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### V.26 The Prime Number Theorem and the Riemann Hypothesis

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How many prime numbers are there between 1 and *n*? A natural first reaction to this question is to define π(*n*) to be the number of prime numbers between 1 and *n* and to search for a formula for π(*n*). However, the primes do not have any obvious pattern to them and it has become clear that no such formula exists (unless one counts highly artificial formulas that do not actually help one to calculate π(*n*)).

The standard reaction of mathematicians to this kind of situation is to look instead for good *estimates*. In other words, we try to find a simply defined function *f*(*n*) for which we can prove that *f*(*n*) is always a good approximation to π(*n*). The modern form of the prime number theorem was first conjectured by GAUSS [VI.26] (though a closely related conjecture had been made by LEGENDRE [VI.24] a few years earlier). He looked at the numerical evidence, which suggested to him that the “density” of primes near *n* was about 1/log *n*, in the sense that a randomly chosen integer near *n* would have a probability of roughly 1/log *n* of being a prime. This leads to the conjectured approximation of *n*/log *n* for π(*n*), or to the slightly more sophisticated approximation