Why shouldn’t this prove the Prime Number Theorem? [on hold]

Denote by $ \mu$ the Mobius function. It is known that for every integer $ k>1$ , the number $ \sum_{n=1}^{\infty} \frac{\mu(n)}{n^k}$ can be interpreted as the probability that a randomly chosen integer is $ k$ -free.

Letting $ k\rightarrow 1^+$ , why shouldn’t this entail the Prime Number Theorem in the form

$ $ \sum_{n=1}^{\infty} \frac{\mu(n)}{n}=0,$ $

since the probability that an integer is “$ 1$ -free” is zero ?

The prime spectrume of integral-valued polynomial ring

Let $ D $ be an integral domain with quotiont field $ K $ and let $ Int (D) $ be the set of all integral-valued polynomials on $ D $ , that is, $ Int (D):=\{f \in K[x]\mid f (D) \subseteq D\} $ . The structure of prime spectrum of $ Int (D) $ is well-known for a Noetherian local one dimensional domain. Is there any characterization for the prime spectrum of $ Int (D) $ without Noetherian local one dimensional assumptions? Thanks for any help.

A strange sequence of numbers that reversed are prime

$ 1, 17, 751, 967, 32953,…$ is a sequence that I invented.

The first term of the sequence divides $ 10^1+12345679$ The second term divides $ 10^{17}+12345679$ .

The n-th term of the sequence a(n) divides $ 10^{a(n)}+12345679$ .

Apart the first term, if you reverse the other four terms of the sequence, you have a prime. I don’t know if other terms of the sequence could be calculated. $ 12345679=37\cdot 333667$ . Is this sequence present in Oeis and has some importance? Do you believe that if other terms exist, reversed are primes?

Safari won’t play Amazon Prime Video on my iMac?

If I try to play videos on my iMac using Safari it won’t play, but it plays fine using Chrome. I’m using macOS High Sierra and I’m running Safari version 12.1.

When I try using Safari I get a black screen with a message as shown below.

                           enter image description here

Can anyone tell me what I need to do to get Safari to play videos?

I don’t think I had Silverlight installed but I don’t know how to tell. I went ahead and installed version 5.1 of Silverlight but again I can’t tell if its installed. When I go to Safari preferences and click on security I get the pane shown below.

enter image description here

Is the language { | p and n are natural numbers and there’s no prime number in [p,p+n]} belongs to NP class?

I was wondering if the following language belongs to NP class and if its complimentary belongs to NP class:

\begin{align} C=\left\{\langle p,n\rangle\mid\right.&\ \left. p \text{ and $ n$ are natural numbers}\right.\ &\left.\text{ and there’s no prime number in the range}\left[p,p+n\right]\right\} \end{align}

I am not sure, but here’s what I think: for each word $ \langle p,n\rangle \in C$ we know that the word belongs to C because there exists a primal certificate – an nontrivial divisor to any of the numbers between $ [p,p+n]$ , though I am not really sure it is in NP.

regarding the complement: I think it is in NP because the compliment compositeness can be decided by guessing a factor nondeterministically. But again I am not so sure about it and I don’t know how to correctly prove and show it.

Would really appreciate your input on that as I am quite unsure and also checked textbooks and internet (and this site) about it.