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Overview of Riemann's Zeta Function and Prime Numbers

Date: 04/11/2006 at 13:52:38
From: Paul
Subject: Riemann's zeta function basic description

Can you please give an overview of the importance of Zeta function and
finding prime numbers?  Why is the Zeta function such a hot topic in
the field of looking for prime numbers?



Date: 04/11/2006 at 15:36:40
From: Doctor Vogler
Subject: Re: Riemann's zeta function basic description

Hi Paul,

Thanks for writing to Dr. Math.  Here's a real big-picture overview of
the importance of the Riemann Zeta Function.  For a description of the
function, refer to

  The Riemann Zeta Function
    http://mathforum.org/library/drmath/view/52831.html 

There is also some talk of it at

  Riemann Zeta Hypothesis
    http://mathforum.org/library/drmath/view/51929.html 

Now, the importance of the Riemann Zeta Function stems from the
ability of its zeros to predict the prime numbers.  In some sense, the
zeros of the Riemann Zeta Function are a kind of frequencies that
generate the prime numbers.  Their locations give us information about
the prime numbers.

To be more precise, there is a simple function called the Chebyshev
function, L(n), which equals

  L(p^k) = log p

when n is the power of a prime number, and L(n) = 0 when n is not the
power of a prime.  (I write L, but it is common to use the capital
Greek letter Lambda to denote this function.)  Then we have another
function called the Mangoldt function P(x) which is the sum of L(n)
for all numbers n with 2 <= n <= x, or the sum of k log p over all
primes p <= x, where k is the largest power with p^k <= x.  In some
sense, this function counts the primes less than x, although it is a
weighted sum.  But if you count small primes separately, and take a
handful of intermediate values of P(x), then you can use this function
to compute exactly the number of primes less than x.  Even when you
aren't as careful, though, it gives you a pretty good approximation to
the number of primes there are less than x.  It turns out that this
rather strange prime-counting function can be computed without
counting primes, by using the formula

  P(x) = x - (1/2)log(1 - x^-2) - log(2*pi) - sum (x^p)/p,

where the final sum is over all of the nontrivial zeros p of the
Riemann Zeta Function.  Now, computing precise values of the zeros of
the Riemann Zeta Function is no easy task, but when x is very large,
then it is easier to compute the first several zeros of the Riemann
Zeta Function, enough to compute P(x) pretty accurately, than it is to
count all of the primes up to x.  Furthermore, if we know some
information about the positions of the zeros p, then we can get
information about the number of primes there are less than x.  For
example, it has been proven that all of the nontrivial zeros have real
part (remember that these are complex numbers) that is strictly
between 0 and 1.  Using this fact and the above formula, you can prove
the Prime Number Theorem, which says that the number of primes less
than x is roughly

  x/log(x),

with Li(x) giving a somewhat more precise approximation.  See also

  Prime Number Theorem
    http://mathworld.wolfram.com/PrimeNumberTheorem.html 

If one were able to prove the Riemann Hypothesis, which is that all of
the nontrivial zeros have real part exactly equal to 1/2, then this
would imply that Li(x) is very close to the actual number of primes
less than x.

There are other questions about prime numbers (like how many twin
primes are there?) that can be reformulated into questions about the
zeros of the Riemann Zeta Function, which means that the more we know
about the Riemann Zeta Function, the more we learn about the prime
numbers.

If you have any questions about this, please write back, and I will
try to explain further.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
College Analysis
College Imaginary/Complex Numbers
College Number Theory

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