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

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Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
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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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A connection between a sequence from the Collatz conjecture and a sequence of densities from $\zeta(k)-1$?

Just for grins, I created lists of first-entries of finite sequences of rank $r$ for the Syracuse problem (Collatz conjecture using only odd numbers) and found these sequences on OEIS. My sequences, ...
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94 views

Series involving the Riemann zeta function

Consider the series: $$\sum_{n=1}^{\infty}\frac{\zeta(2n+1)}{n(2n+1)}$$ We can easily prove that it's a convergent series. My question, is there a way to express this series in terms of zeta ...
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Zeta Zeros and primes / prime powers

The plot $\Re\ x^{Zeta\ Zero} + \Im\ x^{Zeta\ Zero}$ for the first $1000$ Zeta Zeros up to $x = 30$ using the following Mathematica code: ...
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Zeros of “nearby” holomorphic functions

I would appreciate help in how to show that On a compact subset of {$z \in \mathbb{C}: 1/2 < \Re (z) < 1$}: "Given a holomorphic function with an isolated zero, any "nearby" holomorphic ...
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128 views

Mandelbrot set and riemann hypothesis

Has anyone tried to make a connection between the Mandelbrot set and the non-trivial zeros the zeta function? Looking at the Mandelbrot set, it appears that all points are to the left of the line 0.5 ...
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Integer values of the Riemann function - II

For what value of $n \ge 2$ can we have an real $x > 0$ such that both the numbers $$ \zeta\Big(1+\frac{1}{x}\Big) \text{ and } \zeta\Big(1+\frac{1}{nx}\Big) $$ are positive integers.
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Validity of a functional formula of the Riemann Zeta function across the whole complex plane?

Could someone confirm me the validity of the following formula: $$\zeta\left(z\right)=2^{z}\pi^{z-1}\sin\left(\frac{\pi z}{2}\right)\left(\frac{e^{-\gamma ...
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78 views

A function that generates 'alternating' non-trivial zeros of $\zeta(s)$

I am trying to find a function, that assuming RH, generates subsequent non-trivial zeros $\rho_n$ in an alternating way i.e.: $$\frac12+14.134...i,\frac12-21.022...i,\frac12+25.010...i, \dots$$ or ...
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Theta series and Riemann Hypothesis

in the paper http://www.fuchs-braun.com/media/dd209bf5c2203a87ffff80a3ffffffef.pdf section 2 ' Hilbert-Polya space' page: 180 the author introduce the Theta series $$ F(\phi(x))= ...
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148 views

If RH is false , could this be true?

Let $\zeta(s)$ be the Riemann zeta function. Assume RH is false , is it possible that we have in the critical strip $\zeta(a_1+ti) = \zeta(a_2+ti) = \zeta(a_3+ti) = \cdots = \zeta(a_n+ti) = 0$ For ...
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Are the amplitudes of these frequency spikes equal to 1 when the real part of the complex number “s” is equal to one half?

Over at stack overflow I asked a question about how to plot the Riemann zeta zero spectrum from the von Mangoldt function. Then I asked a question about calculating the Riemann zeta function at the ...
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Is this formula for $ \sum_{n} (n^{2}+z^{2})^{-s} $ correct?

I would like to know if this formula is true: $$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$ I have used the ...
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214 views

What are the local minima in this spectrum?

Edit 6.2.2012: The sequence to be transformed should be f = 0,1,2,3,4,5... which makes the mentioning of the von Mangoldt function less necessary. Edit 5.2.2012: I had the wrong plot of the insignal. ...
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208 views

Why Pólya's method is wrong?

By truncating the Fourier transform, Pólya managed to prove that the Xi function on the critical line was approximately $$\xi(1/2+is) = (2\pi)^2 ( K_{9/4+is/2}( 2\pi) +K_{9/4-is/2}( 2\pi))$$ If this ...
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77 views

L-function of product

Given two varieties of finite type over a finite field, what is the $L$-function of their product in terms of the $L$-function of the factors?
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Riemann and 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 ...
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41 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 ...
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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
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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 ...
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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 $$ ...
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q question regarding the numerical zero finding of Riemann zeta function and actual proof of Riemann Hypothesis

(1)Suppose that people verified in 2004, that all zeros of $\zeta(\sigma+i t)$ with $0<t<T<=10^{22},0<\sigma<1$ are on the critical line ($\sigma=1/2$). (2)Suppose Bob proved in ...
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64 views

Riemann Zeta Function equation

So I'm a amateur mathematician and I was working with the Riemann Zeta Function and I was able to proof this identity. So I'm just wondering if this has already been proven before. For S>0 So I ...
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36 views

Zeta zero sum & reciprocals of prime powers

Below is a plot of $$\dfrac{1}{x}\sum_{n=1}^{75}2\Re\left(\operatorname{Ei}\left(\rho_n\log\left(x\right)\right)\right)$$ where $\rho_n$ is the $n$th zeta zero, with grid lines at primes and prime ...
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71 views

Any complex analysis book with programming assignment and exercises?

All: I had studied complex analysis long time ago. Now, I would like to review some material, particularly about Analytic function, Riemann zeta and Analytic function. I have been a software ...
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47 views

What is the sign of the generalized Stieltjes constants $\gamma_{k}(a)$?

Recall that the Stieltjes constants $\gamma_{k}$ appear as the coefficients in the regular part of the Laurent expansion of the Riemann zeta function about $s = 1$: $$ \begin{align} \zeta(s) = ...
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Riemann functional equation question?

I was looking through the derivation of the Riemann functional equation, and I understand how to obtain the result $$ \pi^{-\frac s2} \Gamma (\frac s2) \zeta(s) = \pi^{-\frac{1-s}{2}} ...
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Plotting the pair correlation function for the zeta zeros /GUE

I am making a shameless request for instructions on how to plot this: from this page. I can see from here that normalizing the zeros is given by ...
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A question about theorem 2 in de Bruin's 1950 paper “The roots of trigonometric integrals”

Theorem 2 of de Bruin's paper titled "The roots of trigonometric integrals" (Duke Math. J., 17 (1950)) is given by: What does it mean by "the function $q(x)$ be regular in the sector...? Does it ...
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Prime Zeta Function

Does $$\sum_{p \text{ prime}} \frac{1}{p^s} \sim \log \zeta(s) \quad \text{as} \quad s \to 1^+$$ imply $$\sum_{p \leq n} \frac{1}{p} \sim \log H_n \quad \text{as} \quad n \to \infty,$$ where $H_n$ is ...
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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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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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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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41 views

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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Laurent Series of Riemann Zeta Function

How do I go about finding the Laurent series of the Riemann zeta function about $z=1$?
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Does the Riemann zeta function tell us about the order theoretic properties of the natural numbers?

The classical Möbius function $\mu(n)$ fulfills the multiplicative inversion formula, e.g. see this thread. Now I see in the theory of posets, they generalize the concept of that function, see ...
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Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
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Are the zeros of the sum/difference of two reflexive, entire functions all on the line $\Re(s)=\frac12$?

Remove the order $1$ pole of $\zeta(s)$ at $s=1$, to create the following entire function: $$z(s):=\zeta(s)-\dfrac{1}{s-1}$$ I like to conjecture that all complex zeros of $z(s) \pm z(1-s)$ in the ...
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How to prove that there are $O(T\ln T)$ zeros in the critical strip of the Riemann zeta function?

Define $F(T)$ as the number of solutions to $\zeta(a+ ti) =0$ for $0\le t\le T$ and $0<a<1$. How to show that $F(T)= O(T\ln T)$? For clarity, $\zeta$ is the Riemann zeta function, $i$ is the ...
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Dirichlet series minus Riemann zeta

Suppose $\{a_n\}$ is a sequence of complex numbers such that the sums $A_n=a_1+\cdots+a_n$ satisfy $$|A_n-nb|\leq Cn^{\sigma}$$ for all $n$, where $b\in\mathbb{C},C>0,0\leq\sigma<1$. Consider ...
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Zeta zeros by recurrence of zeta function, but this is useless isn't it?

One more useless question of mine can't do this site any harm. So here we go. The following Mathematica program converges to most of the riemann zeta zeros, by using an approximation as a starting ...
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How should I prove that Zeta'(x)/Zeta(x)+1/(x-1) is strictly monotonously decreasing on the real line (for x>=0)?

The Riemann Zeta function, like most other complex functions, are much easier to deal with in the real line, since the values are also real, and definitions can be used in a more straightforward ...
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The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
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446 views

Ramanujan Summation not consistent with Riemann's Zeta function?

Wikipedia states that Ramanujan sums and the Riemann Zeta function have the same values for even $k$: $$1 + 2^{2k} + 3^{2k} + \cdots = 0\ (\Re)$$ However, I don't understand how this can be true, ...
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Would be this formula valid ? Zeta regularization and Euler product plus zeros.

$$ \frac{\zeta (0)}{\zeta (s)}= \prod _{p}\prod_{m= -\infty}^{\infty}\left(1-\frac{is\log p}{2\pi m}\right),$$ where $m$ does not run over $ m=0 $ and '$p$' means a product over all the primes :) ...
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108 views

Euler's Basel problem continued… $\zeta(2n)$ expressed in terms of $sinc$?

I have to make a brief intro before comming to my question. To approach the famous Basel problem Euler starts with the $sinc$ function \begin{align}\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + ...
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The partial sum and partial product of $\zeta$function

Taking the partial sum of the $\zeta $ function: $$\zeta^H(s,k)=\sum_{n=0}^k \frac{1}{n^s}$$ and the partial product f the $\zeta $ function: $$\zeta^P(s,j)=\prod_{i=0}^j \frac{1}{1-p_i^{-s}}$$ I ...
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Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

For some exercises with (divergent) summation of the Stieltjes constants I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as ...
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135 views

Approximation of distribution of $\pi_k(n)$ using $\zeta(s)$

Let $\pi_k(n) $ be the number of numbers with k prime factors (repetitions included) less than or equal to n. If we take the sums: $z_1(s) = \sum_{n= 1}^\infty \frac{1}{(p_{1,n})^s},~ z_2(s) = \sum ...