Timeline for New experiments involving Ramanujan primes: Benford's law
Current License: CC BY-SA 4.0
26 events
| when toggle format | what | by | license | comment | |
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| S Oct 20, 2021 at 14:22 | history | bounty ended | user142929 | ||
| S Oct 20, 2021 at 14:22 | history | notice removed | user142929 | ||
| S Oct 16, 2021 at 10:26 | history | bounty started | user142929 | ||
| S Oct 16, 2021 at 10:26 | history | notice added | user142929 | Reward existing answer | |
| Sep 24, 2021 at 17:01 | comment | added | user142929 | Many thanks @so-calledfriendDon , it seems that isn't at my (research) level, but sure that I understand some extracts of the article, and your reference will be very interesting for the professor here. | |
| Sep 23, 2021 at 15:10 | comment | added | so-called friend Don | The following recent preprint ("Dirichlet, Sierpinski, Benford") by Pollack and Singha Roy may be of interest: pollack.uga.edu/DSB.pdf It does not discuss Ramanujan primes but does discuss one sense (pointed out by Bombieri) in which primes obey Benford's law. | |
| Sep 23, 2021 at 14:06 | comment | added | user142929 | Many thanks @MichaelHardy for your edit. | |
| Sep 23, 2021 at 2:50 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Sep 14, 2021 at 13:26 | vote | accept | user142929 | ||
| Sep 2, 2021 at 8:57 | comment | added | user142929 | I add a comment to emhasize that I've read previous interesting comments, also the comment about Haar measure (my knowledges about this subject it are poor, but I thanks such good comments) | |
| Sep 1, 2021 at 8:10 | comment | added | user142929 | Many thanks to you @WillSawin and for the other professors in comments. I need some days to read and understand the comment thread of this post. Any case I refer that I've read/my motivation was the article by Bartolo Luque from the journal Investigación y Ciencia 514 (Julio 2019). About the variation of Wolf's experiment I add the reference if some user or you are interested in it, that is the mentioned article or pages 429-430 and exercise 8.33 from Crandrall and Pomerance Prime Numbers, Springer (2005) for an arithmetic function counting the number of Ramanujan primes over intervals | |
| Sep 1, 2021 at 3:23 | comment | added | Will Sawin | @GerryMyerson This is probably my fault for only being familiar with Benford's law in the context of real life phenomena (say, those approximately described by a power law), not the mathematical version for sequences like Fibonacci. I thought the $n^{-s}$ version was a natural abstraction of the real-life version to mathematical sequences of polynomial growth, say. | |
| Sep 1, 2021 at 3:13 | comment | added | Gerry Myerson | @Will, I've never seen Benford defined with any weights on the terms other than weight $1/n$ for each. People write that, say, the Fibonacci sequence follows Benford's Law; they don't feel obliged to add such a disclaimer as "if you weight each $n$ by $1$." | |
| Sep 1, 2021 at 2:45 | comment | added | Will Sawin | @GerryMyerson Sure, we could consider a probability distribution where each $n$ is weighted by $n^{-s}$ (times a constant multiple to make the total probability $1$), look at the frequency of each starting digit, and then take the limit of that frequency as $s \to 1$. | |
| Sep 1, 2021 at 2:43 | comment | added | Gerry Myerson | @Will, are there inequivalent ways to define Benford's Law? | |
| Sep 1, 2021 at 0:56 | comment | added | R W | Benford's law is essentially nothing but an expression for the invariant (Haar) measure on the multiplicative group of positive reals. Therefore, asking whether a certain distribution satisfies Benford's law is the same as asking whether this distribution is asymptotically invariant in the multiplicative scale. | |
| Sep 1, 2021 at 0:39 | answer | added | JoshuaZ | timeline score: 4 | |
| Sep 1, 2021 at 0:26 | comment | added | Will Sawin | @GerryMyerson Sure, but I find "No, this sequence doesn't satisfy Benford's law, because it has polynomial growth and no sequence with polynomial growth satisfies Benford's law" unsatisfying, as it's telling you much more about how you've chosen to define Benford's law than about your sequence. | |
| Aug 31, 2021 at 23:52 | comment | added | Gerry Myerson | It wouldn't be too hard for OP to include the definitions of "Ramanujan prime" and "Benford's Law" instead of making others users chase them offsite, would it? | |
| Aug 31, 2021 at 23:48 | comment | added | Gerry Myerson | @Will, I don't see the problem here. You look at the first $n$ terms of a sequence; you count how many begin with the digit one (say); you divide that by $n$; you see whether the quotient has a limit as $n\to\infty$. If it does, and if that limit is $\log2$, then the sequence satisfies Benford (for the digit "one"). For the sequence of natural numbers, the limit doesn't exist. | |
| Aug 31, 2021 at 21:22 | comment | added | Will Sawin | Well, here is a question for you: Is there a Benford's law for the first digits of natural numbers? Can you state it? How would you test it empirically? | |
| Aug 31, 2021 at 16:25 | comment | added | user142929 | Good afternoon @WillSawin , and many thanks for your comment, I ask if it does make sense a Benford's law for the sequence of first digit of Ramanujan primes. (On the other hand and unrelated to our concern, as you see in my introductory paragraph I evoke a different experiment, I didn't it, thus if you want to study the Fourier transform of such sequence as did Marek Wolf and is explained in the book of Crandall and Pomerance, feel free to study if in the sequence of Ramanujan primes there is noise) | |
| Aug 31, 2021 at 15:09 | comment | added | Will Sawin | I think the answer depends on what distribution over Ramanujan primes you choose. | |
| Aug 31, 2021 at 14:54 | history | edited | user142929 | CC BY-SA 4.0 |
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| Aug 31, 2021 at 14:49 | comment | added | user142929 | I've added the tag (st.statistics) instead of other tags as (analytic-number-theory), feel free to edit the tags as you consider, or improving the grammar of the body. | |
| Aug 31, 2021 at 14:48 | history | asked | user142929 | CC BY-SA 4.0 |