# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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