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,…, 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 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.
The reason we aspiringly anticipate the involvement of design in occurrence and unfurling of prime numbers is the case of glorious Riemann Hypothesis (RH):
“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.
- 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
* A tighter way of saying this is π (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.