Questions tagged [multiplicative-number-theory]

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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 ...
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3 votes
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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 ...
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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/...
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7 votes
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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 ...
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4 votes
1 answer
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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+\...
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4 votes
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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_{...
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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 $$\...
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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 ...
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2 votes
2 answers
417 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_{...
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8 votes
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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)\...
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9 votes
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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(...
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2 votes
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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 (...
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2 votes
1 answer
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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 ...
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5 votes
1 answer
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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 ...
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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 $$\...
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4 votes
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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 ...
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1 vote
1 answer
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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!+...
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6 votes
1 answer
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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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10 votes
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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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1 vote
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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-...
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16 votes
1 answer
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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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3 votes
0 answers
308 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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0 votes
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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 ...
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