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Results tagged with riemann-zeta-function
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user 7402
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.
19
votes
Accepted
$\zeta(0)$ and the cotangent function
This is not a completely satisfactory answer. I would like a simpler one.
Nevertheless still probably a good exercise in Complex variables.
I will only sketch it.
What we want to show is equivalent t …
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16
votes
Accepted
An integral involving the argument of the Gamma function and the Riemann Hypothesis
We prove that
$$I=-\frac{\pi}{4}(\gamma+\log 4).$$
$$I=\int_0^\infty\frac{t\arg\Gamma(\frac14+\frac{it}{2})}{(\frac14+t^2)^2}\,dt.$$
$I$ is the imaginary part of the complex integral
$$\int_0^\infty …
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16
votes
Accepted
Certain functional equations for the Riemann Zeta function?
Equations of this type are known. You may see, for example, the classical
book "Primzahlen" by Landau paragraph 67. "Continuation of the zeta function by partial integration"
There it is proved the f …
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12
votes
Accepted
Is this differential equation for zeta on the critical line? One can compute it from its der...
The zeta function can be written $\zeta(1/2+it)=Z(t)e^{-i\vartheta(t)}$ where
$Z(t)$ are real analytic and $\vartheta(t)$ is a simple function related to
$\Gamma(s)$. The relation between $\zeta(1/2 …
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11
votes
Explicit formula for Riemann zeros counting function
The formula is given by Guinand as said in his answer by Matthew Watkins but it can be found in page 111 of the paper and reads: assuming Riemann Hypothesis and $T>0$
$$\frac12(N(T+0)+N(T-0))=\frac{T} …
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9
votes
Accepted
Do we know an upper bound for the multiplicity of the non-trivial zeros of Zeta?
It is known that the number of zeros with $T-1 < Im(\rho)<T+1$ is $O(\log(T))$. Therefore
the multiplicity of a zero $\beta+\gamma i$ will be less than $C \log|\gamma|$ for some
absolute constant $C …
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9
votes
Accepted
zeta(3) in terms of derivatives of zeta at 1/2 and pi
In fact for odd $n\ge3$ we have
$$\Bigl.\frac{d^{n}}{ds^n}\log\zeta(s)\Bigr|_{s=\frac12}=
\frac{(n-1)!}{2}\Bigl(2^n L(n,\chi)+(2^n-1)\zeta(n)\Bigr)$$
The proof (due to Voros) is the following:
It …
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9
votes
Is this sum of reciprocals of zeta zeros correct?
The series $\sum_\rho \rho^{-1}$ over the non-trivial zeros is not absolutely convergent, this is proved in Davenport p. 80. But as Davenport says and proves in
page 81-82 the series converges con …
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8
votes
Accepted
A recurrence relation for $\zeta(2n)$ - reference request
I have this recurrence in my collection of problems for my lessons in Analytic Number Theory. I have there the reference:
P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, New York, 200 …
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8
votes
Zeros of the derivative of Riemann's $\xi$-function
The Riemann hypothesis implies that the function $\Xi(z)=\xi(1/2+iz)$ is in the
Laguerre-Pólya class. Therefore it is a limit, uniformly on compact sets, of a
sequence of polynomials with real roots. …
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6
votes
How good can we approximate with algebraic curve the egg shaped vanishing of $\Re \zeta(s)$ ...
The figure of egg is not an elliptic curve.
Since the curve pass through the point $(-2,0)$ and $(1,0)$ the equation will be
of type
$$y^2=(ax+b)(x+2)(x-1)$$
We find the numbers $y_0$ and $y_1$
wh …
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6
votes
Accepted
A closed form of infinite products of complex zeros involving $\Im(\rho_n)$. Does a proof of...
The proposed formula is not true if RH is not true.
Let $\Theta$ be the upper bound of the real parts of the zeros of $\zeta(s)$.
Your product has zeros at $\mu$, $1-\mu$, $\overline{\mu}$ and $\ov …
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5
votes
What happens to $\zeta(s)$ when all its $\Im(\rho_n)$ are "scaled" linearly?
I assume Riemann hypothesis on all this answer.
I want a closed form for
$$f(s,x):= \prod_{n=1}^\infty\Bigl(1-\frac{s}{\frac12+i x\gamma_n}\Bigr)
\Bigl(1-\frac{s}{\frac12-i x\gamma_n}\Bigr).$$
Of co …
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5
votes
What happens to $\zeta(s)$ when all its $\Im(\rho_n)$ are "scaled" linearly?
... continue the above answer
Therefore
$$\prod_\gamma\Bigl(1-\frac{s}{\rho}\Bigr)\Bigl(1-\frac{s}{1-\rho}\Bigr)=
\frac{\frac14+\frac{t^2}{x^2}}{\frac14-\frac{1}{4x^2}}\frac{\pi^{-\frac{it}{2x}}}
{\ …
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5
votes
Confusion about Montgomery's pair correlation conjecture
Assuming the Riemann Hypothesis Montgomery consider the function
$$F(\alpha)=F(\alpha,T)=\Bigl(\frac{T}{2\pi}\log T\Bigr)^{-1} \sum_{0<\gamma, \gamma'\le T}
T^{i\alpha(\gamma-\gamma')}w(\gamma-\gamma' …
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