Monthly Archives: February 2012

The Riemann hypothesis: Zeros of the zeta function in the critical strip

If we can’t yet say for sure that Re(s) = 1/2 for all s such that ζ(s) = 0, what can we say? Progress towards establishing the Riemann hypothesis could be viewed in terms of giving tighter limits on Re(s). … Continue reading

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The Riemann hypothesis: Error estimates for the prime number theorem

The Riemann hypothesis has been just about the most notorious unsolved problem in mathematics since Riemann’s work became widely known, so it’s been researched intensively from many angles. For example, various equivalent formulations have been developed. The proof of any … Continue reading

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The Riemann hypothesis: Proving the prime number theorem

As intriguing as it is to have an actual “explicit” formula for π(x), making use of the formula is another matter. The first term of the formula is Li(x), which is precisely what we want to have as the asymptotic … Continue reading

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The Riemann hypothesis: Counting prime numbers in an interval

Given what he had proven (or thought he had proven) about ζ(s), Riemann went on to give an “explicit” formula for π(x), the number of primes not greater than x. That is, not merely an asymptotic formula, but an exact … Continue reading

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