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

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$\sum \zeta (s)$ converging proof?

I was pondering about the fact that the Zeta function can be represented as an infinite sum, $\zeta (s) = 1/1^s + 1/2^s + 1/3^s +\ ...$, and I thought about the infinite sum of the zeta function ...
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1answer
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An algebraic manipulation of the Zeta function

Consider the following form of the Riemann Zeta function: $s\in\mathbb{C}$, such that: $\left |s \right | > 1$ $\zeta \left ( s \right )= 1+2^{-s}+3^{-s}+4^{-s}+5^{-s}...$ Now, due to the ...
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1answer
59 views

Could someone explain how the Gram series relates to Riemann's function?

I was reading an article on the distribution primes which mentions the following equation for Riemann's function $R(x)$: $$R(x) = \sum_{n=1}^{\infty}\frac{\mu(n)}{n}\text{li}(x^{1/n}) = 1 + \sum_{k=1}...
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1answer
57 views

How to calculate $\zeta(s)$ using residues?

We have the Riemann zeta function $$ \zeta(s) = \frac{\pi}{2} \int_{\gamma} \frac{dz}{2\pi i} \, \frac{1}{z^s} \cot(\pi z) $$ and I want to use the Residue theorem in order to understand why it is ...
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38 views

asymptotic expansion in powers of $ 1/s $

How could I get the coefficients in the Dirichlet series expansion $$ \frac{\zeta ' (s)}{\zeta(s)}= \sum_{n=1}^{\infty} \frac{a(n)}{n^s} $$ for the logarithmic derivative of the Riemann zeta ...
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Problem inside the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$

I am having trouble with one step for the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$ found here. In the below steps, how do we get from $$\frac{1}{\pi{i}}\Bigl(G(z)-2G(2z)\Bigr) = -F(z)+F(-z)$$ ...
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2answers
340 views

Motivation on how does complex analysis come to play in number theory?

I am not sure if this is a appropriate question. If it isn't, let me know and I'll delete it. $\textbf{Background}$ I am an undergraduate student and I'm very interested in number theory. I've tried ...
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1answer
185 views

Riemann zeta function and the volume of the unit $n$-ball

The volume of a unit $n$-dimensional ball (in Euclidean space) is $$V_n = \frac{\pi^{n/2}}{\frac{n}{2}\Gamma(\frac{n}{2})}$$ The completed Riemann zeta function, or Riemann xi function, is $$\xi(s) ...
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19 views

Importance of the real-rooted asympototics of $f_n(z)$ that uniformly converges to Riemann $\Xi(z)$ function

We are learning Riemann $\Xi(z)$ and Riemann $\zeta(s)$ functions. This question is related to an earlier one. (1) Suppose that a family of functions, $f_n(z)$, uniformly converges to Riemann $\Xi(z)$...
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989 views

Are the nontrivial zeroes of the Riemann zeta function countable?

It is known that the set of non trivial zeros is an infinite set. But is it known if it is a countable, or uncountable infinite set?
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138 views

Assuming the Riemann hypothesis, does this integral give the Riemann zeta zeros when increasing Working Precision in Mathematica?

It is probably well known that the Riemann zeta zeros satisfy the following equation: $$\frac{\arg \left(\zeta \left(\rho _n+\frac{1}{1000000000000000}\right)\right)}{\pi }+\frac{\vartheta \left(\Im\...
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Is there a contiguous locus of the equality $a = b$ in $\zeta(\rho + \varepsilon)=a + î b$ in the near of a root?

This is just by an accidental couriosity: In the near of a root $\rho$ of the Riemann's zeta - can there be a continuous line starting from $\zeta(\rho)=0$ to $\zeta(\rho + \varepsilon_j)=a_j + î b_j ...
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1answer
210 views

Argument of the Riemann zeta function on Re(s)=1

I refer to the lovely answer to this question. Is there an exact formula for the argument of the Riemann zeta function? Specifically, I would like to know the arguments along the line Re$(s)=1$. Some ...
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2answers
61 views

Clausen zeta function

For $0 < \theta < 2\pi$, define $$\kappa(x,\theta) = \frac{1}{\zeta(x)}\sum_{n=1}^\infty \frac{e^{ in\theta}}{n^x}$$ for $\Re(x) > 1$. It is easy to see that $$\kappa(x,\theta) = \frac{1}{\...
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Values of Riemann zeta function at small height along Re(s)=1

I am looking for sources which give numerical and theoretical computations of $\zeta(1+it)$ and the completed zeta function (without the $s(s-1)$) $$\xi(1+it)=\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)$$ ...
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When does linear combination of real-rooted entire functions of genus 0 or 1 remain real-rooted?

In our search of a family of entire functions to approximate Riemann $\Xi(z)$ function, we encounter the following family of functions: $$f_m(z,n,b)=\sum_{k=1}^m (-1)^k u_k(z,n,b)\tag{1}$$ where $b&...
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1answer
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On evaluating the Riemann zeta function, including that $\zeta(2)\gt \varphi$ where $\varphi$ is the golden ratio

A week ago, I got the following : For a positive integer $k$, using Cauchy–Schwarz inequality, $$\left(\sum_{n=1}^{\infty}\frac{1}{n^k(n+1)^k}\right)^2\lt \left(\sum_{n=1}^{\infty}\frac{1}{n^{...
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1answer
41 views

what is the value of Chebyshev function at non-integer value?

What is the value of Chebyshev $\psi(x)$ function at non-integer values ? For example, what is the value of $\psi(3.56)$? I have seen, in same place, it seems that $$\psi(3.56)=\psi(3)$$ And in ...
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Summation continuation, over no common divisors

Recently i've thought of a new step in my summations. I've been mentioning them in my previous questions. And i know there are certain point were it doesn't work. But most of the time it does. $$\...
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1answer
142 views

Relationship between perfect squares and infinite series (zeta function)

I noticed something when scribbling the zeta function - $$\zeta(3) = \sum_{n=1}^{\infty}\frac{1}{n^3} = 1 + \frac{1}{8} + \frac{1}{27} + \frac{1}{64} + \frac{1}{125} + \frac{1}{216} + \frac{1}{343} + ...
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336 views

I think I found a flaw in Riemann Zeta Function Regularization

I think I may have found a flaw in how Zeta Regularization works. As we all know, it's very famous for proving that $1+2+3+4+...=(-1/12).$ See here (5 rows of equations at the end of this post) •...
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1answer
105 views

Does Riemann Hypothesis imply strong Goldbach Conjecture? [duplicate]

In Andrew Granville's 2007 paper: "REFINEMENTS OF GOLDBACH’S CONJECTURE, AND THE GENERALIZED RIEMANN HYPOTHESIS" He said: "an averaged strong form of Goldbach's conjecture is equivalent to the ...
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27 views

Selberg's theorem, rotational invariance and circle on the Riemann sphere

If I'm not mistaken, Selberg proved that $\vert\zeta(1/2+it)\vert$ is normally distributed. But the normal distribution is known for its rotational invariance property and as a matter of fact, RH is ...
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1answer
51 views

Question about complex analysis in proof in Ingham

This is a detail from a proof in Ingham's Distribution of Prime Numbers, p. 91-92. He forms a Dirichlet integral and assumes for contradiction that the numerator $c(x)\geq 0.$ Then he bounds $f(s)$ in ...
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1answer
100 views

How does Dirichlet regularization of $1 + 2 + 3 + …$ work?

How does Dirichlet regularization assign value $-1/12$ to $\sum_{k=1}^{\infty} k$? Yes, I know that $\zeta(-1) = - 1/12$, a result that follows from the Riemann functional equation $\zeta(s) = 2^s \...
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1answer
50 views

is there a Globally convergent series for Riemann Xi function?

According to Wikipedia, there is a global convergent series for Riemann Zeta function: https://en.wikipedia.org/wiki/Riemann_zeta_function#Globally_convergent_series Is there a similar global ...
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Can this approximate closed form of Apery's constant $\zeta(3)$ be improved?

I know that an approximate closed form is not really a solution. However, I would like to present a method that gives a closed form of $\zeta(3)$ that is accurate to the 5th decimal, hoping that it ...
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66 views

An understandable explanation of the graph of zeta spiral $z(t)=\zeta\left(\frac12+it\right)$

Just as for a graphical real variable, domain and range are studied, their roots, their growth and minimum and maximum, convexity and role that has derivative in previous computations, too its ...
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32 views

Clarify mertens' theorem?

If merten's theorem states this http://mathworld.wolfram.com/MertensTheorem.html (equation on the second line) specifically, then what is the error described as for finite n?
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What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line?

What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line ? Are there particular mechanism to make this happen ? Because of Levinson and Corney's work, we ...
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1answer
106 views

The values of the derivative of the Riemann zeta function at negative odd integers

I would like to know if the values of the derivative of the Riemann zeta function at negative odd integers are computed, i.e. $\zeta'(-n)$ when $n$ is odd. When I look at the page from Wolfram ...
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2answers
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Riemann Hypothesis: Is $1/2$ of critical line same as the $1/2$ of square-root accurately of error term of prime number theorem?

Here is a question about Riemann Hypothesis: Is $1/2$ of critical line same as the $1/2$ of square-root accurately of error term of prime number theorem $?$ In other words, (just for some brain ...
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1answer
59 views

Pick the $\zeta(3)$ contribution from Gamma function countour integral

I edited the post and title. How do we see that given $$ Z= \oint \frac{d \epsilon}{2\pi i} (z\bar z)^{-\epsilon} \frac{\pi^4 \sin 5\pi \epsilon}{\sin^5 \pi \epsilon} \left|\sum_{k=0}^\infty (-z)^k \...
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1answer
85 views

On $\sum (1/\sqrt{n})\cos (t_0 \log n)$ and $\sum (1/\sqrt{n})\sin (t_0 \log n)$, from a zero of $\zeta (s)$ of the form $s_0=(1/2)+it_0$

I assume (as hypothesis, for questions too) that $s_0$ a fixed (nontrivial) zero of the Riemann zeta function $\zeta(s)=\sum_{n=1}^\infty 1/n^s$ has the form $(1/2)+it_0$, Thus for a positive real $...
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How to produce Riemann zeta zero spectrum with the Fourier transform in Mathematica?

All: I post a question generating Riemann Zeta zero spectrum using Mathematica on board of mathematica.stackexchange.com: http://mathematica.stackexchange.com/questions/95294/can-anyone-re-produce-...
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Analytic continuation vs. series convergence near convergence boundary

Citing Wikipedia, the Riemann zeta function is the analytic continuation of $$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} $$ The series itself is only convergent in the right half complex plane ...
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1answer
55 views

Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same?

All: Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same ? They shall be all different, right ? Is there a proof of this statement ? Thank you.
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65 views

Dirichlet series expansion for Zeta(s)

Wikipedia lists a series expansion for $\zeta(s)$ here. How is the Dirichlet series below derived? I apologize in advance if this is a very simple question, I don't know much about Dirichlet series. ...
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Where to find Brun's original combinatoric treatment of Brun Sieve?

I tried to understand Brun's original combinatoric treatment of Brun Sieve. (Unfortunately, I do not understand German), so I could not read Brun's original paper as in following: Viggo Brun (1915). ...
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1answer
76 views

What are some fundamental symmetries of Riemann Zeta (Xi) function are important to Riemann Hypothesis?

What are some fundamental symmetries of Riemann Zeta (Xi) function are important to Riemann Hypothesis ? (Beside the obvious symmetry of Riemann Xi function, s <--> 1-s reflection) IMHO, at the ...
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35 views

What are some recursive properties of Merten function or Summatory Liouville function?

Both Merten function and Summatory Liouville function show some kinds of "scale invariance" properties. (Those functions also display some kind of "periodic" behavior.( Just wonder if those "scale ...
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38 views

Two-point correlations of the Riemann zeta function

In http://www.americanscientist.org/issues/pub/the-spectrum-of-riemannium/5, the author mentions that the function $$P(x) = 1-\left(\frac{\sin(\pi x)}{(πx)}\right)^2$$ seems to be, assuming the ...
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Irrationality of $\zeta(\frac{3}{2})$

Is $$ \zeta\left(\frac{3}{2}\right) = \sum_{n=1}^\infty \frac{1}{n^{3/2}} $$ an irrational number?
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What's the value of $\sum_{k=1}^{\infty}(\zeta(2k+1)-\zeta(2k+2))$?

I'm confused about this. I have this expression $$ \frac{1}{2}=\sum_{k=1}^{\infty}(\zeta(2k)-\zeta(2k+1)) $$ Now if I want claculate $\zeta(2)$ I'll do the apropriate manipulations to get $$ \zeta(2)=\...
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Estimating a sum related to a short Euler product

The Question Is $$\sum_{\substack{n>y\\ p\mid n\Rightarrow p\leq y}}\frac{\Lambda(n)}{n^s\log n}=O(1/\log T)$$ where $y=(\log T)^{100}$ and $T$ is large? Background Assume that $$\log\zeta(\...
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Integral representations of $\zeta(s)$ using the floor/frac functions. How could this one be derived?

Browsing the web, I found quite a few integral representations for $\zeta(s)$ that use the Fractional part {x} or the Floor-function $\lfloor x\rfloor$ e.g.: $$\zeta(s) = \dfrac{s}{s-1} - \frac12+s \...
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3answers
70 views

Computing the tail of the zeta function $\sum_{n>x}n^{-s}$

I want to compute $$ f_s(x)=\sum_{n>x}n^{-s} $$ for some $s>1$ (in my case, $s=3$). Of course $$ f_s(x)=\zeta(s)-\sum_{n\le x}n^{-s} $$ but for $x$ large this is hard to compute. Are there good ...
8
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2answers
144 views

Closed form of $\lim\limits_{n\to\infty}\left(\int_0^{n}\frac{{\rm d}k}{\sqrt{k}}-\sum_{k=1}^n\frac1{\sqrt k}\right)$

Show that $$ L=\lim_{s\rightarrow\infty}\left(\int_0^s\frac{ds'}{\sqrt{s'}}-\sum_{s'=1}^s\frac{1}{\sqrt{s'}}\right) = 1.460\ldots $$ My attempts: To begin, rewriting the limit of the form $$ ...
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1answer
47 views

Is this infinite product for zeta(2) trivial?

I have crafted an infinite product for zeta(2) shown here. Euler's prime product is the only one I'm aware of. In checking Math World, I don't see any products. Is that because they are trivial?
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207 views

Approximate zeros of a (hypothetical) analog of $\zeta(s)$

[Added numbers 11/13.] Motivation (can skip). When prime powers $p_n$ are used to calculate $$y(x) = \sum_{n=1}^{N}\frac{\sin (x \log p_n)}{p_n},\hspace{5mm}(1)$$ for (say) $N= 30,$ $x>5$, at ...