3 ms·
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: Fo
by tzs 3y ago
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] https://arxiv.org/abs/math/0008177 https://arxiv.org/abs/math/0008177
- zeroonetwothree 3y agoWhat a weird coincidence, I was just looking at this paper today for a different reason.
- renonce 3y agoCare to tell us the reason?
- ajkjk 3y agoI've always felt that the Riemann hypothesis' specific formulation, in terms of the zeroes of zeta on the critical strip, is a very annoying way of expressing it. Like it is clearly related to the prime numbers in a really intrinsic way and therefore the most "intrinsic" expression of the problem shouldn't have anything to do with zeta or complex numbers or roots at all... whatever that is, I wish it's what we treated as the big unsolved problem instead of what we got. I wonder what the most pleasing equivalent formulation of the RH would be. Gotta assume that some of them are much more directly about prime numbers. Yours seems pretty good. There are some others on here: https://mathoverflow.net/questions/39944/collection-of-equivalent-forms-of-riemann-hypothesis https://mathoverflow.net/questions/39944/collection-of-equiv... which seem good also.
- tristenharr 3y agoI’m bipolar and when I was first diagnosed with bipolar disorder I was 19 and in my first ever manic episode I dropped out of college and went deep down this particular rabbit-hole in an attempt to solve the Riemann Hypothesis to win the prize. I spent 3 months on it gradually driving myself crazier and crazier. My second company had collapsed out from under me when a sociopathic employee with a gambling addiction had conned me into thinking they were my best friend and meanwhile skimmed all of our profits so it looked like we were barely breaking even right under my nose. (To be fair I was 19) That in combination with a semester of Intro to Philosophy where I read Aristotle and Socrates and a Algebra class I became obsessed with led to me stumbling into the Riemann Hypothesis, and going deep down a rabbit hole in what ultimately turned into my first ever and most intense manic episode and led to my bipolar diagnosis. I hadn’t even finished college Algebra at the time. I basically spent months playing around with equations on Wolfram Alpha. The funny thing is this ultimately was kind of a life-changing moment for me, as I learned programming through it which in combination with a family member dying is what finally pulled me out of the mania, and then I changed my major to Computer Science re-enrolled in college and have since actually gone on to get a degree in CS and a minor in math, sometimes I look back at the giant set of Google Docs I created during that time, mostly in amusement but to this day some things I look at and don’t even understand if it is meaningless or not. A part of me wants to pick it back up, but then I tell myself it isn’t worth the risk, I’ve always felt I am uniquely qualified to understand how the guy in the movie “A beautiful mind” felt when he stopped talking/engaging with the hallucinations. Here’s some links to random Google docs from my ~2017 rabbit hole if anyone is interested. There’s a lot of fairly interesting usage of the Euler-Mascheroni constant and honestly it goes a bit all over the place. I kind of doubt there’s anything here, but if you’ve ever wanted to peek into unchecked mania.. have fun. It was months of insanity. (Sharing these in good faith!) https://docs.google.com/document/d/1CERdyZMlGj76EmMJXt5Zr3pMcBeTeI7uFmIp0aRmX2M/edit https://docs.google.com/document/d/1CERdyZMlGj76EmMJXt5Zr3pM... https://docs.google.com/document/d/1LpKqgCzjo8LrxB547WY5YdVPVsSm6mpw4RT6WGunhGs/edit https://docs.google.com/document/d/1LpKqgCzjo8LrxB547WY5YdVP... https://docs.google.com/document/d/1JvV5NJTYaTHn3eSAShcUtE8Jnn95Eof7raNtRfO87QQ/edit https://docs.google.com/document/d/1JvV5NJTYaTHn3eSAShcUtE8J... https://docs.google.com/document/d/1Ej0bNIpYNT30M31pCxx-LDoe5x3ZJ5OnwP7APcvBT64/edit https://docs.google.com/document/d/1Ej0bNIpYNT30M31pCxx-LDoe... https://docs.google.com/document/d/13nlw3vbxyRGd6pYeAre8c7e5OKmPjjksGaKL422pFfE/edit https://docs.google.com/document/d/13nlw3vbxyRGd6pYeAre8c7e5... https://docs.google.com/document/d/1WChBcGtaUdLckPDqNBVGpP9xaYqpP02GflFMyWbl_IQ/edit https://docs.google.com/document/d/1WChBcGtaUdLckPDqNBVGpP9x... https://docs.google.com/document/d/1_bQbgJ5HIyv_D5ex9TWK2XxH1RDBZx3tTIoZMJeX8zk/edit https://docs.google.com/document/d/1_bQbgJ5HIyv_D5ex9TWK2XxH... https://docs.google.com/document/d/1AomtTAP4GBXpOuArvBYQsHiPRfyAzWZGC-E37v5-Fhw/edit https://docs.google.com/document/d/1AomtTAP4GBXpOuArvBYQsHiP... https://docs.google.com/document/d/1QO8iQ2OmPZkQ7lNS0uUB0uyyWAf2NID85fqfTP9a7Fs/edit https://docs.google.com/document/d/1QO8iQ2OmPZkQ7lNS0uUB0uyy... https://docs.google.com/document/d/1zzedxz1Ya9rT8N4RZZNtpsrrNkLLPfEtOqZBqapNNR0/edit https://docs.google.com/document/d/1zzedxz1Ya9rT8N4RZZNtpsrr... https://docs.google.com/document/d/1vB0Qbqe0yiO0tcxswl2jniYcXjfzTewWEMfXXU0J8rw/edit https://docs.google.com/document/d/1vB0Qbqe0yiO0tcxswl2jniYc... https://docs.google.com/document/d/1K4upBz2awHnKlr5JE1xocntpQCuu7IWJW7iuevJTJXk/edit https://docs.google.com/document/d/1K4upBz2awHnKlr5JE1xocntp... https://docs.google.com/document/d/1XhOIy0RO59CrZEXXPK-QAR_smBhRPewq0LINUw6pAU0/edit https://docs.google.com/document/d/1XhOIy0RO59CrZEXXPK-QAR_s... https://docs.google.com/document/d/1rJx2KHQ0GqZfESqyFzXGLmN_5H0iRxIhJbLJLr9fj5M/edit https://docs.google.com/document/d/1rJx2KHQ0GqZfESqyFzXGLmN_... https://docs.google.com/document/d/19kZrg8HvxZ31-goiwL0XLyPCh3XPVCXaBDmW_67-fHw/edit https://docs.google.com/document/d/19kZrg8HvxZ31-goiwL0XLyPC... https://docs.google.com/document/d/1F8c_gVy47bRw5zZVWKb7x5SVR0vruNk2XbTnOH5vAz0/edit https://docs.google.com/document/d/1F8c_gVy47bRw5zZVWKb7x5SV... https://docs.google.com/document/d/16JolkQcHD3eviZuakfT7iam7zCot3QPoVkD-2jbg7Xs/edit https://docs.google.com/document/d/16JolkQcHD3eviZuakfT7iam7...
- btdmaster 3y agoVery similar: https://news.ycombinator.com/item?id=20387498 https://news.ycombinator.com/item?id=20387498 (The Riemann Hypothesis Says 5040 is the Last) The million dollar prize rules are surprisingly strict. They expect a "natural" proof, in that writing a one-page paper "my computer found counterexample 2^1729-1" may not get you the full prize, maybe instead a tiny acknowledgement prize, and they will wait for a proof of a generalization of the problem. https://www.claymath.org/wp-content/uploads/2022/03/millennium_prize_rules_0.pdf https://www.claymath.org/wp-content/uploads/2022/03/millenni...
- newman123 3y agoIs the log in that equation using a 10 or maybe the natural number e?
- tzs 3y agoMathematicians almost always use log for the natural logarithm (i.e., base e).