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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Visualizing the Pólya-conjecture (Liouville function)
It might be a better idea to post this on cs, but it is math-related...
So the, or rather one Pólya-conjecture is that for all $n > 0$ integers
$$ L(n) = \sum_{k = 1}^n \lambda(k) \leq 0 $$
where $\...
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Liouville Lambda Function and Riemann Hypothesis
What is the exact statement involving the Liouville Lambda function, which is equivalent to Riemann Hypothesis, and true iff RH is true? Can anyone cite the sources for it and/or outline its proof in ...
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I dont understand the last step. I’m trying to understand how equation 10 follows, especially the last delta equation [closed]
I dont understand following steps of a solution where I need to find the Normalization constant $A(E,P,N)$ .
The normalization is given by:
$$
\int \rho(\vec x)d\vec x = 1
$$
where $d \vec x = C_N d^...
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Why do fractal-like patterns appear in this sequence?
I came across this sequence called Digital River, where the next number in the sequence is defined as the sum of the digits of the previous number, plus, the previous number itself.
It caught my ...
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Turán proof that constant sign of Liouville function implies RH
In Mat.-Fys. Medd. XXIV (1948) Paul Turán gives what he says is a proof of the statement that if the summatory $L(x) = \sum_{n\leq x} \lambda(n)$ of the Liouville function $\lambda(n) = (-1)^{\Omega(n)...
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Values of the Liouville function
Let $\Omega: \mathbb{N} \to \mathbb{N}\cup\{0\}$ be the function which counts how many prime factors a number has, with multiplicity. For example, $\Omega(380) = 4$, $\Omega(108)= 5$. More generally, ...
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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 $$ \lambda(n) ...
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Inclusion-exclusion formula for the Liouville Lambda function.
The Riemann hypothesis is equivalent to:
$$\lim_{n\to \infty } \, \frac{\sum\limits_{k=1}^n \lambda (k)}{n^{\frac{1}{2}+\epsilon}}=0$$
according to "The Riemann Hypothesis: A Resource for the ...
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The Dirichlet series for the Liouville function related to the Riemann zeta function
$$\sum_{n=1}^{\infty} \frac{λ(n)}{n^s}=\frac{ζ(2s)}{ζ(s)}$$
Let $λ(n) = (−1)^k$, where $k$ is the number of prime factors of $n$, counting multiplicities. (Liouville function)
for $Re(s)>1$, where $...
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$\sum_{d\mid n}\lambda(d)\sigma(d)=n\lambda(n)\sum_{d^2\mid n}\frac{1}{d^2}$ Solution
Recall that the Liouville function $\lambda$ and $\sigma$ are multiplicative, and the product of multiplicative functions is also multiplicative, thus $\lambda\sigma$ is multiplicative and therefore ...
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There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1)=\lambda(n+2) = +1$;
Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with ...
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Is a bounde entire function of exponential type constant on a any compact?
Let $\phi$ be an entire function of exponential type. That is:
$$\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)|\...
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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 ...
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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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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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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 \...
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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 ...
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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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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.)
...
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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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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 $$M(n)=\...
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Minimal value of summatory function of completely multiplicative functions taking values -1 and 1
Here is a very nice paper http://www.ams.org/journals/tran/2010-362-12/S0002-9947-2010-05235-3/S0002-9947-2010-05235-3.pdf which led me to thinking about the problems below.
Define the Liouville ...
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What about $x\frac{f'(x)}{f(x)}=-\frac{1}{2}\left(\vartheta_3(x)-1\right)$, where $\vartheta_3(x)$ is Jacobi theta function?
If we define for $0<x<1$ $$f(x):=\prod_{n=1}^\infty\left(1-x^n\right)^{\frac{\lambda(n)}{n}},\tag{1}$$ where $\lambda(n)$ is the Liouvile function (and notice is the similar infinite product ...
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Invert: $\sum\limits_{d|n} \mu(d) \lambda(d)=2^{\omega(n)}$
Inverting $\displaystyle\sum_{d|n} \mu(d) \lambda(d)=2^{\omega(n)}$ into $\displaystyle\sum_{d|n} \lambda(n/d) 2^{\omega(d)}=1$ ,where $n \geq1$, by using Mobius Inversion Formula.
I'm able to solve ...
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How quickly can this function be computed?
I can show that $\lambda (n)=i^{\tau(n^{2})-1}$, where $\lambda (n)$ is the Liouville function, $\tau(n)$ is the divisor function, and $i$ is the imaginary unit.
My question is as stated, and what is ...
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Liouville's Theorem Applications
Suppose $a,b>0$ are contants and $F$ is a non-constant function such that $F(z+a)=F(z)$ and $F(z+ib)=F(z)$. Prove $F$ is not analytic in the rectangle $0\leq a \leq b$ and $0\leq y \leq b$
I don't ...
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Computing the first $n$ values of the Liouville function in linear time
Is it possible to compute the first $n$ values of the Liouville function in linear time? Since we need to output $n$ values we clearly cannot do better than linear time, but the best I can figure out ...
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What is the origin of this Riemann Hypothesis equivalent involving the Liouville function?
Peter Borwein (in his 2006 book The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, p. 6) provides an equivalence between the Riemann Hypothesis and this conjecture involving ...
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$\sum_{n=1}^N\lambda(n)[N/n]=[\sqrt{N}]$ Identity involving Liouville Lambda function
I have to prove $$\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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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 ...
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