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

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Some doubts about easy computations involving nontrivial zeros of Riemann's zeta function

On assumption of Riemann hypothesis when I write a complex zero (nontrivial zero) of zeta function as $\rho=\frac{1}{2}+it_\rho$, and I write $x^\rho$ as $\sqrt{x}e^{it_\rho \log x}$, then multiplying ...
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41 views

How to calculate non-trivial zeros of the Riemann zeta function

I wanted to know how riemann calculated some non-trivial zero of the zeta function. Would I like a manual calculation.
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Riemann Zeta Function integral

I was reading about the Riemann Zeta Function when they mentioned the contour integral $$\int_{+\infty}^{+\infty}\frac{(-x)^{s-1}}{e^x - 1} dx$$ where the path of integration "begins at $+\infty$, ...
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30 views

Primes in the zeta denominator

I spotted this while rewriting summation and i'd like to know if it's true. The fact i use prime is because it's an simplified example of a little more complicated verion. I'd like know if i can ...
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117 views

Proof of Riemann Hypothesis

This proof was released this year: http://arxiv.org/abs/1508.00533 Where is the mistake? I just found it and was wondering how obviously wrong it is.
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137 views

Average number of square free divisors for $n\leq x$

Let $d_{sf}(n)$ be the number of square-free divisors of $n$, and let $D_{sf}(k)=\sum_{n=1}^{k} d_{sf}(n)$ denote the corresponding summation function. Mertens showed that the asymptotic expansion of ...
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On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
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What is the effective lower bound on gaps between zeta zeros?

In this question here: Upper bound on differences of consecutive zeta zeros by Charles it is said that: "There are many papers giving lower bounds to: $$\limsup_n\ \delta_n\frac{\log\gamma_n}{2\pi}$$ ...
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Easy computations from a representation for Riemann zeta function, and Riemann zeta zeroes on critical line

The following series for Riemann Zeta function converges for $\Re s>0$ $$\zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}).$$ See for example this site or Wikipedia [1]. ...
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Breaking up integral representations by convergence

A known integral takes the form of $$\zeta(3)=\frac{1}{2}\int_{0}^{\infty} \frac{t^2}{e^t-1}dt$$ Through Wolfram part of the integral converges to $$\int_{0}^{\infty} \frac{t}{e^t-1}dt = ...
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Can you rewrite this function to be independent of the variable $p$?

A while ago Roger L. Bagula came up with the idea to study this function: ...
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122 views

Evaluating an infinite sum involving possibly hypergeometric terms

I was considering the following infinite sum $$ A(n) = \sum_{k=2}^{\infty}\left[\frac{(-1)^{k+n-1}}{k^n}(0k -1)(k-1)(2k-1)...((n-1)k-1) \right] $$ Some cases: $$ A(1) = ...
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Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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57 views

Is this plot of the argument of the Riemann zeta function around ZetaZero(127) correct?

(*Mathematica 8 start*) Clear[n, k, t, z, FL, NZ] N[ZetaZero[127]] NZ[t_] = Arg[Zeta[1/2 + I*t]]/Pi; Plot[NZ[t], {t, 280, 284}] Plot[NZ[t], {t, 282.3, 282.6}] ...
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43 views

Generating function of Riemann zeta function

I want to know about the generating function of the Riemann zeta function which is related with the Laurent expansion at $z=0$. $f(z) := \dfrac{d}{dz} \log(\sin\pi z)$ $f(z) = \dfrac{1}{z} -2\sum ...
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41 views

$\sum \zeta (s)$ 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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54 views

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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47 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 + ...
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44 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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33 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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35 views

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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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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50 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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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 ...
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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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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 ...
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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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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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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) = ...
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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 ...
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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 ...
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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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137 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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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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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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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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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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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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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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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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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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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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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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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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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: ...