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Unformatted text preview: Notes on the second moment method, Erdos multiplication tables January 25, 2011 1 Erdos multiplication table theorem Suppose we form the N N multiplication table, containing all the N 2 products ab , where 1 a,b N . Not all these products will be distinct, since for example ab = ba ; and, for example 2 3 = 3 2 = 6 1 = 1 6. But we might hope that there are enough of them to where these products take up a positive proportion of the numbers up to N 2 as N . That is, one might guess that: Question. Let m ( N ) denote the number of integers of the form ab , where 1 a,b N . Does lim N m ( N ) /N 2 exist, and is it equal to some non-zero (positive) constant? P. Erdos showed that the answer is no; that, in fact, lim N m ( N ) /N 2 = 0. In other words, as N gets bigger and bigger, the set of products ab as above eat up a smaller and smaller proportion tending to 0, in fact of the integers up to N 2 . What was innovative about Erdoss proof was that he did this using probabilistic arguments; and here we will trace through his proof. 2 Markovs inequality and Chebyshevs in- equality The main tools we will need are some elementary estimates in prime number theory, in combination with the following inequality: 1 Chebyshevs Inequality. Suppose that X is a random variable having finite variance 2 and expected value (i.e. E ( X ) = and V ( X ) = 2 ). Then, P ( | X | t ) 2 /t 2 . Another way to express the conclusion here is: P ( | X | t ) 1 /t 2 . The proof of this inequality relies on another inequality called Markovs inequality, stated as follows: Markovs Inequality. Suppose that X 0 and has expected value &gt; 0....
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