Lethbridge Number Theory and Combinatorics Seminar: Nathan Ng
Topic
Inclusive Prime Number Races
Speakers
Details
Let $\pi(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a \pmod{q}$.
A ``prime number race", for fixed modulus~$q$ and residue classes $a_1,\ldots,a_r$, investigates the system of inequalities $$\pi(x;q,a_1)>\pi(x;q,a_2)> \cdots >\pi(x;q,a_r). $$ We expect that this system should have arbitrarily large solutions $x$, and moreover we expect the same to be true no matter how we permute the residue classes $a_j$; if this is the case, the prime number race is called ``inclusive". Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet L-functions. We show that the same conclusion can be reached with a substantially weaker linear independence hypothesis. This is joint work with Greg Martin.
A ``prime number race", for fixed modulus~$q$ and residue classes $a_1,\ldots,a_r$, investigates the system of inequalities $$\pi(x;q,a_1)>\pi(x;q,a_2)> \cdots >\pi(x;q,a_r). $$ We expect that this system should have arbitrarily large solutions $x$, and moreover we expect the same to be true no matter how we permute the residue classes $a_j$; if this is the case, the prime number race is called ``inclusive". Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet L-functions. We show that the same conclusion can be reached with a substantially weaker linear independence hypothesis. This is joint work with Greg Martin.