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My understanding of the Riemann Hypothesis is that it can be reformulated as a statement about prime density. If this result is a concrete outer bound on densit
by whatgoodisaroad 13y ago
My understanding of the Riemann Hypothesis is that it can be reformulated as a statement about prime density. If this result is a concrete outer bound on density, does it have implications on RH?
- fspeech 13y agoThat may be where Zhang is heading.
- tzs 13y agoSpeaking of reformulations of the Riemann Hypothesis, there is a surprisingly elementary statement that is equivalent to the RH. Let H(1) = 1. Let H(2) = 1 + 1/2. Let H(3) = 1 + 1/2 + 1/3. ...and so on. These are called the harmonic numbers. Let S(1) = 1. Let S(2) = 1 + 2. Let S(3) = 1 + 3. Let S(4) = 1 + 2 + 4. Let S(5) = 1 + 5. Let S(6) = 1 + 2 + 3 + 6. The pattern here is harder to spot than the pattern for the harmonic numbers. S(n) for a positive integer n is the sum of the divisors of n. For instance, 6 is divisible be 1, 2, 3, and 6, so S(6) = 1 + 2 + 3 + 6. Conjecture: for any positive integer n, S(n) <= H(n) + exp(H(n)) log(H(n)), with equality only when n = 1. Believe it or not, this conjecture is equivalent to the Riemann Hypothesis! Proof here: http://xxx.lanl.gov/abs/math.NT/0008177/ http://xxx.lanl.gov/abs/math.NT/0008177/
- whatgoodisaroad 13y agoThat is surprising. It's pretty incredible to see a formulation that doesn't involve any advanced Number Theory.
- est 13y agoZhang revealed during an interview that he's already made "astonishing" progress on RH. http://www.changhai.org/community/article_load.php?aid=1383356678 http://www.changhai.org/community/article_load.php?aid=13833... > 這個黎曼猜想，我手上也有一些東西，如果拿出來，也會很轟動，我就是這種習慣，如果沒有完全結果，或者到最後，我覺得我不可能再做了，我也許會把它拿出來，但我現在是不想拿出來的。 He won't publish it until he finalized it.