# Tagged Questions

1answer
68 views

### Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$

The result quoted in the title is usually a stepping stone in the proof of the prime number theorem and I am familiar with the usual argument for this result. The other day my professor was telling ...
1answer
32 views

### Continuation of the Riemann Zeta Function

I am actually aware of the argument showing $\zeta$ has a meromorphic extension to $\mathbb{C}$ with a single pole at $z = 1$. On a recent number theory exam, however, one of the questions asked to ...
0answers
74 views

### Zeta and gamma function relation (question about a paper) [closed]

So was Reading about some zeta function proof and relation to the gamma function. In order to proof the convergence and other things about the zeta function, they start with this theorem ("Teorema ...
4answers
188 views

### Showing that $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$

I need prove the following: ($\zeta(s)$ is the Riemann zeta function) $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$ I really don't know, i have tried, but nothing for now.
2answers
87 views

1answer
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### Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function.

Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
0answers
138 views

### Determining well-definedness for functions

How does one determine well-definedness in analytical continuation for $\Gamma(s)\zeta(s)$ function? Firstly: $$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$$ ...