Distribution of primes and growth rate of $\sigma(n)/n$ over colossally abundant numbers

Question

I am reading this paper written by Jeffrey C. Lagarias "An Elementary Problem Equivalent to the Riemann Hypothesis".

On page 5, it says "Fluctuations in the distribution of primes will be reflected in fluctuations in the growth rate of $\sigma(n)/n$ taken over the set of colossally abundant numbers." It is kind of soft, but I don't follow this sentence.

I understand all before this sentence. After this sentence, the author uses distribution of primes to study $\sigma(n)/n$. So I am wondering how this sentence follows from the context, but any explanation is welcome.

Answer 1

Anyway, the program(s) I am describing use:

$$ \delta = \frac{\log(p^{k+1} - 1) - \log(p^{k+1} - p)}{\log p} $$ given some $\delta > 0,$ the correct exponent for some prime $p$ is $$ \left\lfloor \frac{\log (p^{1 + \delta} - 1) - \log(p^\delta - 1)}{\log p} \right\rfloor \; - \; 1. $$

Take some bound $B$ on any prime power that will be used. For each prime $p$ and exponent $k \geq 1$ with $p^k < B,$ calculate $ \delta = \frac{\log(p^{k+1} - 1) - \log(p^{k+1} - p)}{\log p}.$ Make a list/file, each line has three numbers $ \delta \; \; p \; \; k \; \; .$ The printing out of $\delta$ must be done carefully... Part two, sort that file by decreasing $\delta.$ Step three, read that file, for each line print out the "bumped" prime $p,$ then the prime factorization of that CA number, and the CA number itself if it is not too large.

This method says each CA number is a prime times its predecessor. That holds true as high as anyone has computed, and follows if some conjectures of Siegel are true.

Maybe I can do this again in a couple of hours.

Alright, explicitly bounding the primes was clumsy. Here, instead, I enforce a lower bound $\delta > 0.01$

jagy@gost:~/Desktop/Cplusplus$ ./mse
     0.584963     prime  2  exponent  1
     0.222392     prime  2  exponent  2
     0.0995357     prime  2  exponent  3
     0.0473057     prime  2  exponent  4
     0.0230836     prime  2  exponent  5
     0.0114048     prime  2  exponent  6
     0.26186     prime  3  exponent  1
     0.072858     prime  3  exponent  2
     0.0230453     prime  3  exponent  3
     0.113283     prime  5  exponent  1
     0.0203735     prime  5  exponent  2
     0.0686216     prime  7  exponent  1
     0.0362866     prime  11  exponent  1
     0.0288926     prime  13  exponent  1
     0.0201744     prime  17  exponent  1
     0.0174204     prime  19  exponent  1
     0.0135735     prime  23  exponent  1
     0.0100679     prime  29  exponent  1
jagy@gost:~/Desktop/Cplusplus$ ./mse  | sort -n -r
     0.584963     prime  2  exponent  1
     0.26186     prime  3  exponent  1
     0.222392     prime  2  exponent  2
     0.113283     prime  5  exponent  1
     0.0995357     prime  2  exponent  3
     0.072858     prime  3  exponent  2
     0.0686216     prime  7  exponent  1
     0.0473057     prime  2  exponent  4
     0.0362866     prime  11  exponent  1
     0.0288926     prime  13  exponent  1
     0.0230836     prime  2  exponent  5
     0.0230453     prime  3  exponent  3
     0.0203735     prime  5  exponent  2
     0.0201744     prime  17  exponent  1
     0.0174204     prime  19  exponent  1
     0.0135735     prime  23  exponent  1
     0.0114048     prime  2  exponent  6
     0.0100679     prime  29  exponent  1
jagy@gost:~/Desktop/Cplusplus$ 

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

Next morning, self-contained program. Compare OEIS

1  2   2= 2
2  3   6= 2 3
3  2   12= 2^2 3
4  5   60= 2^2 3 5
5  2   120= 2^3 3 5
6  3   360= 2^3 3^2 5
7  7   2520= 2^3 3^2 5 7
8  2   5040= 2^4 3^2 5 7
9  11   55440= 2^4 3^2 5 7 11
10  13   720720= 2^4 3^2 5 7 11 13
11  2   1441440= 2^5 3^2 5 7 11 13
12  3   4324320= 2^5 3^3 5 7 11 13
13  5   21621600= 2^5 3^3 5^2 7 11 13
14  17   367567200= 2^5 3^3 5^2 7 11 13 17
15  19   6983776800= 2^5 3^3 5^2 7 11 13 17 19
16  23   160626866400= 2^5 3^3 5^2 7 11 13 17 19 23
17  2   321253732800= 2^6 3^3 5^2 7 11 13 17 19 23
18  29   9316358251200= 2^6 3^3 5^2 7 11 13 17 19 23 29
19  31   288807105787200= 2^6 3^3 5^2 7 11 13 17 19 23 29 31
20  7   2021649740510400= 2^6 3^3 5^2 7^2 11 13 17 19 23 29 31
21  3   6064949221531200= 2^6 3^4 5^2 7^2 11 13 17 19 23 29 31
22  37   224403121196654400= 2^6 3^4 5^2 7^2 11 13 17 19 23 29 31 37
23  41   9200527969062830400= 2^6 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41
24  43   395622702669701707200= 2^6 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 43
25  2   791245405339403414400= 2^7 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 43
26  47   37188534050951960476800= 2^7 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47
27  53   1970992304700453905270400= 2^7 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47 53
28  59   116288545977326780410953600= 2^7 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59
29  5   581442729886633902054768000= 2^7 3^4 5^3 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59
30  61   35468006523084668025340848000= 2^7 3^4 5^3 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59 61
31  67   2376356437046672757697836816000= 2^7 3^4 5^3 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67
32  71   168721307030313765796546413936000= 2^7 3^4 5^3 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
33  73   12316655413212904903147888217328000= 2^7 3^4 5^3 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73
34  11   135483209545341953934626770390608000= 2^7 3^4 5^3 7^2 11^2 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$