Human beings have an exceptionally higher level of number sense, the scope of which is well beyond the requirements for the survival of our species. The evolutionary nature and raison d’être of this intricate ability is still widely debated and remains an unresolved mystery. While most researchers suggest a biologically inherent origin for it, some claim it to be a natural consequence of cultural evolution. An interesting study in support of the cultural evolution theory was carried out among the members of an isolated tribal community by Rafael E. Núñez, currently a cognitive science professor at the University of California. It had shown that our sense of numbers is innately calibrated for a logarithmic scale; we give equal importance to small changes in small numbers and big changes in large numbers (the relative scaling of small and large here are bounded by the range of quantities encountered in natural circumstances) and any small changes in large numbers are not obvious to us. This may seem strange for a randomly chosen individual from a post-industrialized culture who has trivialized the sense of numbers to an arithmetic scale through a systematic effort for comprehending numbers and their changes in equal proportion as seen in the arithmetic scale.
Regardless of the nature of our number sense, these two dichotomous theories agree about the presence of an intriguing common denominator- the enthusiasm for numbers that humans have evinced throughout their history. In the pursuit to understand nature, humans attempted to explain the visible part of it in terms of indivisible constituents. This intellectual approach to reduce ordinary things to its irreducible components had heavily influenced the post bronze age civilizations across the world (interestingly the ancient cultures that did not possess phonetic alphabets had tried to interpret nature in a holistic way , for instance, the Chinese culture which had only ideograms explained nature through the philosophy of yin and yang that relied on the wave aspect rather than the particle approach). The intelligentsia of various civilizations had frequently proposed their version of the irreducible building block of nature. For a long time, this lower limit for the scale of things was not accompanied by the idea about the existence of an upper bound until the time of Anaximander. Except for the concept of temporal eternity, which had its root in the religious notion of immortality, the limits of the realistic expanse were not perused to explain the world. The gradual adaptation of the limitlessness from the temporal eternity to the material endlessness took place during the period of Anaximander. He was the first person to theorize about the constituents of the macroscopic world by blending irreducibility with endlessness. Deprived of proper empirical methods for validating such theories to interpret and understand nature rigorously, thinkers of that era were unable to find progress in systematically understanding the material world. But irreducibility and endlessness had found their way into the abstract realm of numbers.
In the domain of integers, the irreducible basic units are called prime numbers. For attributing the notion of endlessness to these building blocks, the need was there to prove their infinite availability. The proof for this came from the works of the legendary mathematician, Euclid of Alexandria. In his seminal work, titled “Elements”, he had proven that the Prime Numbers are infinite. The knowledge that any integer greater than one can be constructed with just prime numbers (later called as Fundamental theorem of Arithmetic) was also present in his work but lacked explicit proof. Despite arriving at the proof for the infinitude of prime numbers, mathematicians were not able to expand the contours of comprehension about the prime numbers for years, as there was no way of mathematically predicting the appearance of prime numbers between the composite integers.
The distribution of Prime numbers is somehow uncannily optimized for subsuming a high degree of randomness, that is present in the narrow ranges, within the orders of astounding regularity that is manifest at considerably longer ranges. Inside the prime distribution, the curious human brain conceives of an ideal balance between chaos and harmony forming a mysterious pattern that has an aesthetic appeal for probing them. Captivated by this perceived aesthetic appeal, generations of mathematicians strove to great extents for decoding the deep patterns that are present inside the prime number distribution. In one of the earliest known attempts, Eratosthenes from ancient Greece had found a method to identify the prime numbers present until a given integer. His method was a kind of brute force calculation that relied on the condition that for an integer n a prime factor p can be found such that p≤√n. With the improvement of our understanding of various areas of mathematics in the centuries that followed, mathematicians realized that the method for predicting the successive primes ought to be reframed from a problem of algorithmic computation into a problem for seeking solutions of a special category of functions. To recast the problem in such a way, a few specific types of functions are desirable. A function that gives a prime number for each integer input, a function whose values for a particular integer input include a set of prime numbers, or a function that gives prime numbers for every integer input.
In the mid-eighteenth century, while intensively working on the test of convergence of some infinite series, Leonhard Euler had discovered a function that was later called as zeta function. He found that the function is well behaved for the values greater than one and after examining the bizarre outcomes of the remaining values, he proved that the zeta function is valued only for values greater than one. In his further analysis of the zeta function, he was also able to establish a connection between the zeta function and the prime numbers (Euler product formula). Some decades later, Carl Friedrich Gauss conjectured relation between the distribution of prime numbers and a well-known function. This relation, referred to as prime number theorem, gave the inference that the prime numbers are asymptotically distributed among the positive integers. An asymptotic relation is true for a sufficiently large sample. In any limit approaching relations like this, the validity of such statements could be proved by reducing the error term associated with the original conjecture. It is here, the later works of Bernhard Riemann come for a rescue. Being a pioneer of complex analysis, he was able to reformulate the zeta functions for complex numbers. Euler had earlier proved that the real zeta function does not have any zeros for real values greater than one. The Riemann zeta functions give zeros for certain negative real inputs (trivial zeros) and a particular positive input that has the real value of ½ (non-trivial zeros). The Riemann hypothesis states that all the non-trivial zeros of the Riemann zeta functions lie on the line passing through real value ½. Armed with this complex-valued zeta function. Riemann was able to refine the prime number theorem whose error estimate depends on the non-trivial zeros of the Riemann zeta function. If the Riemann hypothesis is proved, it offers the best possible error bound for the prime number theorem.
More than any other abstract problem, the prime number distribution, and their underlying patterns are eluding the human attempts to comprehend for a very long period of time. Much progress has been made since the proof of Euclid, yet they still continue to escape the contours of our comprehension. It is no wonder that Leonhard Euler, one of the greatest mathematicians of all time, had once cautiously opined “Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate”. Despite that, the interest in prime numbers is not fading away and there is no sense of complacency with the contemporary number theorists in relentlessly confronting the mysteries of these abstract numbers in the hope that one day human brain can conquer those hidden patterns.
