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

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11
votes
2answers
378 views

Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?

I think I understand intuitively how we can assign a value to the sum of all natural numbers. But of all the proofs that I've seen that show why $\zeta(-1) = -\frac{1}{12}$, none of them use their own ...
2
votes
2answers
146 views

Finding the closed form for $\sum_{n=1}^{\infty }\frac{\zeta (4n)}{\beta^{4n-1}}$ [closed]

Finding the closed-form $$\sum_{n=1}^{\infty }\frac{\zeta (4n)}{\beta^{4n-1}}$$ for $\beta\in(1,+\infty)$. I learned from this site many many important things but I till need more, so I need ...
6
votes
2answers
126 views

The Riemann zeta function $\zeta(s)$ has no zeros for $\Re(s)>1$

I write $\zeta(s)$ for $\Re(s)>1$ as: $\zeta(s) = \prod_{p} (1-p^{-s})^{-1}$ Using this I can show that the Riemann zeta function has no zero for $\Re(s)>1$. I'm however not sure about the ...
0
votes
0answers
17 views

seeking upper/lower bounds of a function $F(m)$ related to Jacobi theta function

I am looking for the upper/lower bounds of function $F(m)$ defined and plotted above. The function is related to Jacobi theta function $\theta(x)$ and its derivative values at $x=1$: ...
1
vote
0answers
39 views

Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
2
votes
0answers
66 views

Riemann's explicit formula for $\pi(x)$

Riemann's explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where ...
2
votes
1answer
54 views

Summation with Zeta function

I'm currently studiyng Zeta function and I don't understand this identity: $$\sum_{n=1}^\infty x \sum_{p=0}^\infty \frac{x^{2p}}{(n\pi)^{2p+2}} = \sum_{p=1}^\infty \pi^{-2p}\zeta(2p)x^{2p-1} $$ I ...
3
votes
1answer
126 views

Cramer and Riemann Conjecture Implication

Cramer's conjecture gives $$p_{n+1}-p_n= O(\log^2 p_n)$$ while Riemann Hypothesis yields just $$p_{n+1}-p_n= O(\sqrt p_n\log^2 p_n).$$ Does Cramer conjecture on prime gaps imply Riemann Hypothesis ...
1
vote
0answers
65 views

Closed form for generating function of Riemann Xi function

What is the closed form for $$f(x)=\ \sum_{k=1}^\infty \frac{\xi(k)x^k}{k!}$$ or $$g(x)=\frac12 \sum_{k=1}^\infty \frac{\xi(k+1/2)x^k}{k!}$$ or $$w(x)=\frac12 \sum_{k=1}^\infty ...
1
vote
1answer
76 views

How to prove that Riemann zeta function is zero for negative even numbers?

Can anyone please explain to me how to prove that Riemann zeta function is 0 for all negative even numbers. In many references , they have just given the statement without any proof. Any explanation ...
1
vote
0answers
48 views

Analytic continuation of the Dirichlet $\eta(s)$ series to $\Re(s) \gt -1$. Why does this work?

Take the known Dirichlet $\eta(s)$ series, $$\displaystyle \eta(s) = \sum _{n=1}^{\infty } \left( {\frac {1}{(2\,n-1)^{s}}} - \frac{1}{(2\,n)^s}\right), \qquad \Re(s)>0$$ and add $\displaystyle ...
4
votes
1answer
114 views

Conway Complex Analysis Book Exercise 8 in the Riemann Zeta Function Chapter

I am studying the book of John B. Conway Functions of One Complex Variable(1978), and in the section of Riemann Zeta Function chapter 7 I couldn't solve the last exercise. Here it is: Let $\zeta (z)$ ...
3
votes
5answers
296 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
56 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
197 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
191 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}}} - ...
-1
votes
1answer
92 views

What's the limit of this sum $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ [closed]

Let $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ and s be a complex variable . $s=\sigma +it $ where $\sigma ,t \in\mathbb{R} $ , Note :I edit the ...
4
votes
0answers
104 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
42 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
48 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
130 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
49 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
61 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
47 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
votes
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
0answers
61 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
76 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
22 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?
3
votes
1answer
131 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
48 views

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

$\def\Re{\operatorname{Re}}$ We start with $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}\qquad \Re(s)>1\tag{1}$$ $$\zeta(1-s)=\sum_{n=1}^{\infty}\frac{1}{n^{1-s}}\qquad \Re(s)<0\tag{2}$$ ...
1
vote
0answers
48 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 ...
0
votes
2answers
64 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
56 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
42 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
157 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
366 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
81 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
225 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 ...
1
vote
1answer
95 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 RH 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
112 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
102 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
vote
0answers
34 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
66 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
157 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
45 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
123 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
88 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 ...