Topics from the Theory of Numbers pp 81-107 | Cite as

# Arithmetical Functions

## Abstract

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.

## Keywords

Dirichlet Series Arithmetical Function Euler Function Natural Integer Perfect Number## Preview

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