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Results tagged with riemann-zeta-function
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The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.
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Are all complex zeros of $\dfrac{\zeta'}{\zeta}(s) \pm \dfrac{\zeta'}{\zeta}(1-s)$ on the cr...
Numerical evidence suggests that all complex zeros (real ones exist as well) of:
$$\frac{\zeta'}{\zeta}(s) \pm \frac{\zeta'}{\zeta}(1-s)$$
reside on the critical line with $\Re(s)=\frac12$.
I made …
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Does the difference of two converging infinite series correctly induce the non-trivial zeros...
The following analytic continuation for $\zeta(s)$ towards $\Re(s)>-1$ derived here:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(s+1+ \sum _{n=1}^{\infty } \left( {\frac {s-1-2\,n}{{n}^{s}}}+ …
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Alternating sums of the non-trivial zeros of $\zeta(s)$.
It is known that the infinite sum of the non-trivial zeros $\rho_n =\beta + \gamma_ni$ of $\zeta(z)$, when taken in pairs that are either conjugated or reflexive (they give the same outcome), converge …
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Wilson's theorem and the Zeta function
$\zeta(s)$ has a direct link to the prime numbers (via the infinite Euler product and the non-trivial zeros in the explicit $\pi(x)$ formula). Wilson's theorem offers a (proven but very inefficient) f …
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Does there exist a closed form for the factors of this infinite product ?
Assume $s,a \in \mathbb{C}, a \pm in \ne 0$.
The following infinite product nicely converges and can be expressed in a closed form:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{a+i n} \right …
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Are these valid expansions of the Riemann $\xi(s)$ function in the Hadamard product?
In this post I derived for $s=a + ti$, that assuming the RH, the following should be true:
$$\displaystyle \frac{\xi(\frac12 - a + s)}{\xi(\frac12 - a)} = \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} …
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Can $\zeta(s)$ for $\Re(s)>1$ be split into two factors that each can be analytically contin...
Assuming the RH and $s \in \mathbb{C}, \rho_n =\frac12 \pm i\gamma_n$, the following (altered) Hadamard product:
$$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n i …
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Are all complex zeros of $\zeta(s) \pm \zeta(-s)$ on the line with $\Re(s)=0$?
My conjecture is that all zeros in the strip $-1 \le \Re(s) \le 1$ of $\zeta(s) \pm \zeta(-s)$ are on the line $\Re(s)=0$.
I did find three complex zeros for $\pm =+$ (i.e. 12 in total) and two comp …
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Do the complex zeros of $\big(\zeta(s-1) -\zeta(s)\big) \pm \big(\zeta(1-s) - \zeta(2-s)\big...
Or stated differently: for $s \in \mathbb{C}$ and with $\chi(s)= \pi^{-s}\,2^{1-s}\,\cos\left(\frac{\pi\,s} {2}\right)\,\Gamma(s)$, do all, except a finite few, of the complex (real ones exist as well …
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Three integral expressions for integer values of $\zeta(s)$. Could these be further reduced ...
In this MSE-question I've asked about three, similarly shaped, integrals for integer vales of $\zeta(s)$ that I found numerically:
$$\zeta \left( 3 \right) =\frac12{\int_{0}^{1} \frac{1}{x}\big(\zeta( …
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Is there anything known about the complex zeros of this integral related to $\zeta(s)$?
The right-hand side of the well known equation:
$$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}} + {x}^{\frac{1-s}{2}}\right)\,\fra …
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Are the complex zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$...
With $s \in \mathbb{C}, a \in \mathbb{R}$,
numerical evidence strongly suggests that the complex zeros in the critical strip of:
$$\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$ …
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Are all zeros of $\dfrac{\Gamma'}{\Gamma^2}(s) \pm \dfrac{\Gamma'}{\Gamma^2}(1-s)$ either re...
In this post it was proven that, except for a finite few, all zeros of $\Gamma(s) \pm \Gamma(1-s)$ are either real or reside on the line $\Re(s)=\frac12$.
I have been searching for similar reflexive …
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Do these infinite series expressing $\zeta(s)$ only (partially) converge at $\Re(s)=\frac12$?
The following analytic continuation for $\zeta(s)$ towards $\Re(s)>-1$ was derived here:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(\sum _{n=1}^{\infty } {\frac {s-1-2\,n}{{n}^{s}}} + \sum …
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The influence of $\chi(s)$ on complex zeros of $\frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \...
I was exploring the formula:
$$g(s)_{\pm} := \displaystyle \frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$$
and found that for all $\Re(s) \ne \frac12$:
$|g(s)_{ …
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