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While solving number theory problems I sometimes I have to use a function that can be defined as

$$ f(n) = \begin{cases} 1 & \text{ if } n = 1, \\ 0 & \text{ if } n > 1. \end{cases} $$

where $ n $ is a positive integer.

For example, using this function, we can define Euler's totient function as

$$ \varphi(n) = \sum_{k=1}^n f(\gcd(k, n))$$

where $ n $ is a positive integer.

Is there a name or notation for such a function already? I just want to make sure that I do not create my own notation for something that already has a popular name or notation in mathematics.

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  • $\begingroup$ You can call it $\delta_1$, or $e_1$ for example. $\endgroup$
    – TheSilverDoe
    Nov 3, 2020 at 14:26
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    $\begingroup$ The Kronecker delta or the Iverson bracket may be what you are looking for. In fact Wikipedia uses the Euler phi function as an example in the article for the Iverson bracket. $\endgroup$
    – player3236
    Nov 3, 2020 at 14:27
  • $\begingroup$ There's no standard name in the sense that that's the name mostly used. It's frequently denoted by $I$, $e$, $\delta$ or (rarer) $\Delta$, and sometimes it's written $\bigl\lfloor \frac{1}{n}\bigr\rfloor$. $\endgroup$
    – Daniel Fischer
    Nov 3, 2020 at 14:27
  • $\begingroup$ I have seen $0^n$ used as an indicator function for $n=0$, so perhaps $0^{n-1}$ for an indicator function for $n=1$ if $n$ cannot be zero. $\endgroup$
    – Henry
    Nov 3, 2020 at 14:29
  • $\begingroup$ Since it is arithmetic you can use $\binom{1}{n}$ maybe. $\endgroup$
    – zwim
    Nov 3, 2020 at 14:49

2 Answers 2

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In Apostol's Introduction to Analytic Number Theory, he calls this function $I$.

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Part of Apostol's rationale is that $I$ acts like the identity element in the group of arithmetic functions (where Dirichlet multiplication is the operator), and so something like $I$ is a good name.

Elsewhere I've seen it called many things. There is no standard name, though there are lots of unambiguous names. If I were king of the notational universe, I might use Kronecker-delta based names like $\delta(\cdot)$, $\delta_1(\cdot)$, or $\delta_{[n=1]}(\cdot)$ --- but (fortunately) I am not king of the notational universe.

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It is usually called an indicator function in signal processing. It is denoted as either $\delta(1)$ or $\mathbb{1}_{n=1}$.

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  • $\begingroup$ If $ \delta(1) $ is the function notation, how is the function usage written? $ \delta(1)(1) = 1 $, $ \delta(1)(2) = 0 $ and so on? $\endgroup$
    – Lone Learner
    Nov 4, 2020 at 3:52
  • $\begingroup$ Yes. I think it should be written with a subscript, i.e., $\delta_k(n)$ which means $\delta_k(n)=1$ if $n=k$ and $\delta_k(n)=0$ otherwise. $\endgroup$
    – unobservable_node
    Nov 4, 2020 at 10:00

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