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Generalizations of RH
Recall again our starting point from Euler:
Why should the numerators all be one? One important way to alter the series is to replace the numerators with functions χ(n) called Dirichlet characters (these can be viewed as functions for which there exists a positive integer k with χ(n + k) = χ(n) for all n, and with χ(n) = 0 whenever gcd(n, k) > 1). The resulting infinite sum L(?,s) is a Dirichlet L-function. Once again we analytically continue the function to one that is meromophic on the entire complex plane. The extended Riemann Hypothesis is that for every Dirichlet character χ and the zeros L(χ,s) = 0 with 0 < Re(s) < 1, have real part 1/2. The distributions of the zeros of these L-functions are closely related to the number of primes in arithmetic progressions with a fixed difference k. Should the extended Riemann Hypothesis be proven, then Miller's test would provide an efficient primality proof for general numbers. See, for example, [BS96 8.5-6].
Another way to generalize Euler's sum is to leave the field of rational numbers, and replace the denominators with the norms of the non-zero ideals (special sets of elements) in a finite field extention of the rationals K (called a number field). The resulting sum is the Dedekind zeta-function of K and can again be analytically continued. These zeta functions also have a simple pole at zero and infinitely many zero in the critical region. The generalized Riemann Hypothesis is again that the zeros in the critical region all have real part 1/2. See, for example, [BS96 8.7].
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