# Questions tagged [multiplicative-number-theory]

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14
questions

**7**

votes

**0**answers

193 views

### Does every multiplicative function have a logarithmic average?

Is it true that for every completely multiplicative function $f:\mathbb N\to\mathbb C$
with $|f(n)|=1$ for all $n$, the logarithmic average
$$\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac1nf(n)\...

**8**

votes

**0**answers

155 views

### Strong uniqueness of Euler's totient function

Let $f:\mathbb N\to \mathbb C$ be some arithmetical function. Define $\varphi_f(n)$ by the following formula:
$$
\varphi_f(n)=\sum_{\substack{k\leq n \\ (k,n)=1}}f(k).
$$
In other words, $\varphi_f(...

**2**

votes

**0**answers

116 views

### Well-known estimate for $L(s,\chi)$ for $\sigma=\text{Re}s\geq 1/2$

This is a very short question.
Let $s=\sigma+it$ be a complex number with $\sigma \geq 1/2$.
In the paper 'Jutila, Matti. "On the Mean Value of $L(1/2, \chi)$ FW Real Characters." Analysis 1.2 (...

**2**

votes

**1**answer

139 views

### Questions about a certain set of primes

Let $A$ be a set of prime numbers, generated by the following procedure. Let $A_0 = \{2\}$. Let $A_n$ be generated by setting $x_n = (\prod_{p_i \in A_{n-1}}p_i) + 1$, and adding all the prime factors ...

**2**

votes

**0**answers

60 views

### Uniformity in Wirsing's Mean Value Theorems

In two important papers of Wirsing, namely "Das asymptotische Verhalten von Summen über multiplikative Funktionen" (1961) and its follow up (1967), several results on mean values of multiplicative ...

**1**

vote

**0**answers

88 views

### Existence of equation about the product of the divisor sum function

Let $\sigma_k(n)$ be the sum of the $k$-th powers of the positive divisors of $n$ and $\mu(n)$ be the Möbius function.
As Arithmetic function - Wikipedia mentioned, there is an equation that $$\...

**4**

votes

**0**answers

149 views

### Math software / language for analytic / multiplicative number theory

What software/languages are the most used to do computations in analytic / multiplicative number theory?
I use Python, Maple, etc., but each time I want to compute expressions like, for $n$ with an ...

**1**

vote

**1**answer

298 views

### Is it possible to find all integer functions which satisfy $f(m!+n!)\mid f(m!)+f(n!)$ and $m+n \mid f(m)+f(n)$?

I'm interesting to know more about multiplicative property of integer functions then I'd like to ask this humble question:
Question:
Is it possible to find all integer functions which satisfy $f(m!+...

**6**

votes

**1**answer

288 views

### 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_{...

**10**

votes

**0**answers

187 views

### 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 $\...

**1**

vote

**0**answers

68 views

### 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-...

**16**

votes

**1**answer

1k views

### 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 ...

**3**

votes

**0**answers

289 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 $k$ ...

**0**

votes

**0**answers

184 views

### 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 ...