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).

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1answer
64 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 $...
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1answer
36 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/...
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1answer
40 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
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1answer
140 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
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2answers
202 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. ...
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0answers
85 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
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1answer
121 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 ...
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0answers
39 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)=\...
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0answers
147 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 ...
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1answer
38 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 ...
5
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1answer
261 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 ...
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1answer
59 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 ...
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1answer
329 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
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4answers
261 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
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0answers
311 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
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1answer
352 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.
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1answer
242 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 ...