# Subtleties in the Distribution of Primes

• Hans Riesel
Chapter
Part of the Progress in Mathematics book series (PM, volume 57)

## Abstract

There are only very few proved results concerning the distribution of primes in short intervals. The prime number theorem tells us that the average density of primes around x is approximately 1/ln x. This means that if we consider an interval of length Δx about x and choose any integer t in this interval, then the probability of t being a prime will approach 1/ln x as x → ∞, if Δx is small compared to x. This implies that the primes tend to thin out as x grows larger; an implication that becomes obvious when considering that the condition for a randomly picked integer x to be composite is that it has some prime factor $$\le \sqrt x$$, and that there are more prime factors $$\le \sqrt x$$ to choose from when x is larger.

## Keywords

Number Series Residue Class Successive Prime White Ball Infinite Product
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