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### A question about $(0,1]$-valued multiplicative functions

Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means
$$
\lim_{N\to\infty}\frac{1}{N} \sum_{...

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### The multiplicative group generated by shifted primes

I am asking for references about the following problem.
In particular, it is still open? If not, what is the state of the art result?
Problem 1. Let $\Gamma$ be the multiplicative subgroup of $\...

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### All all hypo-multiplicative functions linear combinations of quasi-multiplicative functions?

A function is called quasi-multiplicative by many authors if $f(m)f(n)=f(1)f(mn)$, a slight generalization of multiplicativity. (Basically a multiplicative function times a constant is a quasi-...

**12**

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### Least prime in an arithmetic progression and the Selberg sieve

My question concerns a technical step in the proof of Linnik's theorem on the least prime in an arithmetic progression, as presented in Chapter 18 of Iwaniec-Kowalski: Analytic number theory.
The ...

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269 views

### Linear combination of multiplicative functions

Carlitz showed necessary and sufficient conditions for an arithmetic function to be a linear combination of two multiplicative functions. He mentions the possibility of generalizing to $k$ ...

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### Number of biquadrates mod n

Is there an explicit formula for the number of fourth powers mod n?
Finch & Sebah [1] give theorems, partially folklore, for squares and cubes mod n, but I don't know of a similar formula for ...