Let us denote by Z + the set of natural integers; clearly Z +⊂ Z. Functions whose domain is Z or Z + are usually called arithmetical functions (sometimes also number-theoretic functions), regardless of their range. It is, of course, easy to fabricate such functions out of any functions defined over Q, R, or C, simply by considering their restrictions to Z or Z +. But this rarely leads to interesting results. So for instance if we restrict the function y = x 2 to x ∈ Z, we simply obtain the sequence of squares; if we restrict y = sin πx to x ∈ Z, we obtain y = 0 for all x∈Z, and so on. It is much more interesting to consider functions that have Z or Z + as their natural domain, which means that we cannot give them a simple, sensible interpretation unless the independent variable is an integer. So for instance it makes sense to speak about the number of divisors of an integer π, but no simple meaning can be attached to the number of divisors of π, or of e, or of i. We already met with some arithmetical functions. One of them, the Legendre-Jacobi-Kronecker symbol, has been discussed in Chapter 5; another is the number of divisors of an integer; still another one is Euler’s Φ-function. In the present chapter we shall study these and also a few other arithmetical functions. In addition, we shall discuss two functions that are not, properly speaking, arithmetic functions, being defined over the reals; their connection with arithmetical functions is so close, though, that this seems the logical place to study them.
KeywordsDirichlet Series Arithmetical Function Euler Function Natural Integer Perfect Number
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