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),
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:
(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
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.
Seeing the looming “½” takes exceedingly complex renderings like
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.
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.
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.
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’s 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.
- John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, A Plume Book, 2003
- 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.
- Gourdon (2004), The 1013 First Zeros of the Riemann Zeta Function, and Zeros Computation at Very Large Height: http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf
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