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3

Valter's answer is completely correct, but I'll just briefly expand on it to address the specific values you ask about. The place to go, really, is the Wikipedia page Particular values of Riemann zeta function, which lists mosts of the values of $\zeta(s)$ (which, as Valter explained, equals $$\zeta(s) := \sum_{n=1}^{+\infty} \frac{1}{n^s}$$ when ...

6

The true fact is the following. Consider $$\zeta(s) := \sum_{n=1}^{+\infty} \frac{1}{n^s} \quad \mbox{with s\in \mathbb C and } Re \:s >1\:. \tag{1}$$ That function, with the said complex domain, is well defined (the series absolutely and uniformly converges) and is a complex analytic function. As a consequence of a well-known theorem on analytic ...

0

Holographic renormalization for non-conformal branes, i.e. non-AdS/non-CFT systems, was systematically developed in this paper by Kanitscheider, Skenderis and Taylor. They even work it out for the example of the Witten model, which is the background of the Sakai-Sugimoto model. The key principle that allows one to extend the formalism of holographic ...

1

In $d=4$ and $n=3$, you the following relation $w = 4 - E - V$, where: $w=$ supercifial degree of divergence $E=$ external legs $V=$ number of vertices So, if you replace the values of $d$ and $n$, you will get $w<0$, therefore you won't have $UV$ divergences in that diagram.

1

By starting from the Fermi theory and requirement of the tree-unitarity of this theory (it is similar to renormalizability, but only on a tree level) you may build theory of electroweak interactions (even with Higgs boson). I'm only show you how does it work. Fermi theory predicts growth the matrix element of neutrino-lepton scattering as $E^{2}$ (or ...

3

I think this basically sums up the program for what quantum gravity is. The modern viewpoint is that general relativity (and really just about any quantum field theory) is an effective field theory, and the full theory of quantum gravity must provide an ultraviolet completion. As explained in the Donoghue review suggested by bechira (another good review is ...

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