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

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Explicit formulas for Fourier coefficients from its Tayor expansion

In my research, I need to determine unique coefficients $a_k$ in terms $b_k$: $$\sum_{k=0}^n a_k \cos\left(\frac{k}{n+1}t\right)+O\left(t^{2n+1}\right)=\sum_{k=0}^n b_k t^{2k}.$$ This problem showed ...
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fractional part of Riemann zeta $\sum_{s=2}^\infty \{\zeta (s)\}=1$

$$\sum_{s=2}^\infty \{\zeta (s)\}=1$$ where $\zeta (s)$ is Riemann zeta, $\{x\}$ denotes the fractional part of the real number $x$ The problem was proposed by Michael Th. Rassias ...
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How to evaluate Bessel functions $K_{i x+\frac{9}{4}}(2\pi)+K_{i x-\frac{9}{4}}(2\pi)$ with $x>1.5*10^8$ in Mathematica 7?

I am trying to evaluate Bessel functions $K_{i x+\frac{9}{4}}(2\pi)+K_{i x-\frac{9}{4}}(2\pi)$ with $x>1.5*10^8$ in Mathematica 7. This function is the first Polya approximation to Riemann ...
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1answer
66 views

Investigating the convergence of a series using the comparison limit test, Part II [duplicate]

I posted this question earlier, but as I don't know if a comment reply or edit will refresh this so people actually see, I'm going to post it again in hopes that someone knows what's going on. Here's ...
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English translation of two papers by Polya on real zeros of Fourier transform approximation to Riemann $\xi$ function

I am looking for English translation of the following two papers by Polya: [1] G. Polya, Bemerkung über die Integraldarstellung der Riemannschen $\xi$-Funktion, Acta Math. 48(1926), 305-317; ...
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a comparison of the distribution of zeros of Riemann $\zeta(1/2+it)$ against those for a trial function $\omega(1/2+it)$

Here you can find a comparison of the distribution of zeros of Riemann $\zeta(1/2+it)$ against those for a trial function $\omega(1/2+it)$. Notice from the last plot that the number of zeros in the ...
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1answer
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Product of zeta and its conjugate

Suppose we have the zeta function $\zeta(s)$, and we want to multiply it by its complex conjugate $\zeta(s)^*$. Since $\zeta(s)^* = \zeta(s^*)$, we get $\displaystyle \zeta(s)\cdot\zeta(s)^* = ...
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1answer
39 views

Evaluation of Riemann-Stieltjes integral in Laurent expansion of zeta function

I'm probably being really stupid but in a proof of the Laurent expansion of the Riemann zeta function the quantity \begin{equation} S_r(t) = \sum_{n \leq t} \frac{(\log (x/n))^r}{n} \end{equation} is ...
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1answer
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Calculating the residues of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$

Calculating the poles of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$, where x is a fixed real number. I am trying to calculate the poles of this function at the trivial zeros of $\zeta$, ...
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Real zeros of the zeta function

How does one show that the negative even integers make up all the real zeros of the zeta function? That is, how does one show that there are no real zeros on the interval [0,1]? I am aware that you ...
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Dirichlet L function

The function is defined here - http://en.wikipedia.org/wiki/Dirichlet_L-function If $\chi$ is primitive and $\chi(-1)=1$ how do I show that $L$ has infinite number of zeros in the critical strip
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Invalid use of the analytic continuation of the Riemann zeta function?

Watching this video on You Tube I got the impression that some sciences (in this case physics) use the analytic continuation of the Riemann zeta function without justification. Maybe this is just my ...
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56 views

Generalized Riemann Hypothesis : Zeros of Dirichlet L function and its functional equation

Let $\chi$ pe a primitive character modulo q with $\chi(-1)=1$ ; L is the Dirichlet - L function Define, $\xi(z,\chi)=(q/\pi)^{z/2}\Gamma(z/2)L(z,\chi)$ Show that $L(z,\chi)$ has infinitely many ...
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1answer
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Does this relative primes formula violate the feasibility of picking truly random numbers?

I read a question on this site recently that fascinated me by pointing out that you can't truly pick a random number from an infinite set. I can't find the answer now, but it was shown that you have ...
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Different methods of calculating $\zeta(s)$'s Laurent series.

Initially, I thought that calculating$$\int_\gamma \frac{\zeta(z)}{(z-1)^n}dz$$ directly, where $n \in \mathbb{Z}$ and $\gamma$ is an anticlockwise contour around $z=1$ with winding number $1$, would ...
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$\zeta(2 + it) = \zeta(2-it)$

Let $\zeta(s)$ denote the Riemann zeta-function. Show that $\zeta(2 + it) = \zeta(2-it)$ for all real t. Give some hints how to do this one.Thanks in advance.
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riemann zeta zeros some predictable, some not

Presuming the Riemann Hypothesis, the non-trivial zeros of the zeta function occur when both $\Re\{\zeta (s)\}$ and $\Im\{\zeta(s)\}=0$, where $s=\frac{1}{2}+it$. and since ...
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691 views

Derivative of Riemann zeta, is this inequality true?

Is the following inequality true? $$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$ This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches ...
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Could Fibonacci numbers be related to Riemann zeros?

this is my question can tghe fibonacci numbers $$ F_{n+2} =F_{n+1} +F_{n} $$ be related to the zeros of the Riemann zeta function ?? i heard that in the webpage ...
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Riemann zeta, why are the residues either zero or one?

One more question, probably equally simple to answer but I don't know how this is true either: Why is the residue of Riemann zeta zero - trivial or non-trivial: $$\text{residue}\left(\zeta ...
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1answer
86 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
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262 views

A conjecture relating Multiple Zeta Values and the Polya Enumeration Theorem

Let me state my motivation. I believe that the Polya Enumeration Theorem and Multiple Zeta Values (the classic being the Basel problem and the values of the Riemann zeta function at the even ...
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About Riemann's zeta function

Is the riemann zeta function analytic? If so can it be expressed as a power series? Does it have a ratio of convergence ? Could it be said to have a center point of its ratio of convergence at ...
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1answer
83 views

Riemann Zeta circularity?

In this post I show: $$\prod _{p\text{ prime}} \frac{p^s}{p^s-1} = \zeta(s).$$ Wolfram Alpha shows an alternate form for the primes: $$\frac{p_n{}^2}{p_n{}^2-1}=\frac{\left(\sum _{k=1}^{2^n} ...
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1answer
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Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
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The zeros of $2\,\xi(s)-1$. Is there anything known about the curves they lie on?

Take the well known integral: $$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}-1} + {x}^{\frac{-s}{2}-\frac12}\right)\,\psi(x)\, ...
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1answer
66 views

Contour approach to Riemann zeta functional equation

I have a question regarding Riemann's first proof of the functional equation that was given in his paper on the Riemann zeta function. I am an undergraduate working on a fairly short undergraduate ...
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67 views

How to prove $\sum_{n=1}^{\infty}\frac{\mu (n)}{n^{s}}=\frac{1}{\zeta (s)}$?

How can we prove this equation? $$\sum_{n=1}^{\infty}\frac{\mu (n)}{n^{s}}=\frac{1}{\zeta (s)}$$
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Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 ...
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Logarithmic derivative of Riemann zeta, is this derivation correct?

Let matrix $T_2$ be defined below as the Dirichlet inverse of the Euler totient function as a function of the Greatest Common Divisor (GCD) of row index $n$ and column index $k$; $$T_2(n,k) = ...
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Riemann zeta function - Euler product formula

I want to prove that $$ \frac{1}{\zeta(s)}=\sum_{n=1}^\infty \frac{\mu(n)}{n^s}.$$ I know that the standard proof works with the Euler product formula $$\zeta(s)=\prod_{p \ \text{prime}} ...
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Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
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Calculating $\pi$ via the $\zeta$ function?

I was fooling around, trying to come up with a rapid way to compute $\pi$. Then I remembered that we always have: \begin{equation} \zeta(2n)=c\pi^{2n}, \end{equation} where $n$ is a positive integer ...
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Is there an expression for $(S/k)$ where $S=\sum_{n=1}^\infty n$ and $k \in \mathbb{Z}$?

Given that $S=\sum_{n=1}^\infty n=-1/12$ (for an explanation see this question or this video from Youtube) For example if $k=4$: $(S/4)=1/4+2/4+3/4+1+5/4+6/4+7/4+2+9/4...$ Please edit to improve ...
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Find the regularized sum $\frac{1}{\ln(2)}+\frac{1}{\ln(3)}+\frac{1}{\ln(4)}+…$

By considering the integral Zeta function $$F(s)=s+\frac{1}{2^s\ln(2)}+\frac{1}{3^s\ln(3)}+\frac{1}{4^s\ln(4)}+...$$ Evaluate $$\frac{1}{\ln(2)}+\frac{1}{\ln(3)}+\frac{1}{\ln(4)}+...$$ EDIT: ...
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Why is $\zeta(s)\neq0$ for $\operatorname{Re}(s)=0$?

I have a question concerning the Riemann zeta function for a project I've been working on. Why is it that $\zeta(s)\neq0$ for $\operatorname{Re}(s)=0$ (that is, there are no non-trivial zeroes of the ...
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Asymptotic expansion of $\zeta(s)$

It is well known that $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
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Trivial zeros of the Riemann Zeta function

A question that has been puzzling me for quite some time now: Why is the value of the Riemann Zeta function equal to $0$ for every even negative number? I assume that even negative refers to the ...
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Find the sum $\sum_{n = 1}^{\infty}(-1)^{n + 1}\log(1 + (1/n))$

I started as follows $$\begin{aligned}S &= \sum_{n = 1}^{\infty}(-1)^{n + 1}\log\left(1 + \frac{1}{n}\right)\\ &= \sum_{n = 1}^{\infty}(-1)^{n + 1}\sum_{k = 1}^{\infty}(-1)^{k + 1}\frac{1}{k ...
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Zeta function for negative integers

I already proved that $\zeta(z)=\frac{1}{\Gamma(z)}\int_0^\infty\frac{t^{z-1}}{e^t-1}dt=\frac{\Gamma(z-1)}{2\pi i}\int_{-\infty}^0\frac{t^{z-1}}{e^{-t}-1}dt$ Now the Benoulli numbers are defined by ...
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297 views

Problem in Divergent series

Lets try to evaluate $$\frac{(-1)}{1^s}+\frac{(-1)^2}{2^s}+\frac{(-1)^3}{3^s}+...$$ $$=\frac{e^{\pi i}}{1^s}+\frac{e^{2\pi i}}{2^s}+\frac{e^{3\pi i}}{3^s}+...$$ $$=\frac{1}{1^s}(1+\frac{\pi ...
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1answer
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Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$

The result quoted in the title is usually a stepping stone in the proof of the prime number theorem and I am familiar with the usual argument for this result. The other day my professor was telling ...
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103 views

Continuation of the Riemann Zeta Function

I am actually aware of the argument showing $\zeta$ has a meromorphic extension to $\mathbb{C}$ with a single pole at $z = 1$. On a recent number theory exam, however, one of the questions asked to ...
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107 views

Non-trivial zeros off critical line

If non-trivial zeros lay off the critical line (as shown in the picture below), would they have to come in fours rather than conjugate pairs (as the diagram shows)? I am presuming they would, ...
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Upper bound on $\zeta(s)$

I'd like to know an upper bound for $\zeta(s)$ in the critical strip, and hopefully one that is not too difficult to prove. For instance, ...
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Question about the zeros and poles of the PrimeZeta function.

The Euler product over all primes, $$\displaystyle \zeta(s) := \prod_{p\in\mathbb{P}} \dfrac{1}{1-\dfrac{1}{(p)^s}}$$ is only valid for $\Re(s) >1$. However, when taking the log on both sides ...
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Laurent Series of Riemann Zeta Function [closed]

How do I go about finding the Laurent series of the Riemann zeta function about $z=1$?
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1answer
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Analytic Continuation of Riemann Zeta Function

How do we show that this holds for $\operatorname{Re}(z)>0$ (and not $1$) $$\zeta(z)= \sum_{i=0}^{m-1} n^{-i} + \frac{m^{-z}}2 +\frac{m^{1-z}}{z-1} -z\int_{m}^{\infty} ...
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1answer
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Logarithms, prove this limit.

Mathematica knows that: $$\log (n)=\lim_{s\to 1} \, \left(1-\frac{1}{n^{s-1}}\right) \zeta (s)$$ Kind of tautological starting with logarithms, but I would like to know better why this limit works: ...
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1answer
44 views

Sum of zeta(2s) fractions without pi^(2s) in the numerators

$$ \sum _{n=1}^{\infty } \sum _{r=1}^{\infty } (\pi r)^{-2 n}=\frac{1}{2} (1-1 \cot(1)) $$ $\frac{1}{2} (1-1 \cot(1))$ is not in OEIS, so it doesn't seem to be well known. Q1: Would this info be of ...