Category Archives: Riemann hypothesis

The Riemann hypothesis: Dedekind zeta functions

The prospects for generalizing the Riemann hypothesis don’t seem good. It isn’t at all clear what stronger conclusions might be true – since some which have been considered have turned out to be false. But then, everything discussed here so … Continue reading

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The Riemann hypothesis: Generalizations of the Riemann hypothesis

For any given mathematical statement, whether it is an established theorem or an unproven conjecture, there is almost always some way to make it “stronger”. One way to make a statement stronger is to make its conclusions more precise. For … Continue reading

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The Riemann hypothesis: The Lindelöf hypothesis

In spite of the strong numerical evidence in favor of the Riemann hypothesis, all attempts to prove it rigorously using techniques of classical analysis have fallen far short. For example, the Hadamard zero-free region actually excludes only a small part … Continue reading

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The Riemann hypothesis: Equivalents of the hypothesis

One reason that the Riemann hypothesis is so important to number theorists is that, as we’ve noted, it implies the smallest possible error estimate in the prime number theorem. In other words, it gives as much information as possible about … Continue reading

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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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