Tagged Questions
0
votes
4answers
74 views
Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent? [duplicate]
Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?
6
votes
1answer
191 views
how to understand $\log\zeta(s)$ (Riemann zeta function)?
I know that if a function $f$ is analytic and has no zeros in a simple connected region, then we can define $\log{f}$ making it analytic in that region.
Let's assume $Re(s)>1$.
Is $\zeta(s)$ ...
7
votes
2answers
213 views
Intuitive explanation with rigorous details why zeta has infinitely many zeros?
I have seen a proof outline that $\zeta$ has infinitely many zeros on the critical line here but what I really want is:
Simplest possible (least "magic") argument that explains why zeta has ...
5
votes
2answers
206 views
Why does zeta have infinitely many zeros in the critical strip?
I want a simple proof that $\zeta$ has infinitely many zeros in the critical strip.
The function $$\xi(s) = \frac{1}{2} s (s-1) \pi^{\tfrac{s}{2}} \Gamma(\tfrac{s}{2})\zeta(s)$$ has exactly the ...
15
votes
1answer
393 views
Books about the Riemann Hypothesis
I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. Here is my list:
The Riemann Hypothesis: A Resource ...
1
vote
2answers
113 views
Sum of Stieltjes constants
Does anyone know of any papers or resources dealing with the following question: For which values of $s=\sigma+it$ does the following sum of Stieltjes constants hold,
...
5
votes
1answer
180 views
Derivative of the Riemann zeta function for $Re(s)>0$.
The Riemann zeta function can be analytically continued to $Re(s)>0$ by the infinite sum
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}.$$
Can we differentiate this with ...
1
vote
1answer
130 views
Two Representations of $\log \zeta$
I was looking for representations of $\log \zeta$ and found these two:
$ \displaystyle \log\zeta(s)=\color{red}{s}\sum_{n>0} \frac{P(ns)}{n\color{red}{s}}$ from here [$\color{red}{s}$ inserted ...
3
votes
3answers
275 views
Other functional equations for $\zeta(s)$?
For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ ...
11
votes
1answer
216 views
What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?
We have the following evaluations:
$$\begin{aligned}
&\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\
&\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
20
votes
5answers
815 views
What is so interesting about the zeroes of the $\zeta$ function
The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$
The ...
6
votes
2answers
577 views
Calculating the Zeroes of the Riemann-Zeta function
Wikipedia states that
The Riemann zeta function $\zeta(s)$ is defined for all complex numbers $s \neq 1$. It has zeros at the negative even integers (i.e. at $s = −2, −4, −6, ...)$. These are ...
9
votes
2answers
316 views
How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?
I mean the Laurent series at $s=1$.
I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$,
based on the integral formula given in Wikipedia. ...
1
vote
1answer
226 views
about the riemann zeta function and the prime counting function
i have posted this question on MO, and they referred me to post here .
one starts with the formal definition of zeta :
$$\displaystyle \zeta (s)=\prod_{p}\frac{1}{1-p^{-s}} $$
then :
$ \ln(\zeta ...
24
votes
1answer
845 views
Are these zeros equal to the imaginary parts of the Riemann zeta zeros?
The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$:
$$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot e^{-\frac{x}{2}})$$
can be plotted with ...
4
votes
1answer
150 views
Zeta function identity
How does one prove the zeta function identity
$$\sum_{s=2}^{\infty}\left(1-\sum_{n=1}^{\infty}\frac{1}{n^s}\right)=-1 \;?$$
19
votes
1answer
452 views
Upper bound on differences of consecutive zeta zeros
The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...
2
votes
1answer
149 views
Dirichlet character over Riemann zeta function
Let $\chi$ be a Dirichlet character mod q and let $$L(s,\chi)=\sum_{n\leq x} \frac{\chi(n)}{n^s}.$$ What is the value of $\displaystyle\lim_{s \rightarrow 1} \frac{L(s,\chi)}{\zeta(s)}$ for principal ...
8
votes
1answer
138 views
Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function.
Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
0
votes
0answers
143 views
Has it been ruled out that the Riemann hypothesis fails for only finite number of zeros?
Has it been ruled out that the Riemann hypothesis fails, but fails only for finite number of zeros?
1
vote
1answer
64 views
Another question about proof that $\zeta(s) \neq 0$ for $\Re(s) = 1$
This is a question distinct from but related to the question I wrote here: Question about proof that $\zeta(s) \neq 0$ for $\Re(s) = 1$, so assume the same things that I wrote there.
The paper then ...
4
votes
1answer
136 views
Question about proof that $\zeta(s) \neq 0$ for $\Re(s) = 1$
I'm following this paper: http://mathdl.maa.org/images/upload_library/22/Chauvenet/Zagier.pdf
Define $\Phi(s) = \displaystyle\sum_p \frac{\log p}{p^s}$.
By taking a logarithm and differentiating ...
5
votes
0answers
169 views
How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
5
votes
2answers
378 views
An identity involving the Möbius function
$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$
for $s>1$.
How do I prove this identity?
2
votes
1answer
202 views
Why are Gram points for the Riemann zeta important?
Given the Riemann-Siegel function, why are the Gram points important? I say if we have $S(T)$, the oscillating part of the zeros, then given a Gram point and the imaginary part of the zeros (under the ...
10
votes
2answers
309 views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...
11
votes
2answers
601 views
Logarithmic derivative of Riemann Zeta function
Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$?
I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes
...
5
votes
2answers
515 views
Why does the Riemann zeta function have zeros in the complex plane? How is it possible to find them?
I ask this because, according to Euler's product formula, Riemann's zeta function =(1/something), so how could that be zero?
Also, how could one find zeros that are on the negative side and find a ...
7
votes
1answer
221 views
Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$
It seems that the $n$th cumulant of the uniform distribution on the interval $[-1,0]$ is $B_n/n$, where $B_n$ is the $n$th Bernoulli number.
And also $-\zeta(1-n) = B_n/n$, where $\zeta$ is Riemann's ...
8
votes
2answers
388 views
Values of the Riemann Zeta function and the Ramanujan Summation - How strong is the connection?
The Ramanujan Summation of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $$\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 ...
8
votes
2answers
430 views
Trig identity to zeta function identity?
The inequality
$$\zeta(s)^3 | \zeta(s + it)^4 \zeta(s + 2it)| \ge 1$$
follows from
$$3 + 4 \cos(\theta) + \cos(2 \theta) \ge 0$$
How is that done? What is the relationship between zeta and the ...
16
votes
4answers
2k views
Riemann zeta function at odd positive integers
Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the ...
3
votes
3answers
563 views
On Zeta function zeros in the critical strip
I have been reading about Riemann Zeta function and have been thinking about it for some time.
Has anything been published regarding upper bound for the real part of zeta function zeros as the ...
