5
votes
1answer
101 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 ...
2
votes
1answer
43 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 ...
4
votes
4answers
190 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.
1
vote
2answers
91 views

Estimating the integrated Tchebychev function and calculating its error

I would like to understand how to derive (2) from (1) below. Problem: If $\psi_1$ is the integrated Tchebychev function below $$\psi_1(x)=\frac{1}{2\pi i} ...
2
votes
1answer
237 views

Riemann Zeta Function Manipulation

The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$ (i) show that for $Re z> 1$, we have $$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty ...
8
votes
1answer
169 views

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 ...
0
votes
0answers
139 views

Determining well-definedness for functions

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