Is there any Regularity in the Distribution of Prime Numbers at the Beginning of the Sequence of Positive Integers?

Abstract

Prime numbers are the multiplicative building bricks of the number system. According to the fundamental theorem of arithmetic, every integer number larger than 1 is either a prime or the product of a unique set of primes. In what follows, by an integer we will understand a positive integer. In multiplicative number theory each integer is a word, more exactly a commutative juxtaposition of primes. In this coding process each prime is employed according to a rigid rule (the gap between the consecutive multiples of a prime p is just p) and the set of prime numbers is like an alphabet that is self-generating in order to make the resulting code nondegenerate. But how are the prime numbers themselves generated? Contemplating successive gaps between consecutive primes or the number of prime factors of consecutive integers, we can only notice an apparently chaotic behavior of the prime numbers leading us to believe that their distribution law must be very complicated. There are two different ways of looking at prime numbers: globally and algorithmically. From an algorithmic point of view the process of generating prime numbers is relatively clear. The prime-number sieve, attributed to the ancient Greek scholar Eratosthenes, was one of the first step-by-step methods invented for distinguishing primes from composites among the numbers up to some predetermined limit: Take the number 2, eliminate its multiples; the next prime is 3, eliminate its multiples; the next prime is 5, eliminate its multiples, etc. Today, checking whether or not an integer is a prime is one of the first computer programs learned in. any programming language. Eratosthenes' sieve simply tells us what to do, step-by-step, for selecting the primes in a given set of consecutive integers without revealing any regularity in the distribution of primes. Those unhappy with an algorithmic approach have tried several ways to approach a global understanding of the behavior of primes. Many papers have dealt with the asymptotic behavior of different functions depending on primes. There is a rich literature on the subject (see for instance [15], [17], [3], [2]) using very subtle mathematical techniques. To give only one example, let 7T(x) denote the number of primes not exceeding the positive real number x. According to the celebrated prime number theorem (PNT), we have 7T(x) = x/ln x, (x -> oo), which means that the ratio of the two functions, namely wr(x)/(x/ln x), converges to 1 as x grows without bound, proved independently by J. Hadamard [9] and C. J. de La Vallee Poussin [14] using tools involving functions of complex variables. PNT is a superb example of extracting asymptotic order from chaos.