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ROHIT GUPTA
SPACE MONKEY

Rohit Gupta (@fadesingh on Twitter) is a mathematician who thinks the universe is a hologram projected by a microscopic disco ball.

Zeroes and prime numbers ascend in the mist at Zeta Valley

veryday, and on most nights, I stand under a mist that covers the side of a mountain. A ghostly landscape overlays the physical city of Bangalore with something taller than Mount Everest, and farther than the moon. The slope of this mountain at any point x is related to the logarithm of x (see the figure). Somewhere inside the belly of this giant is a fire, hidden carefully underneath a basket, designed to burn for eons.

A prime number can only be divided by itself or 1, such as 2, 3, 5, 7, 11, 13, 17...and so on. The curve simply represents the number of prime numbers below a certain x. So when x=100 then the number of primes below 100 is 25, and if x=1000, the count is 168. Here are some more values of π(x), as the y-axis counter is known (not related to Pi, the ratio between the circumference and diameter of a circle)

In his famous essay, "The First 50 Million Prime Numbers", Don Zagier writes: "For me, the smoothness with which this curve climbs is one of the most astonishing facts in mathematics."

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We can use a combination of the Möbius function with other functions to extract new patterns. In a beautiful way, this process feels like sculpting with invisible grains of sand rather than calculation.

And this smoothness is the reason this slope is very slippery, making prime numbers an impenetrable mystery. Compare this smoothness with the fact that prime numbers are the atoms of integers, each of which has a unique formula written purely in terms of prime numbers. The formula for 344 is 2x2x2x43, for example - and for the number 1729 (made famous by Ramanujan) the formula is 7x13x19. If only currency notes and coins were all based on this knowledge, the world would be a significantly different place. Just as the chemical elements were arranged neatly in a periodic table by Mendeleev, Bernhard Riemann made an observation about prime numbers in 1859 that befuddles the world to this day. To climb this invisible mountain, we take the help of invisible machines - functions and formulae.

One of these machines is known as the Möbius function, whose task is to behave like a dye. You give the Möbius function any number, and the function will stamp a particular color on the number, after seeing what its internal prime number formula is. There are only three possible colors which correspond to the numerical values of (1, 0, -1). Imagine I have a magical kite with Chinese lanterns strung upon it at a distance of 1 meter each, and its end is tied to infinity. Switch the Möbius function on and all the lamps light up according to the number. We can use a combination of the Möbius function with other functions to extract new patterns. In a beautiful way, this process feels like sculpting with invisible grains of sand rather than calculation.

More advanced functions allow one to manipulate a swarm of numbers like the murmuration of starlings in flight. One such function is the infamous Zeta function, which Riemann used to show a hidden pattern in prime numbers. That pattern, known as the Riemann Hypothesis, is the most notorious unsolved puzzle in existence.

This game is not about calculation so much as a sense and feeling for abstract patterns. The longer one fights a problem of this nature, the more solidity the abstract curve assumes. Such an explorer feels something that is sought after by mystics too - a certain kind of unity. The curving roads one walks upon become the traces of an equation on an infinite graph, and the individual becomes - the origin of all curves, the moving zero.

 
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