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  1. Reliability
  2. 6. Arithmetic Semigroups
  3. 1
  4. 2
  5. 3
  6. 4

2. Norm and Related Functions

From Section 1, recall the set \(R = \{\bs{n} = (n_i: i \in I): \sum_{i \in I} n_i \lt \infty\}\) where \(I\) is a nonempty countable set. The corresponding arithmetic semigroup with \(I\) as the set of prime elements is \((S, \cdot)\) where \(S\) is the set of all finite, commutative produces \(x = \prod_{i in I} i^{n_i}\) with \(\bs{n} = (n_i: i \in I) \in R\). The identity element is denoted \(e\) and of course \(i^0 = e\) for \(i \in I\). So the representative of \(x \in S\) as a product of prime elements is unique up to the ordering of the factors. We also studied two semigroups isomorphic to \((S, \cdot)\), namely \((R, +)\) where \(+\) is the usual pointwise addition, and \((T, +)\) where \(T\) is the collection of all finite multisets with elements from \(I\) and \(+\) is multiset addition. Technically, an arithmetic semigroup also has a norm function, and this function provides a structure that goes beyond the sequence of prime exponents. Of course, the results in this section could be formulated in terms of the other two semigroups, but that is not usually done. Here is the definition:

A norm \(|\cdot|\) for \((S, \cdot)\) is a function from \(S\) to \((0, \infty)\) with the following properties:

  1. \(|e| = 1\).
  2. \(|i| \gt 1\) for \(i \in I\).
  3. \(|x y| = |x| |y|\) for \(x, \, y \in S\).
  4. \(N(t) \lt \infty\) for \(t \in (0, \infty)\) where \(N(t) = \#\{x \in S: |x| \le t\}\).

By (c), the norm is completely multiplicative.

If \(x = \prod_{i \in I} i^{n_i} \in S\) where \(\bs{n} = (n_i: i \in I) \in R\) then \[ |x| = \prod_{i \in I} |i|^{n_i} \]

In particular from (b) of and from , \(|x| \gt 1\) if \(x \in S_+\). Conversely, it's easy to construct a norm on an arithmetic semigroup: let \(|e| = 1\) and define \(|i| \gt 1\) arbitrarily for \(i \in I\). Then define \(|x|\) by for \(x \in S_+ - I\). The function \(|\cdot|\) satisfies parts (a), (b), and (c) of the definition. If \(I\) is finite, part (d) is satisfied automatically. If \(I\) is infinite, (d) is satisfied for example if \(\prod_{i \in I} |i| = \infty\).

For the standard arithmetic semigroup \((\N_+, \cdot)\) (with \(I = \{2, 3, 5, \cdots\}\), the ordinary set of prime numbers), the standard norm is the identity function on \(\N_+\) so that \(|x| = x\) for \(x \in \N_+\). Hence \(N(t) = \lfloor t \rfloor\) for \(t \in (0, \infty)\).

The norm is an essential part of \((\N_+, \cdot)\) so that the actual positive integers, and not just the sequence of the prime exponents, are part of the theory.

The arithmetic semigroup \((S, \cdot)\) with a single prime element \(i\) in example is isomorphic to \((\N, +)\). Any function of the form \(i^n \mapsto c^n\) with \(c \in (1, \infty)\) is a norm for \((S, \cdot)\). For this norm, \(N(t) \le | \ln t / \ln c |\) for \(t \in (0, \infty)\).

For the following definitions, we assume that the arithmetic semigroup \((S, \cdot)\) has a fixed norm.

Suppose that \(f\) is an arithmetic function for \((S, \cdot)\). The corresponding Dirichlet series \(F\) is the formal series defined by by \[F(t) = \sum_{x \in S} \frac{f(x)}{|x|^t}\] If the series converges for some \(t \in (0, \infty)\), then the series converges for \(t\) in an interval of the form \((t_0, \infty)\).

Details:

Convergence here means absolute convergence, although we will only be interested in nonnegative arithemetic functions.

The Dirichlet series corresponding to the constant function \(1\) is the the zeta function: \[\zeta(t) = \sum_{x \in S} \frac{1}{|x|^t}\]

These definitions are most important for the standard arithmetic semigroup in example .

Coonsider the standard arithmetic semigroup \((\N_+, \cdot)\) with the standard norm.

  1. The Dirichlet series \(F\) corresponding to an arithmetic function \(f\) is the standard Dirichlet series in number theory: \[F(t) = \sum_{x = 1}^\infty \frac{f(x)}{x^t}\]
  2. In particular, the zeta function is the standard Riemann zeta function \[\zeta(t) = \sum_{x = 1}^\infty \frac{1}{x^t}, \quad t \in (1, \infty)\]

For \((\N_+, \cdot)\) there is a one-to-one correspondence between the arithmetic function \(f\) and the series function \(F\). Given \(f\), we compute \(F\), of course, as the series in the definition. Conversely, given \(F\) defined on the interval of convergence \((t_0, \infty)\), we can recover the arithmetic function \(f\) (see the paper by Alan Gut).

An arithmetic semigroup \((S, \cdot)\) with a single prime element \(i\) can be identified with the standard discrete semigroup \((\N, +)\). Suppose that the norm is given by \(|n| = c^n\) where \(c \in (1, \infty)\).

  1. The Dirichlet series \(F\) corresponding to the function \(f\) is given by \[F(t) = \sum_{n = 0}^\infty \frac{f(n)}{c^{n t}}\]
  2. In particular, the zeta function is given by \[\zeta(t) = \sum_{n = 0}^\infty \frac{1}{c^{n t}} = \frac{c^t}{c^t - 1}, \quad t \gt 1 / \ln c\]

We will need one more special function.

The Mangoldt function \(\Lambda\) is the arithmetic function on \((S, \cdot)\) defined as follows: \[\Lambda(x) = \begin{cases} \ln |i| &\text{ if } x = i^n \text{ for some } i \in I \text{ and } n \in \N_+ \\ 0 &\text{ otherwise} \end{cases}\]