Goldbach's comet

The function, studied in relation to Goldbach's conjecture, is defined for all even integers

to be the number of different ways in which E can be expressed as the sum of two primes.

since 22 can be expressed as the sum of two primes in three different ways (

The coloring of points in the above image is based on the value of

An illuminating way of presenting the comet data is as a histogram.

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.

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.

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:[2] where the product is taken over all primes p that are factors of

Of particular interest is the peak formed by selecting only values of

The peak is very close to a Gaussian form (shown in gray).

When histograms are formed for different average values of E, the width of this (primes only) peak is found to be proportional to

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.

The relative heights of the peaks in the total histogram are representative of the populations of various types of

It is interesting to speculate on the possibility of any number E having zero prime pairs, taking these Gaussian forms as probabilities, and assuming it is legitimate to extrapolate to the zero-pair point.

If this is done, the probability of zero pairs for any one E, in the range considered here, is of order 10−3700.

The integrated probability over all E to infinity, taking into account the narrowing of the peak width, is not much larger.

Any search for violation of the Goldbach conjecture may reasonably be expected to have these odds to contend with.