I’ve read mathematics sites on this but I don’t understand it. First there is the Euler Zeta Function, and then there is the Riemann Zeta Function, 1) What’s the difference, and 2) the Riemann Zeta Function has non-trivial zeroes, what do zeroes mean, and 3) what about the trivial zeroes? Signed Always Curious About Higher Math But Seldom Understand It.
I’m not familiar with the Zeta Functions per se, but a function has a zero at some value x iff f(x) = 0 . Finding the places where a function is zero is often useful, and often not particularly easy (there are tables of the zeros of most named functions, though). As for trivial zeros, those are the ones which aren’t very useful, and/or aren’t hard to find. For instance, with many functions, f(0) = 0 , so 0 is a zero of the function, but it’s usually easy to see that that’s the case, and it’s not usually particularly useful to know, so the origin is a trivial zero.
Here’s a short article on zeta functions. It might have a helpful link at the end.
The Euler function is defined for complex s with real part greater than 1 by
[symbol]z/symbol = [symbol]S[/symbol] n[sup]-s[/sup]
In his only paper on number theory, published in 1860, Riemann proved that the definition can be extended to the whole complex plane ( except s = 1). Riemann’s function is equal to Euler’s function where they are both defined.
A zero of a function is a number x such that f(x) = 0.
Riemann proved that ( his extended) [symbol]z[/symbol] has zeros at -2, -4, -6,… These are the trivial zeros.
Call the part of the complex plane with real part between 0 and 1 inclusive the critical strip. The zeros in the critical strip are symmetrical about the line Re(s) = 1/2. Hadamard in 1893 proved that there are infinitely many zeros in the critical strip. Riemann in his 1860 paper made the
CONJECTURE ( The Riemann Hypothesis): All zeros of [symbol]z[/symbol] in the critical strip lie on the line Re(s) = 1/2.
This has been described as the most important unsolved problem in mathematics. It is one of the prize problems of the Clay Mathematics Institute, with $1,000,000 for the solution.
I read somewhere that it had been shown that 40% of the non-trivial zeroes of the Riemann zera function lie on the line Re(s) = 1/2. 40% of infinity doesn’t mean much to me, so what were they talking about?
Let N(T) denote the number of zeros with imaginary part between 0 and T. Let N[sub]0/sub denote the number of zeros with real part 1/2 and imaginary part between 0 and T. Then ( modulo some technicalities)
THEOREM ( Conrey, 1989): For large T,
N[sub]0/sub/N(T) > 0.4
More accurately, he considered quotients of the form
[N[sub]0/sub - N[sub]0/sub]/[N(T+H) - N(T)]
where H depends on T. Thus at any stage you are only considering finite numbers of zeros.
OK, I thought it might be something like that. Thanks so much.
The Riemann hypothesis is equivalent to a statement of purely elementary number theory. I may have it wrong, but it goes something like this. For a positive integer n, let E(n) count the number of square-free numbers (that is numbers not divisible by any square > 1) that have an even number of prime divisors and O(n) the number that have an odd number of prime divisors. Then the RC is equivalent to the assertions that |E(n) - O(n)| does not grow much faster than the square root of n. The precise statement is that the difference is, for all e > 0, less than n to the power 1/2 + e.
It is amazing to me that this purely number theoretic statement is equivalent to a statement about the zeroes of a complex analytic function.