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/
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