All Questions
Tagged with analytic-number-theory riemann-zeta
458
questions
-3
votes
0
answers
19
views
what is the value of d/ds (integral(0,infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt) [closed]
I was reading about the Riemann zeta function and found the integral form of it ζ(s)= 1/(s-1) + 1/2 + 2(integral(0, infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt). When I was trying to ...
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
1
vote
1
answer
55
views
$H(x)$ approximates $\pi(x)$ pretty well. But what are the drawbacks, when compared with Riemann's $R(x)$?
I'm aware of the Gram series which is equivalent to $R(x)$ (Riemann prime counting function):
$$ R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n}). $$
Over the interval $x=2$ to $x=10^4$ the average ...
- 754
1
vote
1
answer
63
views
Write the sum in terms of the Riemann zeta function
I believe it is a question from JHMT.
Write the sum $\sum_{a=1}^\infty\sum_{b=1}^\infty\frac{gcd(a,b)}{(a+b)^3}$ in terms of Riemann zeta function. The answer should be $-Z(2)+\frac{Z(2)^2}{Z(3)}$...
- 17
2
votes
0
answers
65
views
Dedekind zeta function of abelian number fields is a product of Dirichlet L-functions
Can somebody give an explicit (and self-contained) proof of the fact that the Dedekind zeta function of an abelian number field $K$ is a product of Dirichlet L-functions? I.e.,
$$
\zeta_K(s)=\prod_{\...
- 787
4
votes
0
answers
127
views
If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then $b\neq p^k$
If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then prove that $b\neq p^k$ where $p$ is any prime and $k\in\mathbb{N}$
Take $a,b\in\mathbb{N}$ such that $(a,b)=1$. Now if $b=p^k$ then $$p^k\zeta(5)=a$$ So, by ...
- 928
1
vote
0
answers
92
views
Inequalities for the Dirichlet eta function at non-trivial Riemann zeta zeros.
I am interested in these inequalities for sufficiently large $n$:
$$\Large \left(\Re\left(\sum _{k=1}^n (-1)^{k+1} x^{\log (k) c}\right)\right)^2 \leq \left( \Re\left(x^{\log \left(n+\frac{1}{2}\right)...
- 7,438
4
votes
1
answer
140
views
Closed form of $\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))}\, dx\,dy\,du\,dv\,dw$
I need closed form for the integral $$I:=\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))} \, dx\,dy\,du\,dv\,dw$$
Then $$0<I<\int_{(0,1)^5} \frac{1}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)...
- 928
4
votes
1
answer
118
views
$3\frac{\zeta'(\sigma)}{\zeta(\sigma)}+4 \Re \frac{\zeta'(\sigma+i t)}{\zeta(\sigma+i t)}+\Re\frac{\zeta'(\sigma+2 i t)}{\zeta(\sigma+2it)}\leq0$
In our script it is used without proof that
For $\sigma>1$ and $t \in \mathbb{R}$
$$
3 \frac{\zeta^{\prime}(\sigma)}{\zeta(\sigma)}+4 \Re \frac{\zeta^{\prime}(\sigma+i t)}{\zeta(\sigma+i t)}+\Re\...
- 2,308
0
votes
1
answer
82
views
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
1
answer
60
views
Lower bound for the prime zeta function
The prime zeta function is defined for $\mathfrak{R}(s)>1$ as
$P(s)=\sum_{p} \ p^{-s}$, where $p \in \mathbb{P}$.
It is well-know this series converges whenever $\mathfrak{R}(s)>1$.
Now, ...
- 85
1
vote
1
answer
62
views
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 $$\zeta(s)=\...
- 135
0
votes
1
answer
85
views
Understanding the proof of Theorem 10.2 in Montgomery & Vaughan's Multiplicative Number Theory
In Theorem 10.2 of the book of Montgomery & Vaughan's Multiplicative Number Theory there are two claims comes without any explanation:
1- For $0 < u < \infty$, $(u + a)^{s−1} ≪ |a|^{σ−1}$ ...
- 281
1
vote
0
answers
42
views
Convergence of Riemann zeta function [duplicate]
I am wondering whether the series $$\zeta(s) = \sum_{n=1}^\infty n^{-s}$$ converges for $s$ with $\mathsf{Re}(s)=1$ and $\mathsf{Im}(s) \neq 0$. Note that I use the representation of $\zeta$ as an ...
- 1,803