# Questions tagged [multiplicative-number-theory]

The multiplicative-number-theory tag has no usage guidance.

29
questions

2
votes

1
answer

113
views

### Sum of $\frac{1}{(\delta_1,\delta_2)}$ with congruence restrictions

In the course of my work, I encountered the following sum ($(x,y)$ stands for the GCD of $x$ and $y$):
$$L(Q)=\sum_{\substack{\delta_1,\delta_2\leq Q\\\delta_1\equiv0\ (a)\\\delta_2\equiv0\ (b)}}\frac{...

3
votes

0
answers

41
views

### What lower bounds are known on growth of the distribution of the abundancy index?

Let $a(n)=\sigma(n)/n$ be the abundancy index of $n$ and let $F(x)$ be the distribution function of this index: i.e., the proportion of integers $n$ with $a(n)\leq x$. (This function is well-defined ...

4
votes

2
answers

155
views

### "Efficient" way to build a table of multiplicative orders modulo $p$ of a fixed integer $a$

Given an integer $a$, I would like to build a table of entries $(p, \text{ord}_p(a))$, where $p$ runs over the prime numbers not dividing $a$ and not exceeding a fixed parameter $P$, and $\text{ord}_p(...

1
vote

0
answers

44
views

### Classes of functions which are not almost divisible by a part of the Euler phi function

Given, $f$ and $g$ as functions with domain and range in the integers, we will
write $S(f,g)$ to be the set of $n$ such that $f(n)|g(n)$. We will write $f^2$ to just be $f(n)^2$.
We will say that $f$ ...

6
votes

1
answer

147
views

### Mean value of the divisor function over Piatetski-Shapiro sequences

Let $c>1$, $c\not\in\mathbb{Z}$ and consider the sum
$$
\sum_{n\leq x} \tau(\lfloor n^c \rfloor),
$$
where $\tau(n)$ is the number of divisors of $n$. I'm almost certain I've seen an evaluation of ...

4
votes

0
answers

127
views

### Average of $\lambda(n+1)$ for $n$ smooth, or smooth-and-rough? What follows?

Let $\lambda$ be the Liouville function, i.e., $\lambda(p_1\dotsb p_k)=(-1)^k$ for $p_1,\dotsc,p_k$ not necessarily distinct.
There is a conjecture (due to whom?) that there are infinitely many primes ...

5
votes

1
answer

183
views

### Results using a certain kind of identity

Recently, I've been reading about asymptotics for smooth numbers as well as smooth numbers in arithmetic progressions. One of the ideas I find especially pleasing among some of these results is the ...

3
votes

0
answers

225
views

### Numbers made up of primes from a given set

Take a set $\mathcal P$ of primes and denote by $\langle \mathcal P\rangle $ the set of all natural numbers composed of primes from $\mathcal P$. If
\[ \sum _{p\in \mathcal P}\frac {1}{p}\]
converges ...

0
votes

0
answers

66
views

### Function multiplicativity

Is this a multiplicative function? $ \displaystyle \sum_{a=1 \atop (a,q)=1}^{q}e^{-2\pi i\frac{a}{q}c_q(da)F}=T(q)$,where $d,F\in \mathbb{N}$ and $c_q(da)$ - Ramanujan's sum https://en.m.wikipedia.org/...

7
votes

0
answers

138
views

### Recovering basic information about perfect numbers from a Dirichlet series

The following question is inspired mostly by this question, answer and the comment by Wojowu there
A naive approach to understanding odd perfect numbers is to make a Dirichlet series where the $n$th ...

4
votes

1
answer

350
views

### On a sharp version of Halasz's theorem

I am trying to read the following paper by Granville-Harper-Soundararajan and I had a few questions regarding the paper. They prove, for $S(x):=\sum_{n\leq x}f(n)$ and $F_x(s):=\prod_{p\leq x}\left(1+\...

4
votes

0
answers

119
views

### Consecutive integers which are products of Fibonacci numbers

Let $F_1, F_2, \dots$ be the sequence of Fibonacci numbers, and let $\mathcal{M}$ be the set of positive integers expressible as a product of Fibonacci numbers (OEIS A065108).
One can prove that
$$F_{...

2
votes

0
answers

126
views

### Question regarding proof of Mertens' estimates in Montgomery-Vaughan's "Multiplicative number theory"

I have been trying to read Theorem 2.7 of Montgomery-Vaughan's "Multiplicative number theory" volume 1, and there is an issue I have run into: in the proof of subpart (e), when they write
$$\...

4
votes

1
answer

265
views

### Average of sums of $r_2(n^2+d^2)$

Let $r_2(n)$ denote the number of ways in which a positive integer $n$ is expressed as the sum of two squares (integers). I would like to know if there is a result that gives the exact behavior of (as ...

2
votes

2
answers

544
views

### Convergence of Euler product and Dirichlet series in the same half-plane?

I'm crossposting this from math.stackexchange because I think it might be inappropriately research-level for the community over there.
Suppose we have an Euler product over the primes
$$F(s) = \prod_{...

8
votes

0
answers

277
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)\...

10
votes

0
answers

214
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

136
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

150
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 ...

5
votes

1
answer

151
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

90
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

162
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

316
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

301
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

206
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

77
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 ...

4
votes

0
answers

312
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 $k$ ...

1
vote

1
answer

313
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 ...