics: The Language of Prime Numbers

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Along the theme of earlier post of mathematics as being a universal language of the reality itself, here we shall peek into the revelatory window of prime numbers—for their simplicity and uncertainty at the surface, alongside the intricacy and perfection underneath.

Underneath the uncoordinated display, the prime numbers incite well-structured tones—of mathematics and the universe in their finest resolutions

For their unbreakability primes are viewed as atoms of mathematics—they construct all other numbers of the natural domain. But their appearance at the surface appears arbitrary, for the lack of a recognizable pattern in their structure or intermediary spacing. In the landscape of numbers, the prime numbers crop without any fabric of symmetry, which mathematics and the universe otherwise blatantly seize in their manifestation or flow. Starting from 2, 3, 5, 7, 11, 13, 17, 19, 23, ¼,  Euclid of Alexandria around 300 BC showed that these asymmetric entities stretch to infinity—of which first 100 billion or so are crunched.

The Concept and a Deep Underlying Order

Neat schemes of reality often emerge in the territories elusive and outwardly inconsequential, and take subtler outlooks and deeper visualizations. The correspondence of antimatter, the underpinning of chaotic system, the essence of entropy, and the design in fractals of nature are some examples where principle plays underneath what seems a haphazard display. But, nowhere is this more obvious than in the instruction of prime numbers. It took both the magnetizing appeal of prime numbers and the sharp visionary intellect of the followers to stumble upon the spotless tone that underlie their superficial irregularity. In the abysmal subtleties of their materialization not only does reside a well-pressed systematic structure, its code is both mesmerizingly suggestive and hauntingly captivating.

Never get caught up with the deceptive lack of pattern—concept actually, in math or otherwise.

The number of primes up to a given max N is shown to be N/ ln N (ln: the natural log)* by a relatively analytical theorem known as the Prime Number Theorem, which was proven independently by Jacques Hadamard and Charles de la Vallée in 1896 employing elaborate mathematical measures. The theorem implies that prime numbers thin out as we climb up the number ladder. The clarity of thinning though becomes apparent only at gigantic magnitudes, seen over logarithmic scales (as log function in the above formula suggests). This is slightly reflected at the onset: There are 25 primes to count 100, and 168 to 1,000 (instead of 250 if it were a regular distribution). Then there are 1,229 to 10,000, 9,592 to 100,000, and 78,492 up to a 1 million: the number of primes isn’t expanding proportionally. The tapering effect can be appreciated for large series of crunched primes at a site like primes.utm.edu. Albeit lightly, the Prime Number Theorem brings to light that underneath the mixed up display, the constitution of prime numbers and their mechanics appears to be a parameterized layout, but so far after centuries of effort a clear logic behind the mechanism remains obscure. But not, if we take the Riemann Hypothesis 1, 2, 3 to be not only authentic, but also natural.

Fig1_PrimesEd

The reason we aspiringly anticipate the involvement of design in occurrence and unfurling of prime numbers is the case of glorious Riemann Hypothesis:

“All non-trivial zeros of the zeta function have real part one-half.”

Incredibly simple, isn’t it? The statement is more like a tip of the iceberg though (my thoughts on conveying its potential to general audience), with not only immense and consequential cues lurking under it, it takes up full range of elements from basic arithmetic functions, analysis, calculus, analytic number theory, advanced algebra, probability, statistics, and a fair share of visionary mathematical sense—tailored in place 1 by Carl Friedrich Gauss, Leonhard Euler, Lejeune Dirichlet, and indeed Bernhard Riemann, who was also the one to conceive this interpretation.

Granting the well-groomed and weighty diagrammatic this statement brings forth—so much as to make the hypothesis a self-evident truth—how its intricate circuitry plays challenges even the shrewdest of mathematicians.

But before we question what the prime numbers tell us about the real universe (is it even possible?) and how Riemann hypothesis connects to the field of prime numbers, we need to first delve a little into the articulation of this Riemann message itself, and I will be back with that shortly.

Fig2_Primes

—————————————

* A tighter way of saying this is p (N) ≈ Li (N), where π is the Prime counting function (up to N), Li is logarithmic integral, ≈ is “tends to approximately equivalent” as N gets larger, that the ratio π (N)/ Li (N) tends to 1 as N gets bigger and bigger.

References:

  • John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, A Plume Book, 2003
  • Marcus du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, Harper Perennial, 2002
  • Roland van der Veen and Jan van de Craats, The Riemann Hypothesis, Mathematical Association of America, 2016

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The Upcoming 2017 Solar Eclipse, Sweeping America on its Totality

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The simple mechanics of total solar eclipse exposes deep-seated fundamentals of spacetime. Total solar eclipse occurs in an event of earth, moon, and sun alignment such that moon fully blocks out the sun, casting its shadow on earth on the zone of totality. What remains on sky is sun’s corona shimmering behind the bulbous moon: Includes a rendering imaginatively known as diamond ring. On August 21, 2017 we will transit such a mesmerizing and momentous (literally!) event, and the eclipse experts, chasers and broadcasters have their bits and takes on this. Here are some genuine picks  (1, 2) for those interested in details, and here is an interactive map of the upcoming totality. This year the ASP (Astronomical Society of the Pacific) is holding its annual meeting just for the purpose of convening the ideas and topics around the wonder of total solar eclipse, particularly toward preparing the upcoming 2017 one. Those interested in cosmic magnificence, and like to partake in grasping the nature of reality, would truly benefit from the event.

As profound as the cosmic phenomenon itself is, total solar eclipse has been pivotal in our understanding of the way universe shapes and continues, and a linchpin in rubber stamping a revolutionary theory to be a truly authentic reality. On the May 29 of 1919, an English astronomer, physicist, and mathematician, Arthur Eddington, captured total solar eclipse on the island of Principe to validate Albert Einstein’s theory of general relativity. General relativity offered to blend gravity in the earlier picture of Einstein’s own special relativity, showing that gravity is the geometry of spacetime itself. The endeavor set out by Eddington and his team pinned the precise bending of light that occurs due to the presence of a massive body, in accordance with the principle of general relativity, thus fully endorsing Einstein’s Magnum Opus. Sun as a massive body too bends light that travel from distant stars, but we cannot verify such bending simply because sun’s intense glare blocks out the positions of distant stars. The shade of a total solar eclipse enables us to measure such deflections in the position of stars, as the sun observes its gravity.

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The ramifications of general relativity are wide and far reaching, many we are still trying to fathom: From the origin of the universe to the existence of black holes (remember the fascinating Interstellar Gargantua), the phenomenon of wormhole, the prodigiously expanding universe to speculations of dark matter and dark energy to the recent detection of gravitational waves that employed state of the art technological sensitivity (10-16 cm in 4 km). General relativity has stood a century of experimental verifications, one recent with the validation of gravitational waves by LIGO (Laser Interferometer Gravitational-Wave Observatory), and some tests are still brewing that involve extraordinary precisions to further endorse general relativity, like appraising the contortions due to the black hole at the center of our galaxy or seeing the free fall of different materials in space missions.

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The theory has shown the way universe propels, but also made our lives efficient on a daily basis. General relativity is a part of GPS navigation that we employ every day. Two well crafted titles that shed light on this deeply enriching theory are 1) The Perfect Theory by Pedro Ferreira, and 2) Big Bang by Simon Singh.

The first real validation of general relativity was ticked by the 1919 total solar eclipse. I will be attending the ASP meeting, and in the context of total solar eclipse, I will be speaking on the fundamental architecture of spacetime that the general relativity imparted.

For those interested in cosmic mechanics, deeper universal structure, or just scientific outreach to a wider community, it will be a good venue to participate and connect.

See you soon,

Neeti.

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Einstein in Fiction

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Starting with the elegance of mathematics, here is an article the followers of mathematics will like—the true patrons of ‪mathematics see its reality in the deep-seated concepts.

At the Book Expo America in Chicago this year, as I explored flamboyant publishing setups and flashy book banners, an interesting title The Other Einstein caught my attention, and I was pulled in. After noting that the title refers to Einstein’s wife Mitza Maric as the other Einstein, and that the story narrates of her own potentials in understanding the ways of spacetime that Albert Einstein set forth, I became somewhat curious. I decided to meet up with the author. Even though the book itself is a novel, for it touches spotless territory of spacetime that Einstein established, the story can be seen as rather bold. Anyway, there I was, inquisitive enough to get a copy.

OtherEinstein    PostIX

As I was handed a copy, I spoke briefly with the author on fictionalizing a landscape that is so firmly established and deeply revered, by scientists and laymen alike. The author had her takes on it for the extent of fictionalization, and I was curious enough to give it a try. Fiction isn’t my usual read. Barring a very few known titles, like by Paulo Coelho for instance, I haven’t read much in current fiction. As I said The Other Einstein drew me in, first to just get a copy at the BEA, and then to read it, for the obvious reason. Not only do I have a background in physics, I am an ardent proponent of physics and mathematics for exposing the reality we live in. And for these reasons I am deeply aware of Einstein’s contributions and his legacy, so much so that for me to see that his special relativity theory is referred as being conceived by someone else—even in fiction—seems almost sacrilegious. Having said that, the story is crafted well, and once I started it I was hooked to finish. If the aim was to formulate a page-turner, the title has it.

For us scientists it might have been nicer if the extent of fictionalization was in some way hinted. To the author’s acknowledgement, this fiction weaved some of the real historical bits—time, space etc. Author’s  efforts in assimilating Einstein’s theories, and the scientific structures on which they rest, as it’s penned in the fabric of storyline, is certainly appreciable.

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But the aficionados of pure physics/mathematics, or the sincere advocates of Einstein’s efforts, aren’t probably its best readership target.

See you all soon,

Neeti.

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The Inter-Connection

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I am back. This time it was a longer intermission, after many weeks of steady continuity. That regularity mostly had to do with my being urged on by the efficient Jaymie Shook of the Bohlsen group to write more routinely than I have been. The main reason behind that is to spread the word around on what the subject of my recently released book is about, and I hope I have done a somewhat convincing job.

On my march to spread the word, I also dared to take up something I have managed to cower from thus far—the social media. The thought that a social media presence is a must in order to fetch interest, and target right audience gave me willies. I am zealous about the subject, its scientific order, and mathematical views, love talking about it to an audience, in person, or over e-communication with the people I am acquainted with, and I am passionate to hear their views, what fascinates them most, and ideas. Shooting out tweets, and hurling jottings and condensed utterances on Facebook in a fully open landscape boundlessly seeped with all different opinions, interests, and intentions is something entirely different. And it gave me jitters! I guess such a reaction would be more common in people who have worked all their lives in structured environments of an academic setup, where you cave in comfortably within a premise, relatively sequestered from majority of the outside scenarios. It feels far less risky.

The notion of scientific outreach in an academic institution is itself a very modern, and indeed fruitful, thinking, and many able researchers have caught up with that very well, and take pleasure in popularizing science. Some launch their intellectual views right in the public arena, bypassing the slippage that would be encountered if gone to a specific collegiate field mostly for the interdisciplinarity of their viewpoints.

For the most part I too liked to be tucked in covers, within a well laid out premise. But our scientific quest has come to a point where moving forward necessarily involves large chunks of interdisciplinary views, and takes. And we all are acknowledging that the things are opening up within science, as well as outside of it. The comfort zone on its own is expanding, as we find ourselves plunging into it.

So there it was, I set up a twitter account and started tweeting, opened a Facebook page, and went buzzing, connected with Goodreads, and put up giveaways, and tried to be at LinkedIn more often. The exposure has been better than anticipated. And it is satisfying to see how many original thinkers, and established academics take time and effort to be there in a common open ground, constantly twitting, pitching and improvising. That most of the genuine organizations are in a constant update of their face, voice, and initiative. Their tone isn’t always as weighty, and the cadence at times exceedingly popularizing. And at times I have myself felt that they have gone a bit too far. But I think at a common level that incites to be curious and creative, and importantly there is a conduit to connect to them, and discover new and fascinating places that would have lay hidden without the cause of social media.

IMG_0041_TwitterIMG_FacebookIMG_LinkedIn

I connected to a few, and discovered many new. It is productive, informative, and in a strange way real. We discuss and follow numeric, abstract, real and mysterious ways to mathematics (and mathematicians!), the articulations of space-time, including about the recent discovery of gravitational waves, and the interconnected black holes, keep abreast of up-to-date scientific findings in all flavors, once in a while take in the humor part (which is mostly indispensable), philosophy (not the wacky type but the resolving kind that is essential) and indeed some of the current affair outside of science, and personal flavors.

Shedding hesitation is a tough work, but I guess it is worth pursuing one’s and parallel interests in the growing web of virtual space-time.

I am sure many of you already are trekking the cyber social landscape. You can join me there, on Twitter, Facebook, LinkedIn, GoodReads.

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Also, don’t miss out on having a chance of grabbing a gift copy of my book. Find the “giveaway” in the widget area below. If you win one, I would very much welcome your response, thoughts, curiosities, and even a review. Thank you!

Let me know if you have any questions at nsinha@magnifieduniverse.com, or writemailac@yahoo.com.

Thanks!

See you all soon,

Neeti.

 

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The Title and its Storyline

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Continued from the preceding post…

Foremost, we can’t keep from commemorating the 2016 Abel prize awarded to Andrew Wiles of Oxford University, for proving that the Fermat’s Last Theorem is indeed true (in the year 1995). Congratulations to Andrew Wiles, and Pierre de Fermat! Fermat did claim (in the 17th century) to have surmounted proving his own elegant equation by noting “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” The methodology Andrew Wiles employed is too advanced for the time of Fermat. Inspired at the age of ten, Andrew Wiles decoded the mystery of Fermat’s Last Theorem in the year 1995, a truly uphill task that was interspersed with a humiliating pitfall that ultimately lead to the glory and catharsis, as his humbled tears rolled out upon meeting the wish.

Whether or not did he have the proof (we will never know), Fermat would have cheered the breakthrough, and recognition.

Here is my take on it:

Well, I am more excited than many, first because of the Oxford University backdrop in the recognition, but mostly because it involves the elegance and depth of Fermat’s Last theorem, and seeing it to be accurate.

I delight in the simplicity of its statement (the equation), yet the far reaching and deep insights it casts. I include the insightful cadence of this equation in my book.

The excerpt from the book, following which is the award link:

Excerpt, Pg. 56: Physical Laws of the Mathematical Universe: Who Are We? (about the book: www.magnifieduniverse.com/aboutbook)

“Fermat’s Last Theorem: An Enigma, or Not

For its blunt accuracy and transparency, even though we didn’t have a valid proof at the time it was stated, Fermat’s last theorem became a cliché mathematical citation, appearing regularly in didactic and popular genres alike.5,6 The statement is elegantly simple, but the meaning conveyed is both sharp and profound. Drafted by a French mathematician, Pierre de Fermat, in the year 1637, it states,

FigVI

              where n is the exponent of 3and up. The phrasing tells us that the sum of two exponentiations cannot give rise to an exponentiated entirety for the powers of three and up. For example, 32 plus 42 structures into 52, but 33 plus 43, in accordance with Fermat’s theorem, does not evolve into an entirety of x3—3-D-fold. Fermat’s equation applies for any numerical grade—in fact, tellingly, for any digital combination—as long as the power is 3 or higher.”

The award; The recognition

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Back to the storyline, and the central points of discussions:

Universe Needing to Inflate

The abrupt inflation of universe in our cosmic history, its interrelatedness with the detection of gravitational waves, and seeing the necessity and order of the event of inflation itself

            “As enigmatic as it may sound, the scenario of expeditious growth does have healthy outlooks to support of the way we envisage the universe based on scientific judgments.”

In the Name of Science

The question of how do we amass interest and enthusiasm in science, its concepts and methodology. Then move further to have us all interested in seeking the true order of reality.

Interstellar

Do not miss out, if you like edutainment, especially with small dosages of science. You might pick up serious bits without having to try!

Grothendieck’s Deep Visions

The gravity of mathematics, and its followers: Alexander Grothendieck as an ardent devotee of anything deep and mysterious in mathematics

Continued in the next…

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Grothendieck’s Deep Visions

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Alexander Grothendieck isn’t a household name in academic community, and at a general level hardly anyone would ever have heard of him. Among mathematicians he dwelt as persona of profound brilliance and finesse, colored with uncanny idiosyncratic taints. Not just efficient in clearing up of the most convoluted mathematical renderings, he held clear workings of the most abstractly mathematical landscapes—of algebraic geometry and topology—and advanced them to the level of fathomable depictions for us all.

As in many cases in the modern history, his deep mathematical insights seemingly came with a price. Apart from deep-seated crannies of complex mathematics, he, seemingly by choice, remained mostly disconnected from anything in the rest of the world, even the simpler branches of mathematics.

I read somewhere an anecdote about him on prime numbers, which he chose not to be worthy of attention. He apparently addressed, hopefully unwittingly,  number 57 to be a prime number, which it isn’t (it factorizes to 3 and 19, and is thus not a prime number). Since then in mathematics community 57 is referred as Grothendieck’s prime.

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Neeti Sinha

Magnifieduniverse.com

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