arXiv:math/0008177math.NT
This paper shows the equivalence of the Riemann hypothesis to an sequence of elementary inequalities involving the harmonic numbers H_n, the sum of the reciprocals of the integers from 1 to n. It is a modification of a criterion due to Guy Robin.
Linked by 2 identifiable commenters in 9 comments across 8 threads, from Oct 2011 to Sep 2026.
Here's a fairly easily understood problem, which if you could solve it would make you famous in the mathematical world and win you a million dollar prize: For a positive integer n: Let H(n) = 1 + 1/2 + 1/3 + ... + 1/n Let D(n) = the sum of the positive integers that divide n.…
It's nice to see something other than e^(ᴨi)=-1. If I had to get a math tattoo, I think I'd go for lim n→∞ Q_n^(1/n) = e^(ᴨ^2/(12 log 2)). That comes from a theorem proved by Khinchin and Lévy. Khinchin proved that for almost all real numbers if you take the sequence of…
This cites a 2013 paper by Jeffry Lagarias. Lagarias also has a 2001 paper [1] with another elementary equivalent to the Riemann Hypothesis. Let s(n) be the sum of the positive integers that divide n. For example, s(6) = 1 + 2 + 3 + 6 = 12. Let H(n) = 1/1 + 1/2 + 1/3 + ... +…
There are some interesting statements that are equivalent to the Riemann Hypothesis. What "equivalent" means is that if the statement is true then RH must be true, and if RH is true then the statement must be true. Here's one I find particularly nice. Let s(n) = the sum of the…
Speaking of the harmonic numbers, there is an interesting conjecture. Let H_n be the n'th harmonic number. Let s(n) by the sum of the positive divisors of n. For example s(6) = 1 + 2 + 3 + 6. Then it is conjectured that: s(n) <= H_n + exp(H_n) log(H_n) with equality only in the…
Speaking 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…
Here's an open problem that is not hard to understand (high school math is sufficient), but whose solution would be one of the most important advances in mathematics in the last 100 years, easily: Let H_n = 1 + 1/2 + 1/3 + ... + 1/n So, H_1 = 1, H_2 = 1 + 1/2, and so on. Let…
I believe that hyperbovine was referring to the article linked by tzs that explains the relation. http://arxiv.org/pdf/math/0008177v2
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> Mathematicians know that you can make problems arbitrarily complex, and declaring problems with large prizes attached to them can lead to a lot of competition and drama. Yes, you can make problems arbitrarily complex. But the prize problems were chosen not just because the…
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