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Similar to a previous question of mine, I’m trying to find equivalents to logarithms in other rings in which we can carry out computations. If we consider an arithmetic function to be a function of the form $f: \mathbb{N}^* \to \mathbb{C}$, the set $\mathbb{A}$ of all such functions is a commutative ring with unity with respect to point-wise addition and Dirichlet convolution: $$(f * g)(n) = \sum_{d \mid n} f(d)g\left(\frac{n}{d}\right)$$

Moreover, the set of all such functions such that $f(1) \neq 0$ is the unit group of the ring $A$. My question is:

Does there exist a non-trivial homomorphism $\Phi: \mathbb{A}^\times \to \mathbb{A}$ - I.e., such that $\Phi(f * g) = \Phi(f) + \Phi(g)$?

If it helps defining the function $\Phi$ (in analogy to logarithms themselves), please feel free to consider the codomain to be $\mathbb{R}$, or even $\mathbb{Q}$.

Does such a function exist? If so, what is it?

Thanks in advance!

Gauss
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  • The usual definition of arithmetic function uses $\Bbb C$ as the codomain. Some important arithmetic functions like Ramanujan sums or Dirichlet characters take complex values. – jjagmath Apr 03 '24 at 23:20
  • @jjagmath I’m aware that is the usual definition - as I noted below in my question, I only wrote the codomain to be the reals in case it would make defining the “log” easier (that is the case with the usual log, after all ;) ). I will edit the question and switch the two, though – Gauss Apr 03 '24 at 23:22
  • please link to your previous question. – kodlu Apr 03 '24 at 23:28
  • @kodlu Edited to include the link – Gauss Apr 03 '24 at 23:33
  • And you probably meant $\Phi(f \circ g) = \Phi(f) + \Phi(g)$ – jjagmath Apr 03 '24 at 23:39
  • @jjagmath Yes, of course! Sorry, I’ve fixed it now – Gauss Apr 03 '24 at 23:57
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    Composition is $f\circ g$ and convolution is $f\ast g$. – coiso Apr 04 '24 at 00:24
  • @coiso Sorry, my bad. Was reading some notes that used the “circle” notation - didn’t even realize there could be a confusion with composition. I’ve fixed it, now – Gauss Apr 04 '24 at 00:38

1 Answers1

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Just as functions $f:\Bbb Z\to\Bbb C$ with pointwise addition and additive convolution (and upward-closed support) correspond to formal Laurent series $\sum f(n)x^n$ with typical addition and multiplication, we can think of arithmetic functions $f:\Bbb N\to\Bbb C$ with pointwise addition and multiplicative convolution as "formal Dirichlet series" $\sum f(n)n^{-s}$ with typical addition and multiplication. Thus, arithmetic functions are the number-theoretic analogue of generating functions in combinatorics.

In this context, if $f(1)=1$ we have the Newton-Mercator expansion

$$ \sum_{n=1}^\infty\frac{\Phi(f)(n)}{n^s}= \ln\Big(\sum_{n=1}^\infty \frac{f(n)}{n^s}\Big)=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\Big(\sum_{n=2}^\infty \frac{f(n)}{n^s}\Big)^k $$

so therefore $\phi(f)(1)=0$ and for $n>1$ we ought to have

$$ \Phi(f)(n)=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\!\!\!\!\sum_{\substack{n_1\cdots n_k=n \\ n_1,\cdots,n_k>1}}f(n_1)\cdots f(n_k). $$

Note this sum is always finite. For example, the inner sum is empty when $k$ exceeds the number of prime factors of $n$ (counted with multiplicity). If we write $f\ast g$ for convolution and $f^{{\large \ast}k}$ for iterated convolution, the inner sum can be written as

$$ \sum_{\substack{n_1\cdots n_k=n \\ n_1,\cdots,n_k>1}}f(n_1)\cdots f(n_k) = \sum_{j=0}^k (-1)^j\binom{k}{j} f^{{\large\ast}(k-j)}(n) $$

since we can replace each $f(n_i)$ with $f(n_i)-\delta_1(n_i)$ and then nix $n_1,\cdots,n_k>1$ from the summation conditions, then expand. (Equivalently, inclusion-exclusion.)

If $f(1)\ne1$ the usual branch-cut discourse for $\ln z$ ($z\in\Bbb C$) applies, but in any case we can normalize the Dirichlet series and set $\ln\big(\sum f(n)n^{-s}\big)=\ln f(1)+\ln\big(\sum \frac{f(n)}{f(1)}n^{-s}\big)$, which means $\Phi(f)(1)=\ln f(1)$ and $\Phi(f)(n)=\Phi(\frac{1}{f(1)}f)(n)$ for $n>1$ otherwise.

coiso
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