American Journal of Mathematical Analysis. 2013, 1(1), 8-13DOI:
Abstract: This paper determines proper factors for old and novel logarithmic functions previously used in asymptotic formulas, to make them conservative lower bounds for the “thumb-of-rule” estimation of the number of representations of an even number 2n as a sum of two odd primes (Goldbach’s conjecture). Numerical experiments up to 2n = 500,000 show that, in the graph of the number of prime-pairs versus 2n, the ratio of the ordinate of lowest “cloud” points over the aforementioned functions tends asymptotically to values between 0.61 and 0.74. One of the three formulas proposed takes the simple form 4n/[3(lnn)2], which is a conservative lower bound for the number of representations of an even number 2n as a sum of two odd primes.