Questions tagged [multiplicative-number-theory]

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7
votes
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
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0answers
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
1answer
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
1answer
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
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0answers
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
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0answers
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
1answer
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
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0answers
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
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0answers
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 ...