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Unformatted text preview: ARITHMETICAL FUNCTIONS III: ORDERS OF MAGNITUDE 1. Introduction Having entertained ourselves with some of the more elementary and then the more combinatorial/algebraic aspects of arithmetical functions, we now grapple with what is fundamentally an analytic number theory problem: for a given arithmetical function f , approximately how large is f ( n ) as a function of n ? It may at first be surprising that this is a reasonable and, in fact, vital ques- tion to ask even for the elementary functions f for which we have found exact formulas, e.g. d ( n ), ( n ), ( n ), ( n ) (and also r 2 ( n ), which we have not yet taken the time to write down a formula for but could have based upon our study of the Gaussian integers). What we are running up against is nothing less than the multi- plicative/additive dichotomy that we introduced at the beginning of the course: for simple multiplicative functions f like d and , we found exact formulas. But these formulas were not directly in terms of n , but rather made reference to the standard form factorization p a 1 1 p a r r . It is easy to see that the behavior of, say, ( n ) as a function of n alone cannot be so simple. For instance, suppose N = 2 p- 1 is a Mersenne prime. Then ( N ) = N- 1 . But ( N + 1) = (2 p ) = 2 p- 2 p- 1 = 2 p- 1 = N + 1 2 . This is a bit disconcerting: N + 1 is the tiniest bit larger than N , but ( N + 1) is half the size of ( N )! Still we would like to say something about the size of ( N ) for large N . For instance, we saw that for a prime p there are precisely ( p- 1) primitive roots modulo p , and we would like to know something about how many this is. Ideal in such a situation would be to have an asymptotic formula for : that is, a simple function g : Z + (0 , ) such that lim n ( n ) g ( n ) = 1. (In such a situation we would write g .) But it is easy to see that this is too much to ask. Indeed, as above we have ( p ) = p- 1, so that restricted to prime values ( p ) p ; on the other hand, restricted to even values of n , ( n ) n 2 , so there is too much variation in for there to be a simple asymptotic expression. This is typical for the classical arithmetical functions; indeed, some of them, like the divisor function, have even worse behavior than . In other words, has more than one kind of limiting behavior, and there is more than one relevant question to ask. We may begin with the following: Question 1. a) Does ( n ) grow arbitrarily large as n does? b) How small can ( n ) /n be for large n ? 1 2 ARITHMETICAL FUNCTIONS III: ORDERS OF MAGNITUDE Part a) asks about the size of in an absolute sense, whereas part b) is asking about in a relative sense. In particular, since there are ( p ) = p- 1 elements of ( Z /p Z ) , the quantity ( p- 1) p- 1 measures the chance that a randomly chosen nonzero residue class is a primitive root modulo p . Note we ask how small because we....
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