Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

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3
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5answers
308 views

How are Zeta function values calculated from within the Critical Strip?

We note that for $Re(s) > 1$ $$ \zeta(s) = \sum_{i=1}^{\infty}\frac{1}{i^s} $$ Furthermore $$\zeta(s) = 2^s \pi^{s-1} \sin \left(\frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s)$$ Allows us to ...
0
votes
0answers
69 views

Error with Zeta functional equation [duplicate]

I was trying to prove $$1 + 2 + 3 +\cdots = -\frac{1}{12}$$ Using the following $$\zeta(s) = \sum _{i=1}^{\infty} \left [\frac{1}{i^s} \right]$$ Thus: $$\zeta(-1) = \sum _{i=1}^{\infty}\left [i ...
4
votes
0answers
101 views

An infinite series that gives $f(s)=s$. How could it be explained more easily?

This question loosely builds this one. Equate the following two infinite series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{4\,(s-1)} \left(1+s+\sum _{n=1}^{\infty } \left( {\frac ...
2
votes
1answer
57 views

Functional equation for the $\zeta$-function, bounding a contour

In one of my textbook the following problem is written: Proving the functional equation for the $\zeta$-function: $$\zeta(z) = 2^z\pi^{z-1}\sin\frac{\pi z}{2} \Gamma(1-z)\zeta(1-z) \qquad ...
8
votes
2answers
259 views

Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
8
votes
0answers
198 views

Are these known telescoping series for $\zeta\left(\frac12\right)$?

There are many known telescoping series for $\zeta(s)$ and I was playing with the following two: $$\displaystyle \zeta(s) = \frac{1}{(s-1)} \left(\sum _{n=1}^{\infty } \left( {\frac {n}{(n+1)^{s}}} - ...
4
votes
0answers
113 views

Symmetry and the zeta function

It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it ...
2
votes
0answers
55 views

Riemann vs. Ihara's $\zeta$ Function Variable Question

The Euler product for the Riemann zeta function $\zeta(z)$ implies that $$ \log\zeta_R(z)=\sum_{m>0}\frac{P(mz)}{m} \tag{R}, $$ whereas the Ihara zeta function for a graph $G$--all can be ...
0
votes
0answers
51 views

Evaluate $\lim_{n\to1^+}\left({\zeta(n)-\dfrac{1}{n-1}}\right)$ [duplicate]

Let $$x=\lim_{n\to1^+}\left({\zeta(n)-\dfrac{1}{n-1}}\right)$$ where $\zeta$ is Riemann zeta function. What is the value of $x$? At $n\to1^+$, $\zeta(n)\to\infty$ and $\dfrac{1}{n-1}\to\infty$, so ...
3
votes
1answer
139 views

Extending the zeta function to semiprimes, etc.

The Riemann Zeta function is defined for $s > 1$ as \begin{align} &\prod _{n=1}^{\infty}\dfrac{1}{1 -\ p_{n}^{\ \ -s}}\\ \end{align} It is possible to extend the zeta function to semiprimes ...
1
vote
0answers
50 views

Where is the fault in this approach for transforming this Dirichlet series?

Mathematica knows that: $$\lim_{s\to 1} \, \zeta (s)\left(-2^{1-s}-3^{1-s}+6^{1-s}+1\right)=\sum _{n=0}^{\infty } \left(\frac{1}{6 n+1}+\frac{-1}{6 n+2}+\frac{-2}{6 n+3}+\frac{-1}{6 n+4}+\frac{1}{6 ...
1
vote
0answers
70 views

bounds of Riemann $\zeta(s)$ function on the critical line?

I vaguely remembered that $$0\leq|\zeta(1/2+i t)|\leq C t^{\epsilon},\qquad t>>1,\epsilon>0$$. Is this bound correct? Thanks- mike
0
votes
0answers
60 views

Is this proof show that the derivative of zeta function has no zeros in the critical strip?

suppose that :( $k \circ \zeta )(s)$= $(\zeta \circ k )(s) \neq 0 $.....$(1)$ ,and suppose that $ k(s)=\zeta(1-s)$ where : $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. and ...
0
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0answers
20 views

Can there be a Dirichlet series that gives the functional inverse of the Riemann zeta function?

Can there be a Dirichlet series that gives the functional inverse of the Riemann zeta function? I will delete this question if it gets downvoted.
1
vote
1answer
105 views

Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?

From this question: http://mathoverflow.net/questions/134600/zetax-in-terms-of-zetax-zeta1-x-gamma-psi it seems that Zeta can be expressed through its derivative: $$\zeta(1-x) = ...
1
vote
1answer
80 views

Zeta and Gamma function regularization with $\omega=1/0$

I have recently read about zeta function regularization, a way of ascribing values to functions having simple poles in a point and to divergent series. The values obtained are the same as those ...
0
votes
0answers
32 views

Can you provide a lower bound on $|\zeta(\sigma+it)|$ for $\sigma$<0?

To prove the theorem 9.12 of "The theory of the Riemann Zeta-Function", I have to show that $|\zeta(\sigma+it)|$ > $A|t|^c$ for $\sigma$<0 and sufficiently large t. How can I show this?
4
votes
1answer
142 views

Is there a metric in which $1+2+3+4+\cdot$ converges to $-\frac1{12}$?

It is well known that the sum $1+2+3+4+\ldots$, which tends to infinity in the regular sense, can be assigned the value $-\frac{1}{12}$ by different means, e.g., zeta regularization or different ...
2
votes
0answers
51 views

symmetrized partial sums for $\zeta(s)$ and $\eta(s)$ in the critical strip

$\def\Re{\operatorname{Re}}$ We start with $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}\qquad \Re(s)>1\tag{1}$$ $$\zeta(1-s)=\sum_{n=1}^{\infty}\frac{1}{n^{1-s}}\qquad \Re(s)<0\tag{2}$$ ...
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vote
0answers
49 views

Relationship between partial sum of Riemann zeta functions over even integers and the harmonic series

How do you prove that ${\sum_{n=1}^{k}(\zeta(2*n)/n)-H_k(1)}$ tends to $\ln(2)$ as integer $k$ tends to infinity where $H_k(1) = \sum_{n=1}^{k}{1\over n}$? Is this result well known? Please give a ...
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vote
2answers
71 views

Looking for series representations of Riemann's zeta function valid for $\sigma<1$.

I've been looking for series representations of the Riemann's zeta function $\zeta(s)$ valid for $\sigma< 1$, with $s=\sigma + t i \in \mathbb{C}$. I'm interested, preferably, in series ...
0
votes
1answer
21 views

Proving a certain function involving the Riemann-Zeta function is non-increasing

Show that $ f(x) = \frac{\zeta(x -2)}{\zeta(x-1)} \qquad x > 3, $ where $\zeta$ is the Riemann-Zeta function, is non-increasing. My attempt was to use $\zeta(s) = \frac{1}{\Gamma(s)} ...
1
vote
1answer
62 views

Infinite series and the Riemann zeta function

I have two questions concerning infinite series in the context of the Riemann zeta function. Given the properties of infinite series, why can't we regroup the terms in $\zeta(0)$ in such a way as to ...
0
votes
2answers
44 views

Euler and the factorial function

I recently purchased H. M. Edwards' book entitled The Riemann Zeta Function. In the early pages of the volume, concerning the factorial function $\Gamma$, Edwards notes that "Euler observed that ...
8
votes
2answers
166 views

Taylor's results on zeros of the linear combination of Gamma-completed Riemann Zeta functions

Let $s,z$ be two complex variables, $\zeta(s)$ be the Riemann $\zeta$-function. Let \begin{equation} \zeta_1(s)=\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s), \end{equation} be the ...
5
votes
2answers
384 views

A Deviation from a Conventional Proof of the Basel Problem

There's been many topics on the Riemann-Zeta function, specifically $\zeta(2)$.$$\zeta(2)=\sum_{n=1}^\infty\frac{1}{n^2}=\int_0^1\int_0^1\frac{1}{1-xy}dA$$This is the Basel Problem. Taking the ...
5
votes
2answers
121 views

Prove that $\zeta(4)=\pi^4/90$

I am asked to "use the calculus of residues" to prove that $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^4}{90}$$ I think I can do this given the Laurent series for $\cot z$ ...
9
votes
0answers
239 views

Is this similarity just a coincidence?

Here is the function $-1/x$: If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $\tan x$. But if we do the same only in one direction, we get ...
5
votes
2answers
217 views

Is this claim true$(\xi \circ k)(s)=(k \circ \xi )(s)=0$ $\implies$ $k(s)=\zeta(s)=0 $ is true if and only if the Riemann Hypothesis is false?

It is well known that $\xi(s)=\xi(1-s)$ is a verified functional equation for all complex $s$, where $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. let $k(s)=\xi(1-s)$ and ...
4
votes
1answer
113 views

$0=\frac{13+13^2+13^3+\cdots}{1+2+3+\cdots}$ using infinite sums?

This is not homework, just curiosity. My question arose from the apparent absurdity that $\zeta(-1)=-\frac{1}{12}$, even though $\sum_{n=1}^\infty \frac{1}{n^z}$ only makes sense when $Re(z)>1$. ...
2
votes
1answer
106 views

Contradicting statements about the Riemann zeta function at positive odd integers

I have found two contradicting statements about the value of $\zeta(k)$ when $k=2n+1$ and $n\in\mathbb{Z_0^+}$. Which one is correct? "The Riemann zeta function for odd integers has no known ...
1
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0answers
37 views

Why the poisson summation formula works

If we put the function $ f(x)= |x|^{s-1} $ inside the Poisson sum formula and consider that $ \sum_{n=1}n^{z-1}= \zeta (1-s) $ then we can easily give a proof of Riemann's functional equation $$ ...
3
votes
3answers
67 views

How to prove that $ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 $?

I would like to prove that $$ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 \quad .$$ However, I am not sure which form of the Riemann-zeta function I ought to pick in order to compute this limit. I ...
3
votes
3answers
164 views

My (divergent) summation of the zetas with sets of cofactors give systematically errors of simple integer differences. What am I missing?

This is a "fiddling" in a small project of mine with which I'm concerned from time to time for three years now. I try to focus on the core of the problem, please ask if more context is needed. ...
3
votes
0answers
49 views

a question related to using Hurwitz theorem to bound the locations of zeros of Riemann zeta function

Let $F(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the Gamma-complete version of the Riemann $\zeta$ function. Let $f(z)=F(1/2+i z)$. So it is known that all zeros of $f(z)$ are in the strip $S_{1/2}$ ...
4
votes
1answer
135 views

Proof of $\sum_{n=1,3,5,\ldots}^{\infty}\frac{1}{n^4}=\frac{\pi^4}{96}$

I came across with the infinite series $$\sum_{n=1,3,5,\ldots}^{\infty} \frac{1}{n^4}= \frac{\pi^4}{96}$$ when calculating a problem about an infinite deep square well in quantum mechanics. ...
1
vote
1answer
100 views

Solving $\sum_{n=1}^{\infty} \frac{1}{n^2}$ using the fourier series.

Please do NOT solve the problem, I just need some help, not a full solution. I would like to try this myself. Find $\zeta(2) = \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}$ The fourier series for ...
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0answers
78 views

An inverse Fourier transform of Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\tag{1}$$ The functional equation for $\zeta(s)$ is ...
1
vote
2answers
79 views

Is there a formula for $k\pi ^n$, if $n$ is an odd number and $k$ is a rational number?

I always find the formula for $k\pi ^n$ when $n$ is an even number and $k$ is a rational number, but I did't find for an odd number.
5
votes
0answers
110 views

Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$ P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right) $$ I know that without the power one could ...
4
votes
2answers
76 views

Limit of Riemann Zeta Function as Imaginary Part tends to Infinity

Is it true that $$ \lim_{n\to \infty} \zeta(2+ni) =1 ?$$ If not, what is the value of the limit? What about the same but with other real parts other than 2?
6
votes
2answers
120 views

Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$

For some time I've been playing with this kind of sums, for example I was able to find that $$ \frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\left( \zeta(2k+1)-\beta(2k+1)\right) $$ where $$ ...
3
votes
1answer
91 views

Riemann Zeta Function at $s = 1 + 2 \pi i n / \ln 2$

I am aware that the function defined by $$ \zeta(s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots $$ for $Re(s)>1$ can be extended to a function defined for $Re(s)>0$ by writing $$ ...
1
vote
1answer
98 views

Why zeta(2) in these inifinite sums?

The infinite sum of the reciprocals of these two sequences have zeta(2) in the result. The value is not in OEIS. A000326 A002411 Edit---rolled back the changes. Both $\frac{1}{2}$ and $2$ are ...
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0answers
21 views

q question regarding the numerical zero finding of Riemann zeta function and actual proof of Riemann Hypothesis

(1)Suppose that people verified in 2004, that all zeros of $\zeta(\sigma+i t)$ with $0<t<T<=10^{22},0<\sigma<1$ are on the critical line ($\sigma=1/2$). (2)Suppose Bob proved in ...
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votes
0answers
67 views

How i could show that this inequality true or false: $|\zeta(s)| \leq |2^s \dfrac{1}{1-2^{-s}}|\leq {1}$ for $0< \sigma <1$?

Is this inequality true: $|\zeta(s)| \leq |2^s \dfrac{1}{1-2^{-s}}|\leq {1}$ for $0< \sigma <1$ ? note :$s=\sigma + it$, where $\sigma, t\in \mathbb{R}$. I would be interest for any replies ...
1
vote
2answers
42 views

zero distribution of the Fourier kernel $\Phi(u)$ for Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ The functional equation for $\zeta(s)$ is equivalent ...
3
votes
0answers
86 views

How prime numbers are related to special functions?

We know that the Riemann zeta function is defined as $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},$$ for all $\Re(s)>1$. Because of Euler product formula we also know that $$\zeta(s) = ...
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votes
1answer
96 views

the definition of Riemann zeta function

I've read that the Riemann zeta function for $0<s<1$ is defined : $$\lim_{x\rightarrow\infty} \left(\sum_{n \leq x}\frac{1}{n^s}- \frac{x^{1-s}}{1-s}\right)$$ I don't know how to prove that ...
0
votes
1answer
74 views

Plot zeros of partial sum of zeta Riemann with Maple

I want to plot the zeros of the partial sum of the Riemann zeta function with Maple. Some hint?? Thanks!!!