Questions tagged [liouville-function]

Problems including the Liouville function, $\lambda(n)$, which is equal to $(-1)^k$, where $k$ is the number of prime factors of $n$ (with multiplicity).

Filter by
Sorted by
Tagged with
3
votes
1answer
146 views

Dirichlet transform of $e^{(2 \pi i / 3) \Omega(n)}$

The Dirichlet transform of the Liouville function $\lambda(n)$ is famously $$ \sum_{n=1} \frac{\lambda(n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)}\tag{1}$$ The Liouville function is defined by $$ \lambda(n) ...
0
votes
0answers
30 views

Inclusion-exclusion formula for the Liouville Lambda function.

The Riemann hypothesis is equivalent to: $$\lim_{n\to \infty } \, \frac{\sum\limits_{k=1}^n \lambda (k)}{n^{\frac{1}{2}+\epsilon}}=0$$ according to "The Riemann Hypothesis: A Resource for the ...
0
votes
1answer
107 views

The Dirichlet series for the Liouville function related to the Riemann zeta function

$$\sum_{n=1}^{\infty} \frac{λ(n)}{n^s}=\frac{ζ(2s)}{ζ(s)}$$ Let $λ(n) = (−1)^k$, where $k$ is the number of prime factors of $n$, counting multiplicities. (Liouville function) for $Re(s)>1$, where $...
-1
votes
1answer
53 views

An entire function $f $ such that $f(x+iy)=f(3x+i2y).$ [closed]

Let $f(z)$ entire function suppose for any nonzero z satisfied the equation $f(z)=f(x+iy)=f(3x+i2y)$ , then f must be constant
3
votes
1answer
91 views

$\sum_{d\mid n}\lambda(d)\sigma(d)=n\lambda(n)\sum_{d^2\mid n}\frac{1}{d^2}$ Solution

Recall that the Liouville function $\lambda$ and $\sigma$ are multiplicative, and the product of multiplicative functions is also multiplicative, thus $\lambda\sigma$ is multiplicative and therefore ...
8
votes
1answer
167 views

There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1)=\lambda(n+2) = +1$;

Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with ...
-2
votes
1answer
55 views

Is a bounde entire function of exponential type constant on a any compact?

Let $\phi$ be an entire function of exponential type. That is: $$\exists M, c>0,\quad \forall z\in \mathbb{C}\quad |\phi(z)|\leq M e^{c|z|}. $$ On any compact set $|z|=R$, we have: $$|\phi(z)|\...
1
vote
0answers
67 views

prove you have found all such functions

find all possible entire functions f with the property that $|f(z)|\le2|z|+1$ for all $z\in C$. Prove that you have found all such functions. First of all I am self studying complex analysis so sorry ...
3
votes
1answer
233 views

Question on Divisor Sum over the Liouville Function $\lambda(d)=(-1)^{\omega(d)}$

This question assumes the following: $\nu(n)$ is the number of distinct primes in the factorization of $n$, $\omega(n)$ is the number of prime factors counting multiplicities in the factorization of $...
0
votes
1answer
87 views

Is $\lambda (n/d)$ is also multiplicative?

Let $\lambda$ denote the Liouville $\lambda $- function. We know that $\lambda$ is multiplicative if we define it for integers $n$. It is defined here: https://math.stackexchange.com/posts/3245975/...
0
votes
1answer
271 views

Liouville function and perfect square 2.

As a proof of the second part of part(b) of this question : Liouville function and perfect square I have the solution given below: But I can not see how this solution explains the case when $n =5 \...
2
votes
1answer
223 views

Questions on Convergence of Explicit Formulas for $f(x)=\sum\limits_{n=1}^x a(n)$ where $a(n)\in\{\left|\mu(n)\right|,\mu(n),\phi(n),\lambda(n)\}$

This question is a follow-on to my earlier question at the following link. What is the explicit formula for $\Phi(x)=\sum\limits_{n=1}^x\phi(n)$? This question pertains to the explicit formulas for ...
3
votes
2answers
245 views

What is the closed-form of $\sum_{n=1}^\infty\lambda(n)\log\cosh\frac{1}{n}$, where $\lambda(n)$ is the Liouville function?

Let $\lambda(n)$, for integers $n\geq 1$, be the Liouville lambda function, defined by $\lambda(n)=(-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime factors of $n$, counted with multiplicity. ...
1
vote
0answers
97 views

Generalisation of the Liouville function as irreducible representations for $(\mathbb{N},\cdot)$?

These are only going to be a soft questions. And I thought this question is also a case for MO, so I have posted a duplicate there (Does that comply with the etiquette here? In case not I am sorry.) ...
1
vote
1answer
130 views

Is it possible to deduce that the limit $\lim_{x\to-\infty}\sum_{n=1}^\infty\lambda(n)\frac{x^n}{\Gamma(n)}$ is finite?

Let $\lambda(n)$ for integers $n\geq 1$ the Liouville function, see its definition for example from this Wikipedia. And we denote with $\Gamma(n)$ the particular values of the gamma function over ...
2
votes
0answers
58 views

About a more efficient way of evaluating $L(n):=\sum_{k=1}^n\lambda(k)$, where $\lambda(n)$ is the Liouville function, than this definition of $L(n)$

Let for integers $n\geq 1$ the Möbius function $\mu(n)$, and $\lambda(n)$ the Liouville function (see the definition in this Wikipedia). We consider also the corresponding summary functions $$M(n)=\...
4
votes
0answers
166 views

Minimal value of summatory function of completely multiplicative functions taking values -1 and 1

Here is a very nice paper http://www.ams.org/journals/tran/2010-362-12/S0002-9947-2010-05235-3/S0002-9947-2010-05235-3.pdf which led me to thinking about the problems below. Define the Liouville ...
1
vote
1answer
50 views

What about $x\frac{f'(x)}{f(x)}=-\frac{1}{2}\left(\vartheta_3(x)-1\right)$, where $\vartheta_3(x)$ is Jacobi theta function?

If we define for $0<x<1$ $$f(x):=\prod_{n=1}^\infty\left(1-x^n\right)^{\frac{\lambda(n)}{n}},\tag{1}$$ where $\lambda(n)$ is the Liouvile function (and notice is the similar infinite product ...
3
votes
1answer
568 views

Invert: $\sum\limits_{d|n} \mu(d) \lambda(d)=2^{\omega(n)}$

Inverting $\displaystyle\sum_{d|n} \mu(d) \lambda(d)=2^{\omega(n)}$ into $\displaystyle\sum_{d|n} \lambda(n/d) 2^{\omega(d)}=1$ ,where $n \geq1$, by using Mobius Inversion Formula. I'm able to solve ...
1
vote
1answer
69 views

How quickly can this function be computed?

I can show that $\lambda (n)=i^{\tau(n^{2})-1}$, where $\lambda (n)$ is the Liouville function, $\tau(n)$ is the divisor function, and $i$ is the imaginary unit. My question is as stated, and what is ...
1
vote
1answer
413 views

Liouville's Theorem Applications

Suppose $a,b>0$ are contants and $F$ is a non-constant function such that $F(z+a)=F(z)$ and $F(z+ib)=F(z)$. Prove $F$ is not analytic in the rectangle $0\leq a \leq b$ and $0\leq y \leq b$ I don't ...
7
votes
4answers
306 views

Computing the first $n$ values of the Liouville function in linear time

Is it possible to compute the first $n$ values of the Liouville function in linear time? Since we need to output $n$ values we clearly cannot do better than linear time, but the best I can figure out ...
2
votes
0answers
406 views

What is the origin of this Riemann Hypothesis equivalent involving the Liouville function?

Peter Borwein (in his 2006 book The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, p. 6) provides an equivalence between the Riemann Hypothesis and this conjecture involving ...
4
votes
1answer
520 views

$\sum_{n=1}^N\lambda(n)[N/n]=[\sqrt{N}]$ Identity involving Liouville Lambda function

I have to prove $$\sum_{n=1}^N\lambda(n)[N/n]=[\sqrt{N}]$$ I tried using the approach in this question but I don't know how I'll get $\sqrt{N}$. Please help.
1
vote
1answer
367 views

Prove $\lambda(n)=\sum_{d^2|n}\mu(n/d)^2$ and $\mu^2(n)=\sum_{d^2|n}\mu(d)$

$\lambda(n)$= $\sum_{d^2|n}$ $\mu(n/d)^2$ and $\mu^2(n)$= $\sum_{d^2|n}$ $\mu(d)$ Having a little bit of trouble here.Can I use the fact that $\sum_{d|n}\lambda(n)$ is a characteristic function for ...