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Ομιλία
Multiplication tables

Dimitris Koukoulopoulos
Univ. Iliinois at Urbana-Champaign

18:35, Τετάρτη, 1 Ιουλίου 2009, Αίθουσα Ζ301

In 1955 Erdos posed the following problem: how many distinct integers appear in the $N\times N$ multiplication table, that is how many integers may be written as a product $ab$ with $a\le N$ and $b\le N$? In 2004 Ford established the correct order of magnitude of this quantity. We will study generalizations of this problem towards two different directions. First, we will show the order of magnitude of the number of integers that appear in the $\underbrace{N\times\cdots\times N}_{k\;{\rm times}}$ multiplication table for an arbitrary fixed $k\ge3$. Furthermore, we will prove that the number of shifted primes $p-1$ that appear in the $N\times N$ multiplication table is at least as much as the expected number up to a multiplicative constant. This result complements upper bounds proven by Ford on the quantity in question and thus establishes its order of magnitude. It is worth mentioning that counting shifted primes in the multiplication table is very closely related to the distribution of prime numbers $\le x$ in arithmetic progressions $1\pmod d$ for $d\le\sqrt{x}$, a problem which is beyond the reach of the Bombieri-Vinogradov theorem.

http://www.math.uoc.gr/analysis-seminar