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

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24 views

On a simplification of zeta functions.

Can you simplify the following product? With the known properties of $\zeta(x)$ $$\color\green ...
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1answer
11 views

What is the meaning of cyclically equivalence classes of multiple indices?

Can you please give me an example for the cyclic equivalence classes of multiple indices on the following set? $$ I(w,d)=\{\ (k_1,...k_d)\mid k_1+\ldots+k_d=w, k_1,...k_d \ge 1\,\}$$ where $ w$: ...
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3answers
201 views

Why $\zeta (1/2)=-1.4603545088…$?

I saw $\zeta (1/2)=-1.4603545088...$ in this link. But how can that be? Isn't $\zeta (1/2)$ divergent since ...
2
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1answer
142 views

Finding a solution to $\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$

Finding ONE solution to: $$\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$$ can apparently be done by iterating the following formula: $$\Large s(m+1)=\frac{\log \left(-\frac{1}{\sum _{n=1}^{k-1} ...
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17 views

We derive the nontrivial zeta zeros from the primes - can we use the same method to derive a set from the zeros, and in general for some set {S}?

The number of primes less than a given $x$ have an asymptotic formula and from that, we get a pretty good approximation. The error term between this approximation and the actual value comes from the ...
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40 views

A problem on simplification $\operatorname{Li_3}\frac{1}{3}$

Can you simplify $$\large\operatorname{Li_3} \frac{1}{3} $$ It might be impotant to note that $$\operatorname{Li_3}\frac{1}{2}=\frac{7\zeta(3)}{8}+\frac{\log^3 2}{6}-\frac{\pi^2\log 2}{12}$$ But I ...
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23 views

Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
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200 views

If these two expressions for calculating the prime counting function are equal, why doesn't this work?

So I've seen some different explanations of how the zeros of the zeta function can predict the prime counting function. The common example is that $$\pi(x)=\sum_{n=1}^\infty ...
4
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2answers
98 views

Did Hardy prove that there are countably, or uncountably many zeros on the line Re$(s)=1/2$ of $\zeta(s)$?

It's known that Hardy proved that there are infinitely many zeros of $\zeta(s)$ on the line Re$(s)=\frac{1}{2}$, but did he prove it's countably infinite? Or uncountable?
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28 views

can this functional of the Hardy Z function be written as an elliptic theta function?

Can H(t), Equation 33 of http://vixra.org/pdf/1510.0475v7.pdf be expressed as an elliptic theta or related function ? $H(t)= {\frac {4\,i\zeta \left( -i/2 \left( i-2\,t \right) \right) \pi \, ...
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Modular Euler product?

We know the Euler product. $$\zeta (s)=\prod_{p}\frac{p^{s}}{p^{s}-1}$$ I wonder if there is formula or any kind of work for this kind of prime product below? $$\prod_{p\equiv a \ (mod \ ...
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1answer
28 views

Seeking possibility of more elementary means of evaluating an improper integral.

It can be shown that $\int_0^\infty -\log{(1-e^{-x})}=\zeta(2)$ by expanding out the integral as $\log(1-z)$, exchanging summation and integration, then summing up the integrals. I am wondering if ...
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1answer
47 views

An Analogous Riemann Integral

$$1=\sum_{n=2}^\infty (\zeta (n)-1)$$ is a fairly well known result W|A validates this result Is there a closed form to the analogous integral: $$\text{?}=\int_2^\infty \text{d}x \, (\zeta(x)-1)$$ ...
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2answers
71 views

Why are the trivial zeros of the Riemann zeta function only negative?

The functional equation of the Riemann zeta function is $$\zeta(s)=2^s\pi^{s-1}\sin(s\pi/2)\Gamma(1-s)\zeta(1-s)$$ clearly $2^s$ and $\pi^{1-s}$ are never equal to zero on the complex plane, and ...
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91 views

Difficult expression of sum

I wanna show that $\sum{\frac{1}{k^{4}}}=\frac{\pi^{4}}{90}$. For this, I know that $$\sin(z)=z-\dfrac{z^{3}}{3!}+\dfrac{z^{5}}{5!}-\dfrac{z^{7}}{7!}+\cdots$$ On the other hand, also know that ...
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134 views

Wouldn't the Riemann hypothesis rule out a formula to predict primes? [closed]

Prime formula: a deterministic way to predict primes. Riemann hypothesis: implies "primes are random". If RH is true will we never have a useful prime formula?
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zeros/poles of Laplace transforms of Dirac combs (Riemann zeta function)

let's define $p_\alpha(n) = \displaystyle\int_1^n x^\alpha dx$ so that $\left\{\begin{array}{lll} p_0(n) &=& n-1 \\ p_{-1}(n) &=& \ln n \\ p_\alpha(n) &=& \frac{\textstyle ...
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1answer
39 views

Convexity of the Riemann-zeta function without derivative

Proving that the $\zeta$ function is convex on $(1,+\infty)$ is pretty simple if we use the derivative, but is there a proof without using derivative? I'm allowed to use just the definition of the ...
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1answer
75 views

Riemann zeta and Dirichlet eta functions, and Cauchy-Riemann equations

Taking the complex argument of the complex number $2$ as $0$, I've computed for complex numbers $s=x+iy$ $$1-2^{1-s}=1-2^{1-x}(\cos(y\log 2)-i\sin(y\log 2)),$$ in the equation ...
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1answer
166 views

Product of two series to get a series decomposition of zeta in the critical strip

$\def\sfrac#1#2{% \small#1% \kern-.05em\lower0.1ex/\kern-.025em% \lower0.4ex\small#2}$I've been working on gaining an intuitive understanding of the analytic continuation of the zeta ...
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1answer
43 views

Does this limit involving the Dirichlet eta function and the Riemann zeta function make sense?

Let $p_n$ the sequence of prime numbers (and you will consider below, too, the sequence $\frac{1}{n}$ with $n>1$). And if it isn't wrong for $0<\Re s<1$ the known equation between Dirichlet ...
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1answer
45 views

What are the conditions on this Riemann-Zeta function functional equation?

I am a huge fan of the Riemann Zeta function's functional equation: $$\large{\color\green{\zeta(x)=2^x \Gamma(1-x)\zeta(1-x)\pi^{x-1}\sin\frac{\pi x}{2}}}$$ I am curious as to what conditions on $x$ ...
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Conjecture $\int_0^1\ln\ln\left(\frac{1+x}{1-x}\right)\frac{\ln x}{1-x^2}\,dx\stackrel?=\frac{\pi^2}{24}\,\ln\left(\frac{A^{36}}{16\,\pi^3}\right)$

I did some numeric experiments with integrals involving double logarithms (because they received much interest both on this site and in published papers, sometimes under names of ...
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36 views

How can I prove that $\zeta(2n) = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}$? [duplicate]

I tried to prove this with Valli's equation: $\frac{sinx}{x} = \prod(1-\frac{x^{2}}{(n\pi)^{2}})$ and use $\frac{d(ln(sin(x)))}{dx} = i + \frac{2i}{e^{2ix}-1}$. Maybe it's better to use Taylor's ...
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98 views

Fault in proof of $\zeta(2) = \frac{\pi^2}{6}$

Consider the proof of: $$\zeta(2) = \frac{\pi^{2}}{6}$$ So the proof assume that (because of Euler decomposition) $$\frac{\sin(x)}{x} = \prod_{n > 0}\left(1 - \frac{x^{2}}{(n\pi)^{2}}\right)$$ ...
6
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1answer
44 views

$L$-function absolutely convergent for $\text{Re}(s) > 1$, condition for $L(s, \chi)$ converging for $\text{Re}(s) > 0$?

I have two questions related to here. Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ...
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2answers
76 views

Product of two absolutely convergent Dirichlet series

We have$$(f * g)(n) = \sum_{d \mid n} f(d)g(n/d).$$How do I see that if the two Dirichlet series$$F(s) = \sum_{n =1}^\infty f(n)n^{-s},\text{ }G(s) = \sum_{n=1}^\infty g(n)n^{-s}$$converge absolutely ...
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1answer
132 views

Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $?

The answer to an integral involved $\frac{\zeta'(2)}{\zeta(2)}$, but I am stuck trying to find this number - either to a couple decimal places or exact value. In general the logarithmic deriative of ...
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1answer
50 views

Can be justified this formula for $\zeta(2n+1)$

Can be justified for integers $n\geq 1$ that $$\zeta(2n+1)=\prod_{\text{p, prime}}\frac{1-\sigma(p^{2n})^{-1}}{1-p^{-2n}}?$$ Truly I don't know if I am wrong another time, when I use for an integer ...
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22 views

Residue of $\frac{\text{cot}(\pi z)}{z^6}$ at $0$

I am trying to compute $\zeta(6)$ = $\sum_1^{\infty} \frac{1}{n^6}$; I generally know how to do this using a residue-based proof, but I am stuck at the last bit, namely calculating the residue of ...
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1answer
40 views

Books on Zeta Regularization Product

Does anybody know some book on zeta regularization, and the zeta regularization product? I'm quite interested on the topic but I would need a book with some review...
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24 views

How to prove an equation involving $\sum_{p,n}\frac{\log (p)}{p^{n/2}}\delta_{\log (p^n)}(y)$

My context is that I could assume as true statements (I say this since claims about the type of convergence could be difficult to me) the first equation in [1], that could be written as the derivative ...
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50 views

Something fishy in the Zeta function

Recently I came across the Riemmann representation of the Zeta function as follows: $$\zeta (s) = (2^s)(\pi^s-1) \sin(\frac {\pi s} 2) \Gamma(1-s) \zeta(1-s) .$$ Now, I went ahead to calculate the ...
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56 views

Is it possible to compute $\Gamma(\rho_k)$ or $\zeta(3+2\omega_k)$, when $\rho_k=\frac{1}{2}+i\omega_k$ is a nontrivial zero of Riemann zeta function?

I believe that I can to use the equation (1) (from reference [1]), a well known equation involving the Gamma function, Riemann zeta function and the theta function $\psi(x)=\sum_{n=1}^\infty ...
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56 views

Why do you need to prove the error term goes to zero for the complete derivation of the Euler Product Formula?

I am doing a project on the Riemann-Zeta Function which begins by examining the Euler Product Formula. I understand the proof up until the point where it is made 'rigorous'. In other words, I ...
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26 views

What is the best (today) value of $c_k$ in $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \le c_k Y^k(t)$

From the Vinogradov-Korobov estimate, we have $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \ll Y^k(t)$ What is the best (today) value of $c_k$ in $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \le ...
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3answers
223 views

How to solve this integral: $\int_{-\infty}^\infty\frac{x^2 e^x}{(e^x+1)^2}\:dx$

I am trying to solve an integral like this: $$ I=\int \frac{x^2 e^x}{(e^x+1)^2} dx $$ And I get this answer: $$ \int \frac{x^2 ...
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1answer
82 views

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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1answer
51 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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1answer
132 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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1answer
138 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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1answer
60 views

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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46 views

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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1answer
29 views

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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34 views

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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3answers
125 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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0answers
32 views

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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1answer
61 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}] ...