An Elementary Problem Equivalent to the Riemann Hypothesis · HackerPapers discussions

An Elementary Problem Equivalent to the Riemann Hypothesis — mentioned by tzs

Sat, 12 Sep 2026 13:48:05 GMT

> 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 solutions appear likely to be very complex (the problem statements aren't necessarily inherently complex--there is a way to restate the Riemann hypothesis that a junior high school student could easily understand, which I'll give below). They were chosen because they were important problems that mathematicians really…

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An Elementary Problem Equivalent to the Riemann Hypothesis — mentioned by tzs

Fri, 16 Feb 2024 22:12:40 GMT

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. E.g., D(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. Prove or disprove that for any positive integer n > 1: D(n) < H(n) + exp(H(n)) log(H(n)) That easy to understand problem turns out to be equivalent to the Riemann hypothesis [1], which is one of the most famous and important unsolved problems in number theory. [1]…

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An Elementary Problem Equivalent to the Riemann Hypothesis — mentioned by tzs

Fri, 31 Jan 2020 19:12:54 GMT

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 convergents of their continued fraction expansion, {P_1/Q_1, P_2/Q_2, ...}, then the sequence {Q_1, Q_2^(1/2), Q_3^(1/3), ...} approaches a limit, which is the same limit for almost all real numbers. Then Lévy determined the value of that limit, which is now called either Lévy's constant or the Khinchin–Lévy constant. If not…

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An Elementary Problem Equivalent to the Riemann Hypothesis — mentioned by tzs

Tue, 09 Jul 2019 16:39:12 GMT

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 + ... + 1/n. The Riemann Hypothesis is true if and only if s(n) < H(n) + exp(H(n)) log(H(n)) for all n > 1. [1] https://arxiv.org/abs/math/0008177

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An Elementary Problem Equivalent to the Riemann Hypothesis — mentioned by tzs

Sat, 07 Jan 2017 20:27:28 GMT

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 divisors of n, for a positive integer n. For example s(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. Let H(n) = 1 + 1/2 + 1/3 + ... + 1/n. The RH is equivalent to the claim that for every integer n >= 1: s(n) <= H(n) + exp(H(n)) log(H(n)) This is due to Jeffry C. Lagarias. Here's his paper showing the equivalence…

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An Elementary Problem Equivalent to the Riemann Hypothesis — mentioned by tzs

Mon, 12 May 2014 02:10:14 GMT

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 case n = 1. What is interesting about this conjecture is that it is true if and only if the Riemann Hypotheses is true! [1] The Riemann Hypothesis is one of, if not the, most important unresolved issues in mathematics. [1] http://xxx.lanl.gov/abs/math.NT/0008177/

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An Elementary Problem Equivalent to the Riemann Hypothesis — mentioned by tzs

Wed, 20 Nov 2013 06:11:09 GMT

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 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…

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An Elementary Problem Equivalent to the Riemann Hypothesis — mentioned by archgoon

Mon, 17 Oct 2011 15:35:56 GMT

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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An Elementary Problem Equivalent to the Riemann Hypothesis — mentioned by tzs

Mon, 17 Oct 2011 11:11:01 GMT

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 divsum(n) = the sum of the divisors of n For example, divsum(10) = 1 + 2 + 5 + 10 = 18, and divsum(8) = 1 + 2 + 4 + 8 = 15. Here's the problem: Prove that for n > 1, divsum(n) < H_n + exp(H_n)*log(H_n) It turns out that this is equivalent to the Riemann hypothesis. Their equivalence is shown here…

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