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

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8
votes
2answers
109 views

Why is this true ? $(1-s)\zeta(s) = \sum_{k=0}^\infty \frac{\Gamma(k+1-s/2) A_k}{\Gamma(1-s/2) k!}$

On the whole complex plane the following formula is valid : $$(1-s)\zeta(s) = \sum_{k=0}^\infty \frac{\Gamma(k+1-s/2) A_k}{\Gamma(1-s/2) k!}$$ where $A_k = \sum_{j=0}^k (2j-1) \zeta(2j+2) (k,j)$ ...
12
votes
1answer
894 views

What exactly *is* the Riemann zeta function? [duplicate]

I'm doing a little project on the $\zeta$ function, and I am at a complete loss of what it is actually doing. I understand it is way over my head, but when I am plugging say $\zeta(1 + i)$ into ...
0
votes
2answers
63 views

$\zeta (1/2 + i) = 0$, correct?

Plugging $\zeta (1/2 + i)$ into Wolfram Alpha yields me some complex number, but I was under the understanding that $\zeta (1/2 + it) = 0$, for all $t$ we have yet calculated... Is Wolfram Alpha just ...
3
votes
0answers
56 views

Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

We all know that Riemann Hypothesis has many equivalent statements. After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as ...
1
vote
1answer
36 views

Putting a bound on sum of primes with a certain property

I'm trying to prove that the Dirichlet density of the set $X$ of primes of the form $p = n^2+1$ is zero. The only way I can think to do it is to put a bound on the sum $\Sigma_{p\in X} \frac{1}{p^s}$, ...
1
vote
2answers
110 views

A proof of $\sum{\mu(n)/n}=0$

I am looking for a proof (or references) of the following statement $$\sum_{n=1}^{\infty}{\frac{\mu(n)}{n}}=0$$ where $\mu$ is the Möbius function. Many thanks !
0
votes
1answer
52 views

Riemann Zeta Function and Including Complex Numbers [duplicate]

I'm a high-school senior attempting to make sense of the zeta function. I know Riemann regularized it to include complex numbers. Apparently, from this we could obtain that the sum of natural numbers ...
7
votes
1answer
121 views

Proving $\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$

Trying to prove $$\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$$ I found by numerical calculation that (when $k$ goes to infinity) $$\sum_{n=1}^{k}\zeta ...
4
votes
1answer
63 views

What is known about the complex solutions to $\zeta(s)=-1$?

I don't recall ever coming across a discussion of the complex solutions to the equation $$\zeta(s)=-1,$$ where $s\in\mathbb{C}$. How many such solutions exist? Is there any literature on this? ...
0
votes
2answers
45 views

Properties of $\underset{k\geq1}{\sum}\frac{1}{\left(2k-1\right)^{s}}$

Is this function $$\underset{k\geq1}{\sum}\frac{1}{\left(2k-1\right)^{s}},\,Re(s)>1$$ well known? In particular I'm interessed about analytic continuation and its zeros and poles. Have this ...
4
votes
1answer
71 views

Strange sum involving $\zeta (3)$

$$\sum _{l=0}^{\infty } \sum _{k=0}^{\infty } \frac{(-1)^{k+l}}{2 (l+1)^3 \left((k+1)^3+(l+1)^3\right)}=\frac{9 \zeta (3)^2}{32}$$ I would like to prove the proposed equality, but I do not know where ...
2
votes
2answers
86 views

Riemann Hypothesis: Proving a relation between $\psi(x)$ and $\pi(x)$

I am trying to prove the following If $\pi(x) = \operatorname{Li}(x) + O(x^{\frac{1}{2}}\log(x))$ then $\psi(x) = x + O(x^{\frac{1}{2}} \log^2(x))$ I have tried using $\psi(x) = \theta(x) + ...
1
vote
1answer
98 views

Curve profile for the logarithm-integral sum term of Riemann explicit formula?

I am considering the following term from the Riemann explicit formula (see here >>>): $$\sum_{\rho(\Im>0)}{\mathrm{li}(x^\rho)}$$ with $\rho$ non-trivial zeros of $\zeta$-function. I have a plot ...
0
votes
1answer
43 views

At what hight is the nth zero Riemann zeta function? [closed]

At what hight is the nth zero Riemann zeta function? It is a mathematical formula but I can not get under it on the Internet (once I found).
4
votes
2answers
133 views

Divisor and Riemann Zeta functions series proofs

Where d(n) is the number of divisors of n, show that $\sum_{n=1}^\infty d(n)z^n = \sum_{k=1}^\infty \frac{z^k}{1-z^k}$ where both sides converge for |z|<1 and show that $\sum_{n=1}^\infty ...
1
vote
0answers
64 views

Has the Riemann Hypothesis been generalized to the Octonions and the Quaternions?

I've noticed that it uses imaginary numbers. I know that sometimes when I have too few dimensions like (-1)^n, dots show where I might expect lines due to imaginary numbers. So perhaps there is a ...
1
vote
1answer
35 views

Understanding a series representation of the logarithm of the zeta function

I am reading through M. Ram Murty's Problems in Analytic Number Theory and have the following question regarding the first step in his proof of Dirichlet's Theorem. Given this definition for the zeta ...
1
vote
0answers
24 views

how to prove $\Phi(t)$ is divergent when $Im(t)=\pi/2$?

The Riemann $\Xi(z)$ function admits a Fourier transform of a even kernel $\Phi(t)=3e^{5t/4}\theta'(e^{t})+2e^{9t/4}\theta''(e^{t})$. Here $\theta(z)$ is the Jacobi theta function. ...
3
votes
1answer
72 views

Moving the integral $Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds$ past Re(s) = 1.

Given the integral $$Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds,$$ I know that the integrand is holomorphic except for simple poles at ...
5
votes
0answers
49 views

$\zeta(2)$ Euler's proof (Basel problem) [duplicate]

At one point Euler assumes that $$\frac{\sin x}{x} = \prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi} \right)\left(1-\frac{-x}{n\pi} \right)$$ Why does he assume that? If we factor random functions ...
2
votes
0answers
43 views

Comprehensive summary of where the function $\pi^{-\frac x\pi}$ can be encountered

I am studying the special functions, including the Riemann Xi and Zeta, and everywhere a function $\pi^{-\frac x\pi}$ pops up, usually as multiplier to the Gamma function. But yet I am not sure this ...
0
votes
0answers
37 views

Explicit analytic continuation of the zeta function and shift operators

Is there a way to compute the radius of convergence of the expansion of the zeta function, e.g. around $a=2+2i$? We have $\zeta(a)=\sum_{n=1}^\infty n^{-a}\approx 0.867.. - i\,0.275.. $, but I ...
5
votes
1answer
440 views

Double sum and zeta function

This is a personal research that came to an end , since the results were not those which were being anticipated. I was unable to come up with a solution therefore I post the topic here: Prove (it ...
6
votes
1answer
106 views

Integral Representation of the Zeta Function

How does one get from this $$\zeta(s)=\sum_{k=1}^{\infty}\frac1{k^s}$$ to the integral representation $$\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$ of the Riemann Zeta ...
11
votes
2answers
391 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
149 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
18 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
68 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
136 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
66 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
78 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
117 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
297 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
207 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
94 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
105 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
48 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
49 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
131 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
63 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
50 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 ...