All Questions
Tagged with complex-analysis riemann-zeta
524
questions
7
votes
1
answer
711
views
What's the easiest way to prove the Riemann Zeta function has any zeros at all on the critical line?
I learned in my intro to complex analysis class that the Riemann Zeta has trivial zeros at the negative even integers; this follows from the Riemann Zeta Reflection identity and the behavior of the $\...
- 1,290
0
votes
1
answer
35
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Termwise Differentiation of Convergent Series Involving the Riemann Zeta Function
Let $\zeta(s)$ be the Riemann zeta function. We know that
$$\zeta(s) = \prod_p(1 - p^{-s})^{-1}, \ \operatorname{Re} s > 1$$
in the sense that the righthand side converges absolutely to $\zeta(s)$ ...
- 4,429
1
vote
1
answer
94
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About the $n$th derivative of the Riemann zeta function on positive even integers
I know there exist a formula for the Riemann zeta function on positive even integers involving Bernoulli numbers.
Do there exist any closed form for the $n$th derivative of the Riemann zeta function ...
- 43
0
votes
0
answers
32
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Stein Complex Analysis Proof of Chapter 6 Proposition 2.5
In Stein's Complex Analysis book, within the proof of Chapter 6 Proposition 2.5, the following claim is made:
For $s = \sigma + it \in \mathbb{C}, n \geq 1$, then
$$\left| \frac{1}{n^s} - \frac{1}{x^s}...
1
vote
0
answers
95
views
Integral involving the zeta function along the critical line
WolframAlpha claims that
$$\int\limits_{-\infty}^{\infty}\frac{|\zeta(\frac{1}{2}+it)|(3-\sqrt{8}\cos(\ln(2)t))}{t^2 +\frac{1}{4}}dt=\pi\ln(2)$$
and I don't know how to show this. I tried removing the ...
- 290
1
vote
1
answer
82
views
May I find $\zeta(-1)$ using the Hankel formula for $\zeta$, but not the reflection formula?
The Reimann zeta function for $\Re(s) > 1$ is
$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s},$$
and we may show that the analytic continuation of $\zeta$ onto $\Re(s) \leq 1, s \neq 1$ is
$$\zeta(s)...
- 3,940
0
votes
1
answer
82
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Estimation of the absolute value of the $n$th non-real zero of the Riemann zeta function
Recently, I have been studying the oringinal proof of the prime number theory by Hadamard. I didn't get it on the estimation of the absolute value of the $n$th non-real zero of the $\zeta$ function by ...
0
votes
0
answers
71
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the zero's of $f(s,a) = \sum_{n=1}^{a-1} n^{-s} $
I was looking at the zero's of
$$f(s,a) = \sum_{n=1}^{a-1} n^{-s} $$
for integer $a>3$ in the strip $0 < \operatorname{Re}(s) < 1$.
Now this clearly relates to the Riemann zeta:
$$f(s,a) + \...
- 16.4k
1
vote
0
answers
33
views
maximal continuation of $\Pi_2(x)$
Consider the functions for $k\in \Bbb N$
$$ \Pi_k(x) := \sum_{n \in \Bbb N} e^{\frac{\log^k n}{\log x}} $$
$\Pi_1(x)$ converges for real $1/e<x<1$.
$\Pi_1(x)$ is a Riemann zeta function i.e. $\...
- 754
1
vote
1
answer
62
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Proving that $\left|\sum_{n<x}\mu(n)\right|\ll x\exp(-c\sqrt{\log x})$ for some $c>0$
Assume that for $\sigma\ge 1-\frac{1}{(\log(2+|t|)^2}$ we have $$|\zeta(\sigma+it)|\gg\frac{1}{(\log(2+|t|))^2}.$$ Using Perron's formula and moving the line of integration to $\textrm{Re}(s)=1-\frac{...
- 135
0
votes
1
answer
36
views
Proving $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and $t\ge 10$
Prove that $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and $t\ge 10$.
The solution given by my lecturer is as follows. Recall the approximate formula for zeta, given by $$\zeta(s)=\...
- 135
0
votes
0
answers
35
views
Estimates of the derivatives of $\Xi(s)$
The $\Xi$ Function is defined by $\Xi(s)=\xi(\frac{1}{2}+is)$, where $\xi(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$.
This is a problem from my homework: since we can write it ...
- 81
1
vote
1
answer
91
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The proof that the real zeta has the same domain as the complex valued zeta
I’m reading Havil’s book, Gamma. In Appendix E on p. 249 he has a proof that the complex valued zeta function has the same domain as the real valued zeta function.
He basically shows that $\zeta(x) = \...
- 437
4
votes
1
answer
105
views
How is $\zeta(0)$ actually $-1/2?$
I was searching around the forum, and I came across: $$\zeta(0) = \lim_{s\to 0}\, 2^{s-1} \pi^s \cdot \frac{\sin(\pi s/2)}{\pi s/2} \cdot \Gamma(1-s) \cdot s\zeta(1-s) = 2^{-1} \pi^0 \cdot 1 \cdot \...
0
votes
1
answer
85
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Understanding the proof of Theorem 10.2 in Montgomery & Vaughan's Multiplicative Number Theory
In Theorem 10.2 of the book of Montgomery & Vaughan's Multiplicative Number Theory there are two claims comes without any explanation:
1- For $0 < u < \infty$, $(u + a)^{s−1} ≪ |a|^{σ−1}$ ...
- 281