Agreed, as Thierry and Tobias say, there are too many recommendations for punditry.

Ouch, I thought at first that this reduced to RH but now I'm not so sure. It would also help if it were translated into English: my dictionary lacks "plz" for instance.

The answer to your question is "yes": $L^p(\mu)$ the set of random variables $X$ on the probability space with probability measure $\mu$ with the property that $E(|X|^p)$ is finite, is very well-studied, in probability theory and in functional analysis.

Right, so you are dealing with a space $L^2(\mu)$ where $\mu$ is a probability measure (why didn't you just say so?). But I still cannot see what your question is. (You say "frequently assumed that" - by whom? - I don't usually assume that a given random variable has zero expectation.)

I can't make head nor tail of this. Random variables (sometimes) have expectations, but vector spaces, to my knowledge, usually don't.

One of these is sometimes called the multiplicative valuation and the other the additive valuation.

$$11/3-3(7/9)=4/3$$ and $\phi(4/3)<\phi(7/9)$.

Pretty much any arithmetic property of $\mathbf{Z}[1/p]$ is readily deduced from the corresponding property of $\mathbf{Z}$. For instance, a Euclidean function is $\phi(p^r a)=|a|$ for $a\in\mathbb{Z}$ not divisible by $p$.

So, it isn't an actual title, and so this reply is not an answer to the original question.

So the question reduces to "is Cahit's claimed proof correct?". Questions of this kind often result in unresolvable disagreement: I vote to close.