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

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17 views

Growth rate of $\zeta(n)^{-1}$

What is the asymptotic growth rate of $\frac1{\zeta(n)}$? Is it polynomial in $n$?
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27 views

Is the absolute value of the intersection of two functions related to the nontrivial zeros always equal to $\sqrt{2}$?

With $\displaystyle \chi(s)=\pi^{-\frac{s}{2}}\,\Gamma\left(\frac{s}{2}\right)$ and $K(s)=\Psi\left(\frac{s}{2}\right)-\ln(\pi)$, with $\Psi\left(s\right)$ the digamma function, then the Riemann ...
-3
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0answers
31 views

Does the Riemann Hypothesis consider mirror symmetry on its non-trivial zeros?

Setting the bottom corners of the square 1 on the center of two intersected circumferences and taking as center of symmetry the center of that intersection, it's possible to project the square 1 ...
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1answer
37 views

Root of the $\zeta(s) = s$

What is the root $s_0$ of the equation $\zeta(s) = s$, where $\zeta(s)$ is Euler zeta function? This point $s_0$ has obvious property: the segment $(1,s_0]$ to the left of it is mapping on the ...
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2answers
41 views

Riemann zeta function at $\Re(s)=0$ and $\Re(s)=1$

Riemann zeta function at $\Re(s)=0$ and $\Re(s)=1$. What happens to the zeta function at these points? For example $\sum_{n=1}^\infty \frac1{n^s}$ is defined for $\Re(s)>1$ and for $\Re(s)>0$ ...
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4answers
71 views

How to prove $\lim_{s \rightarrow \infty} \zeta(s) = 1$?

$\lim_{s \rightarrow \infty} \zeta(s) = 1$ I have seen a proof using the fact $1 \leq \zeta(s) \leq \frac{1}{1-2^{1-s}}$ but this relies on proving the inequality first which is quite cumbersome. I ...
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54 views
+50

how to show this function has zeros interlacing and including those of Riemann zeta

Let $\chi (t) = H \left( - \frac{i}{2} (2 t - 1) \right) = \dfrac{4 i \pi \zeta (t) \left( \left( \ddot{\Psi} \left(\frac{t}{2} \right) - \ddot{\Psi} \left( \frac{1}{2} - \frac{t}{2} \right)\right) ...
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2answers
52 views

Is this true :$\zeta(s-1)=\sum_{n=1}^{\infty}\frac{\gcd(n,n)}{{\operatorname{lcm}(n,n)}^{s}}$?

I would like to give other representation for zeta function using fundemental arithmitic I have got this: $\zeta(s-1)=\sum_{n=1}^{\infty}\frac{\gcd(n,n)}{{\operatorname{lcm}(n,n)}^{s}}$ where ...
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29 views

Reciprocal of the primes proof -Euler product formula

In a book it says take the logarithm of the euler product formula. But it says $$\log\ \zeta(s) = -\sum_{p \in \mathbb{P}} {\log(1-p^{-s})}$$ Why does the product in the euler product formula become ...
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1answer
41 views

Integrating $ x^{\frac{3}{2}} \frac{1}{1 + e^x} $

I'm wondering if this integral can be expressed in some compact form: $$ \int\limits_{0}^{\infty} x^{\frac{3}{2}}\frac{1}{1 + e^x}dx $$ And if not - why? I was thinking that it was somehow ...
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28 views

Euler Product Formula - Zeta Function

For $s \in \mathbb{C}$ and $\sigma = \Re(s)>1$, $$\zeta(s) = \prod\limits_{p \in \mathbb{P}}\left(1 - \frac{1}{p^s}\right)^{-1}$$ My question is: is the above correct? Or should the $s$ be ...
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1answer
56 views

What about $\lim_{x\to 1}\left(\zeta(x)-\frac{1}{x^x-1}\right)=1+\gamma$?

When I type 1 in the box lim x to, and zeta(x)-1/(x^x-1) in the box Function, of this online calculator (Wolfram Alpha) one has as output $$\lim_{x\to ...
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1answer
28 views

What are the equivalent statements of GRH using the Möbius or Liouville functions?

We all know that Riemann Hypothesis can be stated as properties of $\mu$ or $\lambda$, particularly in terms of the random behaviour of those functions with "square root" bounds. Are there similar ...
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21 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
9 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
182 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
132 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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15 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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0answers
36 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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0answers
18 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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49 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 ...
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2answers
74 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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25 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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30 views

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
45 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
52 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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86 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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1answer
123 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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50 views

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
32 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
66 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 ...
5
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1answer
159 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
41 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
42 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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2answers
387 views

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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0answers
32 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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97 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) $$\frac{\sin(x)}{x} = \prod_{n > 0}\left(1 - \frac{x^{2}}{(n\pi)^{2}}\right)$$ ...
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1answer
43 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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1answer
58 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
118 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
48 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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19 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
35 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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1answer
47 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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0answers
54 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 ...
2
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0answers
51 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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0answers
24 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 ...
4
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3answers
212 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 ...