Riemann's Hypothesis Part 2 [Other Maths] [Riemann's Hypothesis Part 1] Riemann's Hypothesis Part 2 Riemann's approximation to pi(x) The Prime Number Theorem states that pi(x) ~ Li(x). This is a fair enough approximation, but it has fairly large error terms. The end result of Riemann's paper on pi(x) was to narrow this error term down significantly. Of course the validity of this result is pending the proof that all zeroes lie on the critical line. Although this is not the precise method Riemann used, it covers his general flow. We begin by defining a jump function S(x) as a function with an unusual technique for counting primes and prime powers. S(x) = +1 if x is prime +1/n if x is p n Riemann then determined a formula relating S(x) to pi(x): This result is obvious - S(x) is the sum of the primes to x, plus 1/2 the sum of the number of prime squares, which is identical to pi(x ), and so on. We require this equation with pi(x) as the main term, so we apply the Mobius Inversion formula: divides n = 1 if n is square-free with an even number of prime factors = -1 if n is square-free with an odd number of prime factors Making the exchange gives: The next step is to make a substitution for S(x). S(X) can be analytically defined (via a lengthy process) as: | |
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