Known primes < 10^n and sum of primes < 10^n

                                                                   Cino hilliard June 08 2008

                            Actual                                    Estimate
Primes < 10^n        Sum primes < 10^n          (10^n)^2/(2*log(10^n)-1.)         Relative error
n  Pi(10^n)
0   0                       0                                        -1
1   4                       17                                        27
2   25                      1060                                      1217                                   -0.149034246
3   168                     76127                                     78030                                  -0.025003555
4   1229                    5736396                                   5740303                                -0.000681230
5   9592                    454396537                                 454011971                               0.000846321
6   78498                   37550402023                               37550193649                             0.000005549
7   664579                  3203324994356                             3201414635780                           0.000596367
8   5761455                 279209790387276                           279007258230819                         0.000725376
9   50847534                24739512092254535                         24723998785919976                       0.000627065
10  455052511               2220822432581729238                       2219671974013732243                     0.000518032
11  4118054813              201467077743744681014                     201381995844659893517                   0.000422311
12  37607912018             18435588552550705911377                   18429088896563917716962                 0.000352560
13  346065536839            1699246443377779418889494                 1698738497929718386117081               0.000298923
14  3204941750802           157589260710736940541561021               157548836041268621469737798             0.000256519
15  29844570422669          14692398516908006398225702366             14689129661980376124875151612           0.000222486
16  279238341033925         1376110854313351899159632866552           1375842784994205257358484419433         0.000194802
17  2623557157654233        129408626276669278966252031311350         129386370761402459063701962912822       0.000171978
18  24739954287740860       12212914292949226570880576733896687       12211046444229022681106668101629074     0.000152940
19  234057667276344607      1156251260549368082781614413945980126     1156092973401062822090532446879592309   0.000136896
20  2220819602560918840     109778913483063648128485839045703833541   109765382979108556087719263460893002218 0.000123252
21  21127269486018731928
22  201467286689315906290
23  1925320391606818006727

Some "heuristic" analysis. Notice that the sum of primes < 10^n
is about the same as Pi(10^2n)=Pi((10^n)^2). So to put things in powers of
10 for a given n, we write n = 10^x.

Then log(n) = x*log(10) and x = log(n)/log(10). So SumP(n) = sump(10^x) ~
Pi(10^2x) = Pi(10^(2*log(n)/log(10)). Now log10(n) = log(n)/log(10) => 10^log10(n) =
10^(log(n)/log(10)) = n. So 10^(2*log(n)/log(10)) = n^2.

1. SumP(n) ~ Pi(n^2).

Theorem 1.0. The sum of primes <  n ->  the number of primes < n^2 as n -> infinity.

We know that x/log(x) is asymptotic to Pi(x), so we try that first with x-n^2 as
perameter in our estimator for SumP(n). SumP(n) ~ n^2/(log(n^2) = n^2(2*log(n)). This
gives us reasonable results bet we also know x/(log(x)-1) gives us better results. So we
plug that into 1. to get SumP(n) ~ n^2/(2log(x)-1). This formula  gives reasonable
approximations to SumP(n). Here we do not need a prime counting function.

Similarly, we know that Li(x) or the logarithmic integral of x gives a better
approximation for Pi(x) and therefore SumP(x). This is illustrated at the bottom of the
next table which shows Li(x) and R(x).

  Sources:

  http://www.research.att.com/~njas/sequences/A006880 

  http://www.research.att.com/~njas/sequences/A046731

Table of sums of primes <= n using R(x) Riemanns approximation of Pi(x)
                                                                                                                                    Absolute
Sum of primes < 10^n                            Using Riemann's approx Pi(x)                                      Relative Error
n b-file From Marc Deleglise                            R(10^2n)
1, 17,                                                          25
2, 1060,                                                       1226
3, 76127,                                                     78527
4, 5736396,                                                  5761551
5, 454396537,                                               455050683                                                  0.0014395928373
6, 37550402023,                                            37607910542
7, 3203324994356,                                         3204941731601
8, 279209790387276,                                      279238341360977
9, 24739512092254535,                                   24739954284239494
10,2220822432581729238,                               2220819602556027015                                  0.0000012743142
11,201467077743744681014,                            201467286689188773625
12,18435588552550705911377,                         8435599767347541878146
13,1699246443377779418889494,                      1699246750872419991992147
14,157589260710736940541561021,                   157589269275973235652219770                     0.0000000259190
15,14692398516908006398225702366,                14692398897720432716641650390
16,1376110854313351899159632866552,             1376110866993765865297692177651
17,129408626276669278966252031311350,          129408626505158941382747240985638
18,12212914292949226570880576733896687,       12212914297619365442544535529374722
19,1156251260549368082781614413945980126,    1156251261026516898000789475178448811
20,109778913483063648128485839045703833541, 109778913489828302826978255361638504017  6.162071097138 E-11
My est
21,10448389020120550001479419524236673460613,10449550362264587535471276942848363541143.

Indeed, we know Li(x) or the Logarithmetic Integral gives superior results for
Pi(x) just like Gauss thought it would. Later Riemann came up with a better
estimator for Pi(x) using Li(x). I scripted these Pari functions for the calculations
that follow.

R(x) = local(j); (sum(j=1,400,moebius(j)*Li(x^(1/j))/j))

Li(x) = local(x);-eint1(log(1/x))