Using Dynamical Systems to Construct Infinitely Many Primes

Euclid's proof can be reworked to construct infinitely many primes, in many different ways, using ideas from arithmetic dynamics. After acceptance Soundararajan noted the beautiful and fast converging formula: $$\tau = a^{1/(d-1)} x_0 \cdot \lim_{n\to \infty} \prod_{m=1}^n \left(\frac{x_m}{ax_{m-1}^d} \right)^{1/d^m}$$Comment: To appear in the American Mathematical Monthl

Squares in arithmetic progressions and infinitely many primes

We give a new proof that there are infinitely many primes, relying on van der Waerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else these ideas have come together in the past.Comment: To appear in the American Mathematical Monthl

Primitive prime factors in second order linear recurrence sequences

For a class of Lucas sequences ${x_n}$, we show that if $n$ is a positive integer then $x_n$ has a primitive prime factor which divides $x_n$ to an odd power, except perhaps when $n = 1, 2, 3 or 6$. This has several desirable consequences.Comment: To Andrzej Schinzel on his 75th birthda

What is the best approach to counting primes?

As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of great mathematics, the techniques and insights permeating many different fields. In this article we will review some of the best techniques for counting primes, centering our discussion around Riemann's seminal paper. We will go on to discuss its limitations, and then recent efforts to replace Riemann's theory with one that is significantly simpler.Comment: To appear in a volume dedicated to the MAA Centennial in 201

The distribution of the zeroes of random trigonometric polynomials

We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that \frac{Z_{N}-\E Z_{N}}{\sqrt{cN}} converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.Comment: 51 pages. We cut the size of the paper to better suit publication. In particular, all the results of empirical experiments were cut off. Some standard results in probability and stochastic processes were also omitted. Numerous typos and mistakes were corrected following the suggestions of referees. This paper was accepted for publication in the American Journal of Mathematics

Large character sums

Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n <= x is o(x); they show that this holds if log x / log log q -> infinity and q -> infinity (q is the size of the finite field).Comment: Abstract added in migration

Beyond the LSD method for the partial sums of multiplicative functions

The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the partial sums of a multiplicative function $f$ whose prime values are $\alpha$ on average. In the literature, the average is usually taken to be $\alpha$ with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak error term, and so we explore here how good an estimate this will imply for the partial sums of $f$, developing new techniques to do so.Comment: Addressed referee's comments; added some references; corrected and simplified the proof of Theorem 9. 26 page