Tangente: What topics did you work on together?**
Gérald Tenenbaum: We mainly worked together on the multiplicative structure of integers. When I began my research, Erdős was one of the few people in the world with any knowledge of the subject now known as the anatomy of numbers.
From my very first steps in the field, I asked myself how to describe a "normal" integer (see box below). The notion of normality in number theory corresponds to that of an "almost sure" variable in probability theory. This means considering sequences of integers (or sets of integers) rather than particular numbers.
The sequence of integers n having fewer than ln n divisors has density 1 (where ln is the natural logarithm function). The same is true of the sequence of integers n having at least (1/2) ln(ln n) prime factors.
Thanks to a famous theorem of Erdős and Kac dating from 1939, we know that the distribution of the prime factors of a normal integer follows a Gaussian distribution. The distribution of divisors, on the other hand, is far more complex and includes many aspects that remain unexplained. After my thesis, I spent five years working on Erdős's conjecture, which states that a normal integer always has two distinct divisors whose ratio lies between 1 and 2. This property of divisors holds, for example, for 15, which is divisible by 3 and 5, but not for 21, whose divisors are 1, 3, 7 and 21. In a paper published in 1984, written jointly with the German mathematician Helmut Maier, we managed to confirm the conjecture. Erdős was known to put a price on some of his conjectures. He paid us $650 for this theorem.
