Cumulative sum of Goldbach Partitions

[Paul Mackenzie]

I noted the following concerning the cumulative sum of Goldbach partitions


C[2N] = sum[ G(2N) ;from 6 to 2N] is greater than pi[2N]*(pi[2N] -1)/2

where
2N is an even number 2N=6,,,,,
C[2N] is the cumulative sum of the goldbach partitions of the even numbers 6,...2N
G(2N) is the number of Goldbach partitions for the even number 2N and
pi[2N] is the number of primes less than 2N.

I checked the above on my computer for values of 2N upto 100,000

What would a proof of this be? I got as far as formulating every possible combinations
of primes (p,q) where p and q are both less than 2N, This lead me to a rough guess of
C[2N] = pi^2[2N] /2 but why is it greater?.

Paul

 

[Paul Mackenzie]

I have been able to prove a weak version of the above.

Let the cumulative sum C[2N] = Sum([G[2N]; 2N= 6 to 2N)
where G[2N] is the number of goldbach partitions for the even number 2N.

Consider every combination of odd primes less than N,
note that the sum of these combination pairs is less than 2N
and note that this set of combination pairs forms a subset
of all combination pairs of odd primes that form the goldbach
partitions constituting the cumulative sum thus
we can state

pi[N]*pi[N] < C[2N]

where pi[N] is the number of primes less than N

though I still am unable to prove

pi[2N]*(pi[2N]-1)/2 < C[2N]

the latter is closer to the function C[2N]

I found another function which appears to be
even more closer to C[2N] for large 2N,
though it does not hold for some small 2Npi[2N]*pi[N] < C[2N]

Any suggestions?

 

[RamaWolf]

To get familiar with this topic, I started to establish the function
Goldbach2 to calculate the number of partitons of 2n; the result
r2(2n) is the number of prime pairs (p,q) with p not greater q;
up to 40 the list looks like this:

r(2n)|2n|Sum(r(2n))
-------------------
1|4|1
1|6|2
1|8|3
2|10|5
1|12|6
2|14|8
2|16|10
2|18|12
2|20|14

Perhaps, you did not use r2(n), but R(2n), which is simply 2*r(2n) if
n is not prime, and 2*r(2n) - 1 else.

Regards

 

[Paul Mackenzie]

To better understand the terminology I have included a word document with a graph with values 2N= 6 to 72,000. It appears the term R[2N] you referred to and G[2N] are interchangeable.

Regards

Paul

 

[RamaWolf]

Below is a table with the following entries:


R2(2*n) is definied as in my previous post as the
        number of binary Goldbach partitions
        (eg R2(10) = 3; the partiitions are {(3+7),(5+5),(7+3)}

C(2*n) := Sum(R2(2*n)), the sum is taken from 4 to 2*n
Pi(k) are the number of primes not exceeding k
H1 := Pi(n) * Pi(n)
H2 := Pi(n) * Pi(2*n)
H3 := Pi(2*n)*(Pi(2*n) - 1) / 2

    2*n    C(2*n)    Pi(n)  Pi(2*n)      H1       H2       H3
-------------------------------------------------------------
      4        1        1        2        1        2        1
      6        2        2        3        4        6        3
      8        4        2        4        4        8        6
     10        7        3        4        9       12        6
     20       24        4        8       16       32       28
     30       50        6       10       36       60       45
     40       78        8       12       64       96       66
     50      117        9       15       81      135      105
     60      158       10       17      100      170      136
     70      199       11       19      121      209      171
     80      252       12       22      144      264      231
     90      312       14       24      196      336      276
    100      361       15       25      225      375      300
    200     1135       25       46      625     1150     1035
    300     2233       35       62     1225     2170     1891
    400     3540       46       78     2116     3588     3003
    500     5139       53       95     2809     5035     4465
    600     6946       62      109     3844     6758     5886
    700     9012       70      125     4900     8750     7750
    800    11274       78      139     6084    10842     9591
    900    13759       87      154     7569    13398    11781
   1000    16349       95      168     9025    15960    14028
   2000    52934      168      303    28224    50904    45753
   3000   105975      239      430    57121   102770    92235
   4000   173401      303      550    91809   166650   150975
   5000   254879      367      669   134689   245523   223446
   6000   349452      430      783   184900   336690   306153
   7000   456847      489      900   239121   440100   404550
   8000   576324      550     1007   302500   553850   506521
   9000   707138      610     1117   372100   681370   623286
  10000   850833      669     1229   447561   822201   754606

H1, H2 and H3 serve as lower estimates for C2(2*n)
and H2 seems to serve best

Regards

 

[Paul Mackenzie]

Hi

I think I have found an upper bound for C[2N] [though it does not hold for some small 2N]. Again an intuitive guess.

C[2N] < 4*Pi[2N]*Pi[N] - 2*Pi[N]*Pi[N] - P[2N]*Pi[2N]

so we have

Pi[N]*Pi[2N] < C[2N] < 4*Pi[2N]*Pi[N] - 2*Pi[N]*Pi[N] - P[2N]*Pi[2N]

I checked this for odd primes upto 2N= 100,000

Regards