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

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A Hamiltonian with smooth term exact to the Riemann zeros

what would happen if one found a Hamiltonian with an smooth level density in the form $$ N(E)= \frac{E}{2\pi}\log\left(\frac{E}{2\pi e}\right)$$ which is exactly the density of the RIemann zeros.. ...
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165 views

How to get the floor function as a Mellin inverse of the Hadamard product of the Riemann zeta function?

The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely : $$\left \lfloor x \right \rfloor=\frac{1}{2\pi ...
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193 views

Is this formula for $\zeta(15)$ true?

Apery gave, $\begin{aligned} \zeta(3) &= \frac{5}{2}\,\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k^3\,\binom {2k}k}\end{aligned}$ J. Borwein and D. Bradley found this can be generalized to ...
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356 views

Can the openness of Gilbreath's conjecture be reduced to proof of the Riemann hypothesis?

As an information theoretician, it's a personal hobby of mine to find elegant analogies to open mathematical problems. After all, they have a profound impact on how I conduct my research and how I ...
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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 ...
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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 ...
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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 ...
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symmetrized partial sums for $\zeta(s)$ and $\eta(s)$ in the critical strip

$\def\Re{\operatorname{Re}}$ We start with $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}\qquad \Re(s)>1\tag{1}$$ $$\zeta(1-s)=\sum_{n=1}^{\infty}\frac{1}{n^{1-s}}\qquad \Re(s)<0\tag{2}$$ ...
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49 views

how to prove that an entire function is positive on the real axis

The error function $\mathrm{erf}(x)$ is defined as: $$\mathrm{erf}(x):=\frac {2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}}dt\tag{1}$$ Let us define the following 3 functions: ...
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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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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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67 views

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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79 views

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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60 views

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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106 views

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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87 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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160 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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215 views

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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224 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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210 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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22 views

On Zero-Free Regions for $\zeta(s)$ and $L(s,\chi)$ with $|t| \le 2$

I'm reading the proof from Hildebrand that for some $c_1 > 0$, the Riemann zeta function $\zeta(s)$ has no zero in the region $\sigma > 1-c_1$, $|t| \le 2$. (Here $s = \sigma + it$ per ...
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Simple Zero of the Riemann Zeta Function

Let $s=σ+it$. Assume that $ζ(s)-1/(s-1)$ has an analytic continuation to the half plane $σ>0$. Show that if $s = 1 + it$, with $t≠0$, and $ζ(s) = 0$ then $s$ is at most a simple zero of $ζ$. I ...
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Divergence Dedekind zeta function

Let $K$ be a number field, $\mathcal O_K$ be its ring of integers, T a positive integer and $N$ the norm function. Give an upper bound (in T) for $$\sum_{I\leq \mathcal O_K: N(I)\leq T} ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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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36 views

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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80 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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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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85 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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Are there any Riemann zeta like functions that may have nontrivial zeros on the critical line but only involves integers up to $N$?

Question: Do there exist any Riemann zeta $\zeta(s)$ like functions $f_N(s)$ that may have all nontrivial zeros (verified via numerical calculation) on the critical line but only involve integers up ...
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50 views

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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77 views

A question about theorem 2 in de Bruijn's 1950 paper “The roots of trigonometric integrals”

Theorem 2 of de Bruijn'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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70 views

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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27 views

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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43 views

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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39 views

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 ...