Timeline for New experiments involving Ramanujan primes: Benford's law
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 20, 2021 at 14:22 | history | bounty ended | user142929 | ||
| Sep 14, 2021 at 13:26 | vote | accept | user142929 | ||
| Sep 1, 2021 at 12:17 | comment | added | Wojowu | The fraction $R_d(x)/R(x)$ should oscillate, with minima near $d10^t$ and maximal near $(d+1)10^t$. I suspect density might exist and satisfy Benford's law in logarithmic density sense. | |
| Sep 1, 2021 at 8:02 | comment | added | user142929 | Many thanks for your answer, and your attention and the attention of yours colleagues (professors) of this post and my recently deleted post about mathematicians (now I was disconnected). I need some days to read and understand the comment thread of this post and your excellent answer. | |
| Sep 1, 2021 at 2:25 | history | edited | JoshuaZ | CC BY-SA 4.0 |
added 152 characters in body
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| Sep 1, 2021 at 2:24 | comment | added | JoshuaZ | @WillSawin You are absolutely correct. I convinced myself when I made that satisfying Benford's law implied that one was growing no more than $x^\alpha$ for any $\alpha$. I did not have a set in mind that satisfied it. You are correct to point out that the much stronger statement about what the set needs to satisfy is in fact true. | |
| Sep 1, 2021 at 2:15 | comment | added | Will Sawin | I don't understand the last comment. Let $S$ be a set of integers and let $f(n)$ be the number of such integers less than $n$. Then if $S$ satisfies Benford's law, we have $f( (k+1) \cdot 10^n)= (1+o(1)) f( k \cdot 10^n)$ for any $1 \leq k \leq 9$, since otherwise the frequency of numbers with leading digit $k$ would be different at the two points. Combining this for $k$ from $1$ to $n$, we get $f( 10^{n+1}) = (1+o (1)) f(10^n)$, i.e. $f(n)$ grows slower than $n^{\alpha}$ for any $\alpha$. So $\alpha<1$ or the lack of a good asymptotic is not really enough. | |
| Sep 1, 2021 at 0:45 | history | edited | JoshuaZ | CC BY-SA 4.0 |
Comment about what would be needed for a set of satisfy it.
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| Sep 1, 2021 at 0:39 | history | answered | JoshuaZ | CC BY-SA 4.0 |