All Questions
Tagged with analytic-number-theory zeta-functions
81
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 ...
1
vote
0
answers
57
views
Periodic zeta function
Let $e(x)=e^{2\pi ix}$ and let $$F(x,s)=\sum _{n=1}^\infty \frac {e(nx)}{n^s}$$ be the periodic zeta function.
What is the functional equation for the periodic zeta function ?: I can find a statement ...
- 1,662
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
2
answers
188
views
Proper Way to Calculate Value of Riemann Zeta function?
I understand that an Analytic Continuation of a function will extend its domain into areas that it previously wasn't defined in.
I've been looking at one of the Analytic Continuations of the Zeta ...
1
vote
1
answer
101
views
An infinite series of powers of fractions.
The series at hand is given by,
$$\sum_{k=1}^{\infty} \left(\frac{2k+1}{k (k+1)}\right)^s$$
I think it converges for $s>1$, but I have not been able to derive a general expression for this series.
...
- 33
1
vote
0
answers
46
views
Is there a useful/meaningful notion of a multi-variable L-function in number theory?
I recently encountered multi-variable generalizations of various classical zeta functions. For example, the multi-variable Riemann zeta function
$$
\zeta(s_1, \ldots , s_r) := \sum_{0 < n_1 < \...
- 470
2
votes
1
answer
103
views
$|\zeta(1/2 + it)|^2 \geq \frac{\log(t)}{\log \log(t)}$
I've tried to solve this exercise for hours but I didn't managed to figure it out.
Show that there exists a sequence $t \to \infty$ for which
$$|\zeta (1/2 + it)|^2 \geq \frac{\log(t)}{\log\log(t)}$$
...
- 1,374
1
vote
1
answer
78
views
$\frac 1{2\pi i}\int_{2-\infty i}^{2+\infty i}\frac{x^s}{s^2}\left(-\frac{\zeta^{\prime}(s)}{\zeta(s)}\right)ds$ as a finite sum of $\Lambda(n)$
[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 13, question 7]
Express
$$\frac 1{2\pi i}\int_{2-\infty i}^{2+\infty i} \frac{x^s}{s^2}\left(-\frac{\zeta^{\prime}(s)}{\zeta(s)}\right)...
- 9,592
1
vote
0
answers
78
views
For arithmetical periodic function $f$, if $\sum_{r=1}^k f(r)=0$, then $S=\sum_{n=1}^\infty \frac{f(n)}{n^{s}}$ converges
[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 12, question 1(b)]
Let $f(n)$ be an arithmetical function which is periodic mod $k$. If
$$\sum_{r=1}^k f(r)=0$$
then prove that the ...
- 9,592
0
votes
0
answers
62
views
"Mollifier" of the Dirichlet L-function
I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
3
votes
1
answer
111
views
Zeros of Riemann's $\xi(t)$
In Riemann's paper he defined $\xi(t)=\Pi(\frac{s}{2})(s-1)\pi^{-\frac{s}{2}}\zeta(s)$, where $s=\frac{1}{2}+ti$. On page 4 he said:
The number of roots of $\xi(t)=0$, whose real parts lie between $0$...
- 91
1
vote
0
answers
58
views
A question from Titchmarsh's book " The Theory of the Riemann zeta function, Theorem 9.16, page 231
I am studying about upper bounds for $N(\sigma, T)$ (zero-density estimates), and while going through Theorem 9.16 in Titchmarsh's book (2nd edition) (page 231), I got a bit stuck in understanding the ...
- 39
6
votes
1
answer
126
views
A few questions about Riemann's Main Formula in the paper On the Number of Primes Less Than a Given Magnitude
Sorry for asking multiple questions these days about the same topic, but the thing is I was doing a school project about Riemann's zeta function so I kind of suffered when reading Riemann's paper On ...
- 101
10
votes
1
answer
280
views
Why do number theorists care so much about how well $\text{Li}(x)$ approximates $\pi(x)$ if it's not our best approximation?
An alleged primary motivator for the RH is so that we can bound the error term $|\text{Li}(x) - \pi(x)|$ by a factor of $O(\sqrt{x}\log x)$. However, I also learned about Riemann's explicit formula $R(...
- 681
0
votes
1
answer
50
views
modified riemann zeta function $\zeta ^*(s)$?
I remember there being a function $\zeta ^*(s)$ where
$$\zeta ^*(s)=\zeta (s), \ s\neq 1$$
$$\zeta ^*(1)=\gamma$$
but now I can't seem to find any record of it, does a function like this exist or am I ...
- 359