Applications of Multizeta Values to Mahler Measure

These notes correspond to a mini-course taught by the author during the program “PIMS-SFU undergraduate summer school on multiple zeta values: combinatorics, number theory and quantum field theory”. Please send any comments or corrections to the author at [email protected]. 1. Primes, Mahler Measure, and Lehmer’s question We start our study by discussing prime numbers. First consider the sequence of numbers Mn = 2 n − 1 for n natural. We may ask, when is Mn prime? We can write (2 − 1) = (2 − 1)(2(s−1)r + 2(s−2)r + · · ·+ 2 + 1). This implies that Mr | Mrs and we need n prime in order for Mn to be prime. Notice that M2 = 3,M3 = 7,M5 = 31,M7 − 127, but M11 = 2 − 1 = 2047 = 23× 89. Therefore the converse is not true. The primes of the form Mp are called Mersenne primes. It is unknown if there are infinitely many of such primes, or if there are infinitely many Mp composite with p prime. The largest Mersenne prime known to date is with p = 57, 885, 161 (it has 17, 425, 170 digits). The search for large Mersenne primes is being carried by the “Great Internet Mersenne Prime Search”: http://www.mersenne.org. Exercise 1. Let a, p be natural numbers such that a − 1 is prime, then show that either a = 2 or p = 1 Exercise 2. Let p be an odd prime. Show that every prime q that divides 2 − 1 must be of the form q = 2pk + 1 with k integer. Looking for large primes, Pierce [Pi17] proposed the following construction in 1917. Consider P ∈ Z[x] monic, and write P (x) = ∏ i (x− αi) then, we look at ∆n = ∏ i (α i − 1). The αi are algebraic integers. By applying Galois theory, it is easy to see that ∆n ∈ Z. Note that if P = x− 2, we get the sequence ∆n = 2 − 1, the Mersenne numbers. The idea is to look for primes among the factors of ∆n. The prime divisors of such integers must satify some congruence conditions that are quite restrictive, hence they are easier to factorize than a randomly given number. Exercise 3. Prove that ∆n is a divisibility sequence, namely, if n | m, then ∆n | ∆m. Then we may look at the numbers ∆p ∆1 , p prime. Pierce and Lehmer observed that the only possible factors of ∆n are given by prime powers p e of the form nk+ 1 for some integer k and 1 ≤ e ≤ deg (P ). It is then natural to look for P that generate sequences that grow slowly so that they have a small chance of having factors. Lehmer [Le33] studied |∆n+1| |∆n| , observed that lim n→∞ |α − 1| |αn − 1| = { |α| if |α| > 1, 1 if |α| < 1, and suggested the following definition: