Goldbach's comet is the name given to a plot of the function g ( E ) {\displaystyle g(E)} , the so-called Goldbach function (sequence A002372 in the OEIS). The function, studied in relation to Goldbach's conjecture, is defined for all even integers E > 2 {\displaystyle E>2} to be the number of different ways in which E can be expressed as the sum of two primes. For example, g ( 22 ) = 3 {\displaystyle g(22)=3} since 22 can be expressed as the sum of two primes in three different ways ( 22 = 11 + 11 = 5 + 17 = 3 + 19 {\displaystyle 22=11+11=5+17=3+19} ).
Properties An illuminating way of presenting the comet data is as a histogram. The function g ( E ) {\displaystyle g(E)} can be normalized by dividing by the locally averaged value of g, gav, taken over perhaps 1000 neighboring values of the even number E. The histogram can then be accumulated over a range of up to about 10% either side of a central E. Such a histogram appears on the right. A series of well-defined peaks is evident. Each of these peaks can be identified as being formed by a set of values of E / 2 {\displaystyle E/2} which have certain smallest factors. The major peaks correspond to lowest factors of 3, 5, 7 ... as labeled. As the lowest factors become higher the peaks move left and eventually merge to give the lowest value primary peak. There is in fact a hierarchy of peaks; the main peaks are composed of subsidiary peaks, with a succession of second smallest factors of E / 2 {\displaystyle E/2} . This hierarchy continues until all factors are exhausted.
The magnified section shows the succession of subsidiary peaks in more detail. The relative location of the peaks follows from the form developed by Hardy and Littlewood:
g ( E ) g a v = Π 2 ∏ ( p − 1 ) ( p − 2 ) , ( 1 ) {\displaystyle {\frac {g(E)}{g_{av}}}=\Pi _{2}\prod {\frac {(p-1)}{(p-2)}}\,,\quad \quad \quad \quad \quad \quad (1)}
where the product is taken over all primes p that are factors of E / 2 {\displaystyle E/2} . The factor on the right is Hardy–Littlewood's twin prime constant
Π 2 = ∏ p > 2 ( 1 − 1 ( p − 1 ) 2 ) = 0.6601618... {\displaystyle \Pi _{2}=\prod _{p>2}\left(1-{\frac {1}{(p-1)^{2}}}\right)=0.6601618...}
Here the product is taken over all primes greater than 2.
Of particular interest is the peak formed by selecting only values of E / 2 {\displaystyle E/2} that are prime. The product factor in equation (1) is then very close to 1. The peak is very close to a Gaussian form (shown in gray). For this range of E values, the peak location is within 0.03% of the ideal Π 2 {\displaystyle \Pi _{2}} . When histograms are formed for different average values of E, the width of this (primes only) peak is found to be proportional to 1 / g p a v ( E ) {\displaystyle 1/{\sqrt {g_{pav}(E)}}} . However, it is a factor of about 1.85 less than the value 2 / g p a v ( E ) {\displaystyle {\sqrt {2/g_{pav}(E)}}} that would be expected from a hypothesis of totally random occurrence of prime-pair matching. This may be expected, since there are correlations that give rise to the separated peaks in the total histogram.
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