Riemann Hypothesis (thing)

On reading book titles such as The Music of the Primes and Prime Obsession, many people wonder: How is the Riemann Hypothesis connected with the primes? Statements of the Riemann Hypothesis such as this:

All non-trivial zeroes of the Riemann zeta function lie on the critical line Re(s)=1/2.

give no help. This writeup attempts to answer this question.

To begin with, we consider the Riemann zeta function, written ζ. Although this is quite complicated to describe, it is derived from an earlier function, the Euler zeta function, which is defined by:

   infinity
     ___
ζ(s)=\    -s = _1_ + _1_ + _1_ +_1_ + ...
     /__ n       s     s     s    s
     n=1        1     2     3    4

One notices that when s=1, ζ(s) is equal to the famed divergent harmonic series, 1 + 1/2 + 1/3 + 1/4 +.... When s<1, the series also diverges, and thus Riemann had to make some changes to adapt it to the complex and negative ranges.

But I digress. The reason this series is named after Euler is a special equality that Euler proved regarding it. He showed that:

   infinity                               
     ___                                    ___      
ζ(s)=\    -s = _1_ + _1_ + _1_ +_1_ + ... = | | __1__
     /__ n       s     s     s    s         | |    -s
     n=1        1     2     3    4           p  1-p

where p represents every prime, i. e.

        s       s       s       s        s
ζ(s)=__2__ * __3__ * __5__ * __7__ * __11__ * ...
      s       s       s       s        s
     2 - 1   3 - 1   5 - 1   7 - 1   11 - 1 

In other words, Riemann's function is based on a function expressible in terms of every prime.

Secondly, one of the most important number theory applications of the Riemann Hypothesis relates to Gauss's Prime Number Theorem. The Prime Number Theorem says that, as x increases, the number of primes less than x (a function denoted π(x)) is approximately equal to the integral with respect to n from 2 to x of 1/ln(n) (a function denoted Li(x)). However, the error term is sufficiently large. Riemann proposed an improved error term equivalent to the sum of:

            1/n
μ(n) *  Li(x   )
       __________
           n

for all integers n greater than 2. This resulted in around 80% improvements on Gauss's function, but only worked if the Riemann Hypothesis were true.² (This is because one of the calculations involved the sum of a function over all roots of the Riemann zeta function.)

Thus, if the Riemann Hypothesis were proven true, it means that we could possibly calculate Gauss's function -- and thus the values of large primes. This could mean a new generation in, for example, encoding of e-mail and suchlike. Not that the error term isn't accepted already, but we are still not sure of its truth.

¹The Mobius function, μ(x), is equal to 0 when x has a square factor, 1 when x is square-free and has an even number of prime factors, and -1 when x is square-free and has an odd number of prime factors.

²For those interested in the Prime Number Theorem, the Riemann Hypothesis was proved equivalent to:

         1/2            1/2
π(x)= O(x   ln(x)) = a(x   ln(x))

for some constant a.