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Abstract: For each integer n at least 2, let \gamma(n) denote the product of the primes which divide n. For example, \gamma(12)=6. We define \lambda(n), ``the index of composition" of n, as the logarithm of n in the base \gamma(n), that is, \lambda(n):=(\log n)/(\log \gamma(n)). In a sense, \lambda(n) measures the level of compositeness of n. Although \lambda(n) can be arbitrarily large, one can show that the average value of \lambda(n) is 1, that is, the average over the first x integers tends to 1 as x tends to infinity. The study of the local behavior of n is more intricate: for example, can two consecutive values of \lambda be both arbitrarily large? This is related to the famous abc conjecture.
This talk will be accessible to graduate students.
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