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Computations on 8000 colossally abundant numbers out to exp(exp(29.710462))

Variation of log(delta(n)) from Best-fit linear function of log(log(n))

Variation of log(delta(n)) from Best-fit linear function of log(log(n))

Reminder:  For a colossally abundant number n, with n > 5040, one can define

delta(n) := exp(gamma)*log(log(n)) – sigma(n)/n               [Briggs, def. 4-2, page 254 ] .

This is as in the paper by Keith Briggs,

Briggs, K.,  2006, “Abundant Numbers and the Riemann Hypothesis“.

sigma(.) is the sum of divisors function, sigma(n)/n is the same as the sum of the inverses of all divisors of n [easy proof], also known as the “abundancy index of n”;  gamma is the Euler-Mascheroni constant 0.57721 …

The Theorem of Guy Robin is that the Riemann Hypothesis is equivalent to :

exp(gamma)*log(log(n)) – sigma(n)/n > 0 , for all  n > 5040. Also see Grönwall’s theorem on the asymptotic growth rate of the sigma function, at Wikipedia:

https://en.wikipedia.org/wiki/Divisor_function#Approximate_growth_rate  .

We can plot how delta(n) changes with n. Here, one continues the study of the variation of log(delta(n)) from a linear (affine) Best-fit function of log(log(n)), as in: in Figure 3 on page 255 of the 2006 paper by Keith Briggs.

The Best-fit line from his paper can be thought of as

y ~=   f(x) = 0.323336  – x/2  .

In ‘x’, we would have log(log(n)), and in ‘y’ we would have f( log(log(n) ) , i.e.:

0.323336 – log(log(n))/2 .

In other words,

log(delta(n)) is very very closely approximated by 0.323336 – log(log(n))/2 .

Therefore, one plots the deviation,

log(delta(n)) – [ 0.323336 – log(log(n))/2 ] .

Negative values of :   log(delta(n)) – [ 0.323336 – log(log(n))/2 ]

correspond to more negative values of log(delta(n)), i.e. when the ‘gap’ delta(n) is “unusually small or tight”. Nevertheless, compared to values of – log(log(n))/2 , or

0.323336 – log(log(n))/2 , the oscillations are tiny, Briggs Figure 3 (page 255), Top.

For 8000 CA numbers, we show the deviation from the same Best-fit line, i.e.

Y =  log(delta(n)) – [ 0.323336 – log(log(n))/2 ] , with  X = log(log(n)),

for CA (colossally abundant)  numbers n with   log(log(n)) up to about 29.710462   , or in other words, the largest CA numbers ‘n’ considered are “close to”   exp(exp(29.710462)) .

David Bernier

Written by meditationatae

June 27, 2015 at 3:19 pm

Posted in History

Memo on extending the CA number calculations out to exp(exp(29))

For the record, the source code, compiled code, table of the first 1,000,000 primes and files used to store intermediate results are held in four directories:

[david2@localhost ~]$ pwd
/home/david2

(above is my home directory on my computer, /home/david2/

About below:

Below, the command `ls -l eratosthenes1[5678]’ has the system list all files and directories in

/home/david2/eratosthenes15 , /home/david2/eratosthenes16, /home/david2/eratosthenes17 and /home/david2/eratosthenes18 .
[david2@localhost ~]$ ls -l eratosthenes1[5678]
eratosthenes15:
total 1004
-rwxrwxr-x. 1 david2 david2  69496 Jun  8 01:21 a.out
-rwxrw-r–. 1 david2 david2    721 Jun  8 00:18 Loop6.sh
drwxrwxr-x. 2 david2 david2   4096 Jun  8 00:20 OLD
-rw-rw-r–. 1 david2 david2  19828 Jun  8 00:13 primes-gen-atkin.c
-rw-rw-r–. 1 david2 david2 458565 Jun 12 15:35 scriptLoop5Out.txt
-rw-rw-r–. 1 david2 david2     97 Jun  8 01:24 scriptLoop6Out.txt
-rw-rw-r–. 1 david2 david2 458662 Jun 12 17:28 test_data_4k.txt
-rw-rw-r–. 1 david2 david2   3225 Jun  8 00:12 time-make-primes6Bscript20a.c

[   primes-gen-atkin.c is the Atkins sieve source code of Dan Bernstein.

time-make-primes6Bscript20a.c  drives the prime sieving, by calling primes-gen-atkin.c .

The use of time-make-primes6Bscript20a.c is in adding logs of primes within intervals of length 10^9 [there will be 4000 of these out to 4×10^12 ],

secondly computing Pi_{p in I}  (1 + 1/p),

where I is an interval of 10^9 consecutive integers ending at k*10^9 for k = 1 (respectively, k =2, 3, 4, 5, …. 4000)’

thirdly, for each k with 1<=k<=4000, we keep track of the largest prime not exceeding k*10^9. ]

[ the compiled program in in a.out above, and the script  Loop6.sh is a bash shell script that invokes a.out with the two parameters needed for each of the 4000 intervals of 10^9 numbers, starting at 1-10^9 and ending at 3999*10^9 to 4000*10^9 ]

[ the output is sent to files: scriptLoop6Out.txt and scriptLoop5Out.txt ; only 1 file is needed. An oversight in editing a previous script had me write to 2 files. The entire output is then in the file:

test_data_4k.txt . ]

eratosthenes16:
total 8980
-rw-rw-r–. 1 david2 david2    1861 Jun 12 17:40 newreadtest58e.c
-rwxrwxr-x. 1 david2 david2  106797 Jun 12 17:44 newreadtest58e.out
-rw-rw-r–. 1 david2 david2  361782 Jun 13 10:44 newreadtest58e.txt
drwxrwxr-x. 2 david2 david2    4096 Jun 12 17:41 old
-rw-rw-r–. 1 david2 david2 8245905 Jun 12 17:26 primes1meg02a_tmp.txt
-rw-rw-r–. 1 david2 david2  458662 Jun 12 17:28 test_data_4k.txt

[ In eratosthenes16, we start from the output test_data_4k.txt from the jobs in eratosthenes15. The file primes1meg02a_tmp.txt contains the “small primes”, the first 1 million.

newreadtest58e.c is used to compute the critical epsilons in the Erdos-Alaoglu formulas relating to CA numbers, epsilon depending on the largest prime not exceeding 10^9 (respectively, k*10^9 for k=2, 3, … 4000). From epsilon_p , we compute the exponents (of 2, 3, 5, …) in the desired CA numbers for the first 10^6 primes. For the p and epsilon_p under consideration, the exponents of small primes such as 2, 3, 5 will exceed 1; but pas the million-th prime, all exponents will be 1. In test_data_4k.txt , we have sums of logs of primes within 4000 intervals I of length 10^9 . Thus, we are tweaking the sums of logs of primes to account for exponents above 1 in the “small primes”, meaning the first million primes .

The output in newreadtest58e.txt includes log(n), for 4000 CA numbers n.

cat /home/david2/eratosthenes16/newreadtest58e.c
#include <stdio.h>
#include <quadmath.h>

int smallp[1000000];
int main(void)
{
int j;
long low, high, count, lastprime;
long lstprimes[4000];
__float128 datalog[4000];
char buf1[100];
char buf2[100];
__float128 r;
__float128 epsilon;
__float128 var01;
__float128 var02;
__float128 var03;
__float128 var04;
__float128 var05;
int power2;
long p;
long smallprime;
__float128 fsmallprime;
__float128 deltasumlog;
__float128 sumlog;
int k;
FILE *in;
FILE *in2;

in = fopen(“/home/david2/eratosthenes16/test_data_4k.txt”, “r”);
in2= fopen(“/home/david2/eratosthenes16/primes1meg02a_tmp.txt”, “r”);
for(j=0; j<4000; j++)
{
fscanf(in, “%ld”, &low);
fscanf(in, “%ld”, &high);
fscanf(in, “%ld”, &count);
fscanf(in, “%s”, &buf1);
fscanf(in, “%s”, &buf2);
r = strtoflt128(buf2, NULL);
if(j == 0)
{
datalog[j] = r;
}
else
{
datalog[j] = datalog[j-1] + r;
}

fscanf(in, “%ld”, &lastprime);
lstprimes[j] = lastprime;
}

fclose(in);
for(j=0; j<1000000; j++)
{
fscanf(in2, “%d”, &smallp[j]);
}

fclose(in2);

for(j=0; j<4000; j++)
{
p = lstprimes[j];
epsilon = logq(1.0Q + 1.0Q/((__float128) p))/logq( ((__float128) p) );
printf(“%ld “, p);
printf(“%.30Qe “, epsilon);
deltasumlog = (__float128) 0;

for(k=0; k<1000000;k++)
{
smallprime = smallp[k];

fsmallprime = (__float128) smallprime;

var01 = expq((1.0Q+epsilon)*logq(fsmallprime))-1.0Q;
var02 = expq(epsilon*logq(fsmallprime))-1.0Q;
var03 = logq(var01/var02);
var04 = var03/logq(fsmallprime);
var05 = floorq(var04) – 1.0Q;
deltasumlog = deltasumlog + (var05 – 1.0Q)*logq(fsmallprime);
}

sumlog = datalog[j] + deltasumlog;

printf(“%.25Qf\n”, sumlog);
}

return 0;
}

end section on eratosthenes16  ]]]]

eratosthenes17:
total 8956
-rw-rw-r–. 1 david2 david2    2163 Jun 14 01:55 newreadtest68a.c
-rwxrwxr-x. 1 david2 david2  109929 Jun 14 01:58 newreadtest68a.out
-rw-rw-r–. 1 david2 david2  338889 Jun 14 17:21 newreadtest68a.txt
drwxrwxr-x. 2 david2 david2    4096 Jun 14 01:55 old
-rw-rw-r–. 1 david2 david2 8245905 Jun 14 01:51 primes1meg02a_tmp.txt
-rw-rw-r–. 1 david2 david2  458662 Jun 14 01:52 test_data_4k.txt

In eratosthenes17 , using newreadtest68a.c as source code, we compute sigma(n)/n for 4000 CA numbers n. For this, we use products of values of (1+ 1/p) for primes p in an interval I of 10^9 numbers, from 1-10^9 and ending at 3999*10^9 — 4000*10^9.

Once again, Erdos-Alaoglu formulas for epsilon_p are used to compute the exponents of 2, 3, 5 … p_1000000 and, e.g. sigma(2^44)/(2^44) in case 2 has exponent 44 in the CA number.

The output, sigma(n)/n for 4000 CA numbers, is in the file:

newreadtest68a.txt  ]]]   //end descr. eratosthenes17 dir

eratosthenes18:
total 1272
-rw-rw-r–. 1 david2 david2  91782 Jun 14 21:39 newreadtest28bb.txt
-rw-rw-r–. 1 david2 david2   1307 Jun 14 21:37 newreadtest48a.c
-rw-rw-r–. 1 david2 david2   1337 Jun 15 03:19 newreadtest50ay.c
-rw-rw-r–. 1 david2 david2 361782 Jun 14 21:44 newreadtest58e.txt
-rw-rw-r–. 1 david2 david2   1291 Jun 14 21:54 newreadtest60a.c
-rwxrwxr-x. 1 david2 david2  72431 Jun 14 21:56 newreadtest60a.out
-rw-rw-r–. 1 david2 david2 193598 Jun 14 21:57 newreadtest60a.txt
-rw-rw-r–. 1 david2 david2 338889 Jun 14 20:05 newreadtest68a.txt
-rw-rw-r–. 1 david2 david2    447 Jun 15 03:43 newreadtest80ax.c
-rwxrwxr-x. 1 david2 david2   7723 Jun 15 03:44 newreadtest80ax.out
-rw-rw-r–. 1 david2 david2  96000 Jun 15 03:51 newreadtest80ax.txt
-rw-rw-r–. 1 david2 david2    446 Jun 15 04:06 newreadtest80ay.c
-rwxrwxr-x. 1 david2 david2   7723 Jun 15 04:08 newreadtest80ay.out
-rw-rw-r–. 1 david2 david2  93598 Jun 15 04:11 newreadtest80ay.txt
drwxrwxr-x. 2 david2 david2   4096 Jun 16 11:45 NEXT

[ In eratosthenes18 , we use the data from the files in eratosthenes17 and eratosthenes16 to put together  sigma(n)/n and log(n) for 4000 CA numbers out to exp(exp(29)), and do computations on the gap between  sigma(n)/n and the upper bound exp(gamma)*log(log(n)) , or Robin criterion, accrding to which if n>5040 abd the Riemann Hyposthesis holds, then     sigma(n)/n <  exp(gamma)*log(log(n)) . ]]] end for dir eratosthenes18

Written by meditationatae

June 16, 2015 at 6:27 pm

Posted in History

Computations on 4000 colossally abundant numbers out to exp(exp(29.017316))

XYrobin4000_Graph

Written by meditationatae

June 16, 2015 at 3:50 pm

Posted in History

About the graph in the previous post

Please compare preceding graph with that in Figure 3 (Bottom) from:
Keith Briggs, “Abundant Numbers and the Riemann Hypothesis”,
Experiment. Math., Volume 15, Issue 2 (2006), 251-256.

Written by meditationatae

June 4, 2015 at 9:36 am

Posted in History

Deviation of log(δ) from a best-fit line at CA numbers

june4

Written by meditationatae

June 4, 2015 at 9:24 am

Posted in History

Output of newreadtest48a.out (compiled code of newreadtest48a.c)

Please compare Briggs [2006] ,  “FIGURE 3″ , bottom,

“Deviation of log(δ) from a best-fit line at CA numbers”

x = log(log(n)), y = deviation, n = a CA number (below).

X  ,  Y

20.72328036823640753487 0.01928812277933495504
21.41641568589025523946 0.00467503027989789420
21.82188330500718220766 -0.01857803449402955915
22.10956936787925566329 -0.01442664754704527536
22.33270529973400392535 -0.00498884443497253787
22.51503213252200525579 0.01088285328216521960
22.66918125498666494740 -0.01429344711957802149
22.80270969051157491845 0.01583390785426878241
22.92049377575773232772 0.00913729536799508871
23.02585918471295815829 -0.00556723123961362092
23.12116444840649185777 -0.01428057706984939257
23.20817449921066280511 0.00370437172240109774
23.28821967654175891721 0.00695132946738601145
23.36232606161474320291 0.00076871960116718725
23.43131916151075415000 0.01274392582876600430
23.49585883297631387013 -0.00335811620823994591
23.55648461116446634664 -0.01401918865700222004
23.61364068083174946700 -0.01980720813223231345
23.66770435981997525392 0.00066812371262036426
23.71899926506496231256 0.01342889380622961036
23.76779089165435028764 0.02332671413910199470
23.81431267672866812923 0.01333914889179610769
23.85876349324037205987 0.00069070384323750305
23.90132424430857878716 -0.00571173508055897435
23.94214458846479657810 -0.01530621147089548173
23.98136301244608917547 -0.00779229482294242842
24.01910460860912145274 -0.00050612455614651609
24.05547313559778042218 -0.00584971396450571242
24.09056458535465025866 -0.00402813259050826630
24.12446382944861821744 0.00184978733575586626
24.15725477500790481481 0.00500468747871845172
24.18900406039539414815 0.00855341561944828078
24.21977677530109895093 0.00710880543130345228
24.24962700368683499273 0.00897526802044472934
24.27861616735687489168 0.01721758292400090709
24.30678871078111296572 0.01728502111745177686
24.33418689203431220228 0.00505581573607876893
24.36085581098907651584 -0.00317651273507625803
24.38683052471922202669 -0.01389580224875100356
24.41214760533638652360 -0.01359983731339108116
24.43684038089749325379 -0.01328490025048106883
24.46093739759440910261 -0.00808735521559941535
24.48446654859477312389 -0.01097859266271281488
24.50745712898584062773 -0.00423585382086897024
24.52993042990824717979 -0.00281351927743313602
24.55191009958961378338 -0.00676672611438055835
24.57341577742556902734 -0.00678176290863876528
24.59446824792602871777 -0.00232793939168685355
24.61508687062889591396 0.00724076768078046394
24.63528975535010829668 0.01459138652831469832
24.65509243522055246399 0.02343962919465075369
24.67451220989210557046 0.02403083869636613927
24.69355914607862320009 0.02476510602326945229
24.71225248429892474537 0.01936959724812572215
24.73060259294691178511 0.01422157735037443301
24.74862050915757732441 0.00646267891978183932
24.76632023644343575493 -0.00119625936111822428
24.78371123240964922626 -0.00783290074076015798
24.80080469222575060657 -0.00982264347274422774
24.81761240601927998110 -0.01062772213634976377
24.83414207564879229287 -0.01501281681593070363
24.85040179984159655498 -0.01559816006397293661
24.86640234745133579072 -0.01414916253087120984
24.88215021604272055895 -0.01308157063814694307
24.89765409584391574821 -0.01176913742088577322
24.91292122441778779685 -0.01004615228596281725
24.92795875882910042239 -0.00626321636077151271
24.94277392797095466828 -0.00043250083998449393
24.95737389169956843041 0.00379326553067476257
24.97176315811246950806 0.00240711939176479920
24.98594682064251999507 0.00333037362889405688
24.99993272582807801568 0.00041147191616931355
25.01372636200693164107 0.00333437651928006131
25.02733104354073044618 0.00468019965036640039
25.04075440491057461972 0.00954811361599500698
25.05400041033472148325 0.01240725795545400513
25.06707346561165736011 0.01148700992872605951
25.07997573672614295330 0.00812244005509031934
25.09271575950642078601 0.00822328933363650466
25.10529396793129410231 0.00634036174376458056
25.11771607519493759158 0.00647613036213673948
25.12998630989126654146 0.00719870651618003790
25.14210791750502601354 0.00769089106740817265
25.15408433954744423032 0.00673559848878251341
25.16591895857865196358 0.00592790780170135880
25.17761505410819406963 0.00436534530766930776
25.18917519999848453533 0.00530761495022338131
25.20060349586301831946 0.00494852557973299769
25.21190357235364681609 0.00575264981372978178
25.22307628466570272372 0.00540890176074215196
25.23412628419565789253 0.00673877261963198342
25.24505573339428604479 0.00756350856045214991
25.25586662373721864625 0.00719010450283171427
25.26656277115204025505 0.00606706441598806813
25.27714457795505163499 0.00353832015321127562
25.28761601612445684370 0.00193079036690103383
25.29797814570111582886 0.00092683782077071434
25.30823536983803243605 -0.00000955730063995683
25.31838782654960287211 -0.00402921628496715144
25.32843788343636584283 -0.00624655296818536190
25.33838873141985672672 -0.00963017551547203904
25.34824130068832049535 -0.01636668103818863445
25.35799773970226713575 -0.02365625362642652664
25.36765830469668036498 -0.02793652121906652882
25.37722857109117440066 -0.03117325783038967091
25.38670606372806581686 -0.03241963495517011342
25.39609455347383700551 -0.02852937530397195286
25.40539682002136391327 -0.02316677122600176607
25.41461321779053106708 -0.01656481408861202274
25.42374577595568906326 -0.00962497529602308774
25.43279575392899602667 -0.00223883497082718035
25.44176497324293742710 0.00310023855356455038
25.45065422828302447064 0.00774648938500373258
25.45946488691907789823 0.01157759669603987466
25.46819902077410435530 0.01287368489057119169
25.47685720762751752471 0.01307474869220360997
25.48544079830176978340 0.01390550752541919926
25.49395184625015442483 0.01417079649853272071
25.50238972567232979420 0.01547185185931291298
25.51075786838600151188 0.01726722645325394794
25.51905633799394756407 0.02149652170617013031
25.52728814401597391680 0.02438800296460544012
25.53545132879721154848 0.02418316074213816744
25.54354932322663367988 0.02272897944731219862
25.55158156650441833947 0.02006116392969212490
25.55954883324730776461 0.01796064782678620801
25.56745424460833942064 0.01800093143704537291
25.57529750246386877143 0.01763789265109372268
25.58307943061516044972 0.01703039617264062808
25.59080251613651090816 0.01403758574350192287
25.59846546146083934062 0.01016730189947602714
25.60606943642823523266 0.00690021457717034485
25.61361661943542288735 0.00492433522156986273
25.62110775982646271138 0.00228903430082709573
25.62854221191365021021 -0.00003306256071324777
25.63592159782177715126 -0.00025481011331418335
25.64324824373167359794 -0.00039506153613687515
25.65052019435039552255 -0.00049580533784309682
25.65774133055464497986 0.00002298977603173750
25.66490940007871916583 -0.00142822202292705175
25.67202670074130266245 -0.00135291662097086363
25.67909382884841078690 -0.00122428956438152014
25.68611106567959008177 -0.00064988412923398385
25.69308024491883688534 -0.00004781034826051598
25.70000079510912893019 -0.00049785164371811218
25.70687381911334646613 -0.00091594682847891327
25.71369992207714236744 -0.00232610706965416966
25.72048032814288851224 -0.00574862723532042068
25.72721442761866773497 -0.01029853978774008985
25.73390291787852994542 -0.01351808425172689019
25.74054699920028135351 -0.01574093471354150831
25.74714744523559508287 -0.01631828629004622578
25.75370442132932676545 -0.01549173654622096542
25.76021864116408794880 -0.01229566661268181719
25.76669167617466585577 -0.01010637948187555295
25.77312284047212830009 -0.00977229111335710165
25.77951300622944831924 -0.01047798987214886156
25.78586188239367945056 -0.01072189048754021061
25.79217111501690177428 -0.01051678372614308013
25.79844066855813848329 -0.01068819177011196754
25.80467154674088480403 -0.01151944018155385153
25.81086318575551888050 -0.01159736941998064843
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Written by meditationatae

June 2, 2015 at 11:00 pm

Posted in History

cat newreadtest48a.c

#include <stdio.h>
#include <quadmath.h>
int main(void)
{
int j;
long lastprime;
__float128 thesigmas[1000];
__float128 thelogs[1000];
char buf1[100];
char buf2[100];
__float128 epsilon;
__float128 ca_log;
__float128 sigma;
__float128 y;
__float128 gamma = 0.57721566490153286060651209008240243104Q;
__float128 f2;
__float128 delta;
FILE *in1;
FILE *in2;

f2 = (__float128) 2;

in1 = fopen(“/home/david2/eratosthenes10/June02/newreadtest28bb.txt”, “r”);

in2= fopen(“/home/david2/eratosthenes10/June02/newreadtest38bb.txt”, “r”);
for(j=0; j<1000; j++)
{
fscanf(in1, “%ld”, &lastprime);
fscanf(in1, “%s”, &buf1);
fscanf(in1, “%s”, &buf2);
epsilon = strtoflt128(buf1, NULL);
ca_log = strtoflt128(buf2, NULL);
thelogs[j] = ca_log;
}

fclose(in1);
for(j=0; j<1000; j++)
{
fscanf(in2, “%ld”, &lastprime);
fscanf(in2, “%s”, &buf1);
fscanf(in2, “%s”, &buf2);
epsilon = strtoflt128(buf1, NULL);
sigma = strtoflt128(buf2, NULL);
thesigmas[j] = sigma;
}

fclose(in2);

for(j=0; j<1000; j++)
{
ca_log = thelogs[j];
delta = expq(gamma)*logq(ca_log) – thesigmas[j];
y = logq(delta) – (0.323336Q – logq(ca_log)/f2 ) ;
printf(“%.20Qf %.20Qf\n”, logq(ca_log), y);
}
return 0;
}

Written by meditationatae

June 2, 2015 at 10:51 pm

Posted in History

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