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New Winners of a "Derbyshire Mug":
Ken Luckhurst & Luis Rodriguez
Read their Derbyshire Mug winning questions, and John's answers, below: |
| QUESTION from Ken Luckhurst |
| Q: It seems to me that one of the more interesting ideas in the book revolves around the Littlewood Violations. As you know this is the kind of thing that gets the attention of Mathematicians when someting as predictable as the differnce between Li(x) - Pi(x) all of a sudden reverses. I believe that a couple of famous mathematicians, I don't remenber who, (was it Littlewood himself ?) have stated that they have doubts about the truth of the RH as a consequence of the existence of Littlewood violations. Does this not seem like a logical place to start for a (dis) proof? Hope this is worth a mug. |
| JOHN'S ANSWER: |
| A: Thanks, Ken. It is indeed a thing that snags the attention. It got mine, and I asked several mathematicians if the Littlewood violations indicated a parallel -- i.e. unexpected exceptions at very high numbers. None of them seemed to think so. The reasons given for not thinking so are those I give at the top of p.357. Now, for sure, any violations of the RH must be way, way up high on the critical line. We know that because we've already explored the "lower" regions. There is certainly no RH violation up to about T = 30 billion. But the Littlewood violations are not seen as a real parallel by mathematicians, in spite of the mathematical connections between the two phenomena. Remember that there is no real match between "big primes" and "high regions of the critical line." In investigating really, really big primes, what counts is the **sum** of terms derived from **all** the zeta zeros. This is certainly a connection; but it is not really a direct connection between "big" and "high." That's how mathematicians see it. I think. |
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| QUESTION from Luis Rodriguez |
| Q: It is known the Littlewood's Theorem: "If and only if Lim. Sup.M(x) = O(x^1/2 + e) then R.H is true." Don't you think that this theorem makes R.H indecidable? Because that means that M(x) obeys the Khinchin's Law of the Iterated Logarithm (Feller pag. 192) and mathematical analysis cannot handle statistical situations. |
| JOHN'S ANSWER: |
A: Thank you for introducing me to Khinchin's Law, which I did not know. I can't see why it would matter here, though. The essence of those "statistical situations" that, as you correctly state, mathematics cannot handle, is that the next number to show up in them is **unpredictable**. Now, while a statistical approach to M(x) (and other number-theoretic issues) can be taken, and is interesting, M(x) is not unpredictable in the same way. It is perfectly well defined, by straightforward arithmetical rules, in a way that (say) Brownian motion is not. The fact that a graph of M(x) **looks** Brownian (and tempts us to statistical inquiries) conceals a fundamental and irreducible difference between the two things. So, no, I don't think the Khinchinian (?) behavior of M(x) tells us anything about the decidability of Littlewood's RH-equivalent. Statistics is fun; but this is fundamentally arithmetic, not true statistics.
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John Derbyshire is author of Prime Obsession, the best-selling book on Bernhard Riemann and the Riemann Hypothesis. In this space, John will answer questions you submit related to Riemann, or mathematics in general.
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