Seminar Details

On the math-fun list, Neil Sloane posed the following problem: Let V(n) denote the integer formed by using the base 10 digits of n in base 11. It is classical that the series sum_n 1/V(n) converges. It is challenging to calculate a good approximation to its value. As a second, related, problem find a good approximation to the subseries sum_p 1/V(p), where the sum is over primes. It turned out that the first problem was efficiently solved by two related methods described by Robert Baillie and Jean-Francois Burnol. However, they do not appear to apply to the second sum, since they both depend, implicitly, on the fact that the language of digits in the first problem is a regular language, and a recent result of Thomas Dubbe shows, in a technical sense, that the digits of primes are poorly approximated by a regular language. In this talk I'll describe attempts at approximating the value of the second sum. They involve fractals, Fourier series, the prime zeta function, and the Karamata inequality. The process of analyzing this was helped, considerably, by experimentation, and seeing structure in graphs of quantities related to the series.