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$\alpha^z=e^{z\log\alpha}$ is multi-valued. Now I am confused with the definition of Riemann zeta function: $$\zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s}, s=\sigma+it$$ because $$n^s=e^{s(\log n+i2k\pi)}$$ where $\log n $ is the natural logarithm, then, Is $\zeta(s)$ a multi-valued function? or, we should think $n^s=e^{s\log n}$ ? many thanks! |
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There is no problem as long as you choose a definite branch of the complex logarithm. As far as I know it is customary always to assume the standard branch (remove the whole non-positive real axis), with $\,\arg\, r=0\,$ for positive real numbers, as any other branch only multiplies the sum by the constant $\,e^{-2s\pi i}\,$ , which doesn't seem to be very interesting for most purposes... |
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Yes, You should specify the branch of the logarithm. The principal branch is the one with a logarithm of a real number remaining real. Consider equation $$ n=e^{\log n + i2k \pi} $$ It really says that the logarithm of a natural number $n \in \mathbb{N}$ is $\log n + i2k \pi$. Here $\log n$ is the "usual" logarithm :D This means, $\log n$ is the value of the logarithmic function on the principal branch, where the logaritm of a real number remains real. That is, $\log_{principal} : \mathbb{R} \to \mathbb{R}$. Conversely, this is also a definition of a principal branch. As You can see, You intuitively knew this, because You already understood that $\log n$ in Your second equation is real. However, $\log n$ in all of these equations can be evaluated on any branch, of course, and $k$ can be any integer. If You are into this, I suggest checking some literature on the Riemann's sphere and Riemann's surfaces. I guess Wikipedia has some info on it. For instance: http://en.wikipedia.org/wiki/Riemann_surface#Examples. This may elucidate the concept of branches and branch points pairs a bit better. Regards. |
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