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Number Theory Seminar

Monday, March 6th, 2023

Time: 4:30 p.m.  Place: Jeffery Hall, Room 319

Speaker: Lauryn Needham (¾ÅÐãÖ±²¥)

Title: Some results on averages of digits of continued fractions

Abstract: For almost all x in (0, 1), the Gauss-Kuzmin distribution tells us that the probability that a random digit of the continued fraction expansion of x equals k is $\log_2(1 + \frac{1}{k(k+2)}$. We might then wonder if $\frac{1}{N}\sum_{n=1}^N a_n$, where $a_n$ is the n-th digit of the continued fraction of x, converges as N goes to infinity for almost all x in (0,1). In fact, it does not, but there are other averages that do. In this talk we will look at a theorem that tells us for which positive arithmetic functions f the limit $\frac{1}{N} \sum_{n=1}^N f(a_n)$ converges as N goes to infinity.