Questions tagged [multiplicative-number-theory]
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36 questions
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Efficiently Updating Matrix Multiplication Result When One Matrix Changes [closed]
Suppose you have two matrices $A \in Z_q^{m\times l}$ and $B \in Z_q^{l\times n}$, and the product $A\cdot B$ has already been computed. Now, matrix $B$ remains unchanged, but a few elements in matrix ...
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Why is $\sum_{n=1}^\infty \frac{\sigma_a(pn)}{n^s}=(1+p^a-p^{a-s}) \zeta(s) \zeta(s-a)$ only when $p$ is a prime number?
I tried to find the summation for $a,b\in N$ and $s>a+1$
$$ \Omega_a(b,s)=\sum_{n=1}^\infty \frac{\sigma_a(bn)}{n^s}$$
where $\sigma_a(n)$ is sum of positive divisors function which defined by
$$ \...
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Is the Conjecture of Representing Integers as Differences of Semiprimes and Primes Extendable to Products of Distinct Primes?
Conjecture:
Let $k$ and $l$ be fixed distinct positive integers ($k≠l$). Then, for every positive integer $n$, there exist prime numbers $p_1,p_2,…,p_k∈\mathbb{P}$ and $q_1,q_2,…,q_l∈\mathbb{P}$ such ...
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Sums of multiplicative functions over residue classes
It was stated in this Shiu, P. work, page 169, Theorem 2, that $$\sum_{\substack{n\le x\\ n\equiv a\pmod k}}d_r^{\ell}(n)\ll\frac{x}{k}\left(\frac{\phi(k)}{k}\log x\right)^{r^{\ell}-1}.$$
Here, $d_r(n)...
user536681
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Is it in theory possible to create a subexponential algorithm for solving discrete logarithms in multiplicative subgroups or within an Integer range?
As far I understand, when it comes to finite fields, Pollard rho and Pollard’s lambda are still the best algorithm for solving discrete logarithms in a multiplicative subgroup/suborder…
Index calculus ...
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Möbius square root function: existence of multiplicative and bounded function
With $\mu$ being the Möbius function, there exist infinite possibilities of square roots. For example, for each $n$ such that $\mu(n)\neq 0$ there is a choice: if $\mu(n)=-1$, we can choose to define $...
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Is it true that $\left\{\frac{\sigma(n)}{\varphi(n)}:\ n\in\mathbb{Z}_{\geq 1}\right\}=\{r\in\mathbb Q:\ r\ge1\}$?
For any positive integer $n$, let $\sigma(n)$ be the sum of all positive divisors of $n$.
Clearly, $\sigma(n)\ge n\ge \varphi(n)$ for all $n\in\mathbb{Z}_{\geq 1}$, where $\varphi$ is Euler's totient ...
- 15.6k
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Sum of $\frac{1}{(\delta_1,\delta_2)}$ with congruence restrictions
In the course of my work, I encountered the following sum ($(x,y)$ stands for the GCD of $x$ and $y$):
$$L(Q)=\sum_{\substack{\delta_1,\delta_2\leq Q\\\delta_1\equiv0\ (a)\\\delta_2\equiv0\ (b)}}\frac{...
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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 ...
- 1,951
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510
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"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(...
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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 $f$ ...
- 6,969
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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 ...
- 1,990
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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 ...
- 20.2k
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210
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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 ...
- 1,990
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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 ...
- 1,381
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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/...
- 1
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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 ...
- 6,969
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384
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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+\...
user147650
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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
$$F_{...
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135
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Question regarding proof of Mertens' estimates in Montgomery-Vaughan's "Multiplicative number theory"
I have been trying to read Theorem 2.7 of Montgomery-Vaughan's "Multiplicative number theory" volume 1, and there is an issue I have run into: in the proof of subpart (e), when they write
$$\...
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277
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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
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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
$$F(s) = \prod_{...
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Does every multiplicative function have a logarithmic average?
Is it true that for every completely multiplicative function $f:\mathbb N\to\mathbb C$
with $|f(n)|=1$ for all $n$, the logarithmic average
$$\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac1nf(n)\...
- 1,701
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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(...
- 7,193
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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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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 ...
- 658
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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 ...
- 14.6k
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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 $$\...
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Math software / language for analytic / multiplicative number theory
What software/languages are the most used to do computations in analytic / multiplicative number theory?
I use Python, Maple, etc., but each time I want to compute expressions like, for $n$ with an ...
- 587
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Is it possible to find all integer functions which satisfy $f(m!+n!)\mid f(m!)+f(n!)$ and $m+n \mid f(m)+f(n)$?
I'm interesting to know more about multiplicative property of integer functions then I'd like to ask this humble question:
Question:
Is it possible to find all integer functions which satisfy $f(m!+...
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A question about $(0,1]$-valued multiplicative functions
Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means
$$
\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 $\...
user40023
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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-...
- 9,114
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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 ...
- 105k
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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 $k$ ...
- 9,114
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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 ...
- 9,114