All Questions
Tagged with liouville-function riemann-zeta
5 questions
2
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0
answers
103
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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 ...
- 565
3
votes
1
answer
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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)...
- 356
3
votes
1
answer
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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 $$ \lambda(n) ...
- 1,756
1
vote
1
answer
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The Dirichlet series for the Liouville function related to the Riemann zeta function
$$\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 $...
- 107
2
votes
1
answer
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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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