what is the Best known Upper bound on Twin Primes?

How does the Cantor set relate to primes and the zeta function?

• It's trivial to dismiss any relationship between these things existing in different domains and having no researched connection, but there is some joy to thinking of it and a bit of mathematics is required in that pursuit. To aid that end, I'll describe some differences and similarities, and pose a casual consideration of their relationship. Aspects of Cantor's ternary set include: It's definition exists upon the real numbers, bound within 0 to 1. It contains no interval of zero length (thus it can be iterated infinitely.) It is uncountable. Each iteration contains the elements of the next and all following intervals. The end points of subsets sets are preserved in following iterations (which gives rise to the counter intuitive property that it contains as many points as the original 0-1 interval, despite the removal of the middle thirds. Any earlier iteration's interval can partitioned evenly the lengths of any later interval, and some mixed combinations thereof (i.e. 1/3 = 1/6th + 1/6th, or 1/6th + 1/18th + 1/18th) It is a prototypical fractal, each iteration having the same form as the prior. Contrastingly, the set of prime numbers are: Described by the natural numbers. Countable (as each ordered prime can be mapped to a natural number.) Infinite, and unbounded. Probabilistically sparse (as the information required to represent the larger numbers is inversely related to probability of a prime existing at that number.) Some properties which seem to suggest some vague relationship to the Zeta function are: The fractal nature of prime partitions, as demonstrated by Ken Ono, relating to the ability to compose earlier iterations from elements in following iterations within the ternary. The unique and infinite set of preserved end points in the Cantor set growing probabilistically less likely to be any randomly chosen point in the 0-1 interval at each successive iteration (though not by the same measure as the growth in the prime number theorem.) The mysteriously beautiful formulations of the zeta function in terms of mapping the integer limit to one composed of primes - in a sense perhaps an expression of iteration. As a naive thought experiment, I wonder if the structure of the zeta function can be considered as reflecting some aspects of the Cantor set's tree like "prototypical fractal structure" in terms of ordering of factorial partitions and prime relationships.   Although the primes extend out to infinity, by considering the zero line of the Zeta along the real axis to be the boundary of the 0-1 domain, might one casually invert the infinite extents into corresponding reals spaced within that domain? Then, let those primes of the zeta be the unique preserved and countable endpoints of the ternary. Finally let zero be the infinite and flip the cantor set on it's head such that it's limits descend into the primes of the zeta. I've transposed the axies in the picture below for clarity, but you'll have to imagine swapping the zero lines with the infinite on your own.   To disclaim, patterns are everywhere, especially in numbers and their structure. We are all aware of how easy it is to fool oneself into seeing that which isn't there, and especially among complex and confusing topics. Many things can be imagined, but so few correct or utilitarian. Yet, with that in mind, I hope it isn't a barrier to the joy of creative thinking about another, and especially in the myriad rainbows of math.

• Answer:

  It doesn't. Not in any reasonable way.