4
votes
2answers
85 views

What is $\zeta(n)$ as $n$ tends to $\infty$? How fast it goes to the limit?

What is $\zeta(n)$ as $n\to\infty$? How fast it goes to the limit?
5
votes
2answers
104 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
4
votes
1answer
72 views

Analytic Continuation of Zeta Function using Bernoulli Numbers

In my complex analysis textbook by Stein and Shakarchi, as an exercise, I am supposed to extend $\zeta(s)$ to the entire complex plane using Bernoulli numbers, but I am stuck. I can prove that $$ ...
2
votes
0answers
39 views

Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
1
vote
1answer
44 views

Why doesn't the functional equation imply that $\zeta(s)=0$ for positive even integers?

The Riemann Zeta Function satisfies the functional equation $\zeta(s)=2^s\pi^{s-1}\sin\left(\dfrac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$. But when $s$ is a positive even number, $\sin\left(\dfrac{\pi ...
5
votes
0answers
77 views

Generating function of the squared Riemann zeta function

It's a well known fact that $$\sum_{k=2}^{\infty} \zeta(k) x^k=-x \psi(1-x)-x\gamma \space (|x|<1) $$ but I didn't meet yet a version for squared Riemann zeta function $$\sum_{k=2}^{\infty} ...
0
votes
1answer
39 views

Analytic continuation of Zeta type function

Can one analytically continue the function (Not equal to the Zeta function) $$Z(s)=\prod_{p}\frac{1}{1+p^{-s}}=\sum_{k=1}^{\infty}\frac{(-1)^{\Omega(k)}}{k^s}$$ Where $\Omega(k)$ is the number of ...
3
votes
1answer
44 views

Why does the imag. part of the graph of $\zeta(n^{ix})$ resemble the tangent function?

If you input $\zeta(n^{ix})$ into the Wolfram Alpha search bar, in the plot, you get an infinitely repeating sinusoidal curve, which resembles the real part, and you get an infinitely repeating ...
2
votes
1answer
36 views

Calculating the residues of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$

Calculating the poles of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$, where x is a fixed real number. I am trying to calculate the poles of this function at the trivial zeros of $\zeta$, ...
2
votes
3answers
87 views

$\zeta(2 + it) = \zeta(2-it)$

Let $\zeta(s)$ denote the Riemann zeta-function. Show that $\zeta(2 + it) = \zeta(2-it)$ for all real t. Give some hints how to do this one.Thanks in advance.
2
votes
1answer
80 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
1
vote
0answers
64 views

Laurent Series of Riemann Zeta Function

How do I go about finding the Laurent series of the Riemann zeta function about $z=1$?
0
votes
0answers
55 views

Specific form of integral representation of the Riemann zeta function

Is there an integral represenation of the Riemann zeta function of the form: $$\zeta(s) = f(s)+c\int_a^b\frac{g(x)}{x^{p(s)}}dx,$$ where $a,b,c\in\mathbb{R}$ with $a\neq b$, $p(s)$ is some ...
1
vote
1answer
50 views

Difference between Meromorphic and Analytic Continuation

We have that $\frac{X}{spec \mathbb{Z}}$, a scheme of finite type. Consider $\zeta(X,s)= \prod_{x\in{X}} \frac{1}{(1-Nx^{-s}}$, with $Nx$ the norm. I didn't catch what my teacher was saying and he is ...
1
vote
0answers
31 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
3
votes
1answer
187 views

Calculating $\zeta(0)$ by the residue of $\zeta(1)$

$$\begin {aligned}\pi^{-s/2}\Gamma(s/2)\zeta(s)=&\pi^{-(1-s)/2}\Gamma((1-s)/2)\zeta(1-s) \\ \zeta(0) =&\frac{\pi^{-1/2}\Gamma(1/2)\zeta(1)}{\pi^{0}\Gamma(0)}=\frac{\zeta(1)}{\Gamma(0)}\end ...
11
votes
2answers
252 views

Integration method for $\int_0^\infty\frac{x}{(e^x-1)(x^2+(2\pi)^2)^2}dx=\frac{1}{96} - \frac{3}{32\pi^2}.$

The following definite integral is obtained directly from Hermite's integral representation of the Hurwitz zeta function. But is it possible to obtain the same result via the residue calculus or ...
3
votes
3answers
138 views

$(1-2^{1-s})\zeta(s)$ is an entire function

Show that $(1-2^{1-s})\zeta(s)$ is an entire function, which is represented by the series $$(1-2^{1-s})\zeta(s)=1-\dfrac{1}{2^s}+\dfrac{1}{3^s}-\dfrac{1}{4^s}+\cdots$$ for $\Re{s}>1$. From the ...
1
vote
0answers
29 views

Dirichlet series minus Riemann zeta

Suppose $\{a_n\}$ is a sequence of complex numbers such that the sums $A_n=a_1+\cdots+a_n$ satisfy $$|A_n-nb|\leq Cn^{\sigma}$$ for all $n$, where $b\in\mathbb{C},C>0,0\leq\sigma<1$. Consider ...
3
votes
1answer
207 views

Analytic continuation of the Riemann zeta function using contour integration

To find the analytic continuation of the Riemann zeta function using contour integration one can integrate $\displaystyle f(z) = \frac{z^{s-1}}{e^{-z}-1}$ around a contour that consists of rays just ...
8
votes
1answer
428 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
7
votes
1answer
197 views

Serie for $\text{Arg}( \zeta (z))$ [closed]

Though I don't know if the formula I've found is useful, I decided to publish it anyway. $$ \text{Arg}( \zeta (z)) = -\sum_ {k = 1}^{\infty}\sum _ {q = 1}^{\infty}\frac {1} {k P_q^{k x}}\text {Sin}( k ...
4
votes
2answers
85 views

A convergent-everywhere expression for $\zeta(s)$ for all $1\ne s\in\Bbb C$ with an accessible proof

I'm looking for a way to define the Riemann zeta function $\zeta(s)=\sum_{n\in\Bbb N_0}n^{-s}$ on the whole complex plane, without having to use analytic continuation, or perhaps more accurately, in a ...
3
votes
1answer
177 views

Elementary bound on the Riemann zeta function

I am currently preparing for a course in analytic number theory and I wanted to get a heads start. In my preparation, I came across the following problem: Show that for $|y|\geq 2$, $|\zeta(1+iy)| ...
30
votes
3answers
623 views

proving that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ ($H_n=\sum_{k=1}^{n}\frac{1}{k}$)
6
votes
1answer
71 views

Evaluating limit $\lim_{m\to{\infty}}\frac{\sum_{k=1}^m\cot^{2n+1}(\frac{k\pi}{2m+1})}{m^{2n+1}}$

How can I prove the following equality? $$\lim_{m\to{\infty}}\frac{\displaystyle\sum_{k=1}^m\cot^{2n+1}\left(\frac{k\pi}{2m+1}\right)}{m^{2n+1}}=\frac{2^{2n+1}\zeta(2n+1)}{\pi^{2n+1}}$$
2
votes
0answers
55 views

Zeros of “nearby” holomorphic functions

I would appreciate help in how to show that On a compact subset of {$z \in \mathbb{C}: 1/2 < \Re (z) < 1$}: "Given a holomorphic function with an isolated zero, any "nearby" holomorphic ...
1
vote
0answers
79 views

How should I prove that Zeta'(x)/Zeta(x)+1/(x-1) is strictly monotonously decreasing on the real line (for x>=0)?

The Riemann Zeta function, like most other complex functions, are much easier to deal with in the real line, since the values are also real, and definitions can be used in a more straightforward ...
0
votes
1answer
101 views

Is there only one analytic continuation of the Riemann zeta function?

If I were to manipulate the zeta function in a 'new way' would I end up with an analytic continuation that is equal to the one know or something completely new for values less than 1 and complex ...
6
votes
1answer
137 views

Questions regarding the Riemann-Siegel $\theta$ Function

My questions are a request, please, for help in understanding some comments in the wikipedia article discussing the Riemann-Siegel $\theta$ function ...
4
votes
2answers
232 views

How to Compute $\zeta (0)$?

Ultimately, I am interested in analytically continuing the function $$ \eta _a(s):=\sum _{n=1}^\infty \frac{1}{(n^2+a^2)^s}, $$ where $a$ is a non-negative real number, and calculating $\eta _a$ and ...
2
votes
1answer
71 views

Show in between steps in this Riemann zeta function equivalence/reduciton

In the answer chosen by the OP in this question I had trouble understanding the steps taken to get the equivalences/reduce the zeta function into another one. Can somebody show me the steps to go from ...
0
votes
0answers
66 views

Zeroes of $s+\sum\limits_{n=2}^\infty \frac{(-1)^{n+1}}{n^s\ln n} $?

Where are the solutions of the equations $$s+\sum\limits_{n=2}^\infty \dfrac{1}{n^s\ln n}=0\quad \text{and}\quad s+\sum\limits_{n=2}^\infty \dfrac{(-1)^{n+1}}{n^s\ln n}=0 ?$$ Since the ...
4
votes
3answers
103 views

Two questions regarding $\mathrm {Li}$ from “Edwards”

I would appreciate help understanding a relation in Edwards's "Riemann's Zeta Function." On page 30 he has: $$\int_{C^{+}} \frac{t^{\beta - 1}}{\log t}dt = \int_{0}^{x^{\beta}}\frac{du}{\log u}= ...
2
votes
2answers
255 views

New tools for complex analysis and application to the Riemann Zeta function?

I've worked as a graphic artist for the past fifteen years, thus I have no relationship with the academic mathematical community. It is therefore difficult for me to check some results. 1. Tools for ...
2
votes
0answers
60 views

Validity of a functional formula of the Riemann Zeta function across the whole complex plane?

Could someone confirm me the validity of the following formula: $$\zeta\left(z\right)=2^{z}\pi^{z-1}\sin\left(\frac{\pi z}{2}\right)\left(\frac{e^{-\gamma ...
3
votes
1answer
211 views

Proving that $\sum (-1)^{n+1} n^{-z}$ defines an analytic function in $Re z>0$

I want to show that the series $\sum_{n=1}^\infty (-1)^{n+1} n^{-z}$ converges to an analytic function for $\Re z>0$. For $\Re z>1$ the terms are dominated by $n^{-x}$ so that we have absolute ...
0
votes
2answers
59 views

What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$?

Let $s$ be complex and $\zeta(s)$ the Riemann Zeta function. What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$ ? I want a series expansion such that ...
16
votes
1answer
424 views

“Orientation” of $\zeta$ zeroes on the critical line.

I am pretty ignorant about complex analysis so please forgive my lack of terminology. I saw a pretty picture (posted below) of the behavior of the Riemann zeta function along the critical line. What ...
3
votes
4answers
3k views

What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
2
votes
2answers
126 views

Analytic continuation of Riemann Zeta funtion

I am reading about zeta function from book by Ingham. In that book the following theorem is given. I am unable to understand what does he mean by finite part of plane in the statement.
3
votes
1answer
126 views

Riemann zeta function at zero

Can the value of Riemann zeta function at 0, $\zeta(0)=-1/2$, be deduced from the identity $E(z)=E(1-z)$, where $$E(z)=\pi^{-z/2}\Gamma(z/2)\zeta(z)?$$
7
votes
1answer
127 views

A question related to Riemann zeta function

Does anyone know why the following statement is correct? Let $f(x)$ be the function whose value on the interval $m\pi<x<(m+1)\pi, m=0,1,2,\cdots$, is $(-1)^m\frac{\pi}{4}$. Let $0<s<1$. ...
9
votes
1answer
742 views

Zeta function zeros and analytic continuation

I'm learning about the zeta function and already discovered the intuitive proof of the Euler product and the Basel problem proof. I want to learn how to calculate the first zero of the Riemman Zeta ...
5
votes
0answers
197 views

Identity involving $\zeta(3)$

This is related to this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that ...
3
votes
1answer
175 views

Discontinuities of $\sum \frac{x^{\rho}}{\rho}$

H. Edwards in his book on the zeta function says that $\sum\frac{x^{\rho}}{\rho}$ converges conditionally "even when $\rho ,1-\rho$ are paired." I tried calculating some terms (n = 500 or so) and ...
5
votes
3answers
120 views

Two trivial questions about zeta function

I have two questions concerning the Riemann zeta function which are rather trivial so if anyone can give me the answers that would be nice, here is what I`m interested in: 1) In the equality ...
8
votes
1answer
487 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 $\zeta(s)$ ...
5
votes
1answer
151 views

Serre's proof that zeta function is meromorphic

I try to understand the proof of Chap. VI, n° 3.1, Prop. 10 in Serre's "A course in arithmetic" (page 70). The goal is to prove that zeta-function can be written as \begin{align*} ...
3
votes
1answer
90 views

How to evaluate $\xi(0)$?

How do I evaluate $\xi(0)$ for the Riemann xi function? I know $\xi(0) = \xi(1)$ and $\xi(0) = \tfrac{1}{2} \cdot 0 \cdot (-1) \cdot \Gamma(0) \cdot \zeta(0)$ $\xi(1) = \tfrac{1}{2} \cdot 1 \cdot 0 ...