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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.
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 …
- 7,024
2
votes
Bounds on horizontal minima of the Riemann zeta function
This is not exactly an answer, but too long for a comment.
Perhaps I have not been sufficiently clear in the comments. In the paper:
T. Kotnik, Computational estimation of the order of $\zeta(1/2+i t …
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2
votes
Do the complex zeros of $\big(\zeta(s-1) -\zeta(s)\big) \pm \big(\zeta(1-s) - \zeta(2-s)\big...
I do not think this is so difficult as the Riemann hypothesis, I will only
explain why this is so without giving complete proof.
First on the line $s=1+it$ the functions are
$$(\zeta(it)-\zeta(1+it)) …
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1
vote
$\sum_{d\leq x} (\mu(d)/d) \log x/d$: elementary estimates?
In chapter 4, p. 15-18 of the book
H. Iwaniec, Lectures on the Riemann Zeta Function, American Mathematical Society, University Lecture Series nº 62, 2014
there is an elementary proof of the prime …
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9
votes
Accepted
Do we know an upper bound for the multiplicity of the non-trivial zeros of Zeta?
It is known that the number of zeros with $T-1 < Im(\rho)<T+1$ is $O(\log(T))$. Therefore
the multiplicity of a zero $\beta+\gamma i$ will be less than $C \log|\gamma|$ for some
absolute constant $C …
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8
votes
Accepted
A recurrence relation for $\zeta(2n)$ - reference request
I have this recurrence in my collection of problems for my lessons in Analytic Number Theory. I have there the reference:
P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, New York, 200 …
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1
vote
Integrals involving $1/|\zeta(1+i t)|^2$: closed expressions?
This is not properly an answer,
after the comments of Tao it is difficult to give an answer. Only an explanation of my comment above. I still think that my series and the integral are equal.
We can w …
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1
vote
Zeroes of real and imaginary parts of $\zeta(1+it)$ separately (if any for $t>1$ say)
The solutions of $\Re\zeta(1+it)=0$ are scarce. They limit small intervals where
$\Re\zeta(1+it)<0$. The probability in the sense of the limit of the
quotient of the measure of the
set $\{0<t<T: \Re\ …
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3
votes
Odlyzko's reformulation of Montgomery's pair correlation conjecture
I will say that the natural normalisation of the zeros of zeta is
$$\tilde\gamma=\frac{1}{\pi}\vartheta(\gamma)$$
where
$$\vartheta(t)=\Im(\log\Gamma(\frac14+\frac{it}{2}))-\frac{t}{2}\log\pi=\frac{t} …
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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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3
votes
Accepted
A expression for the tangent function involving $\zeta(n),n=2,3,\ldots$
When I was writing my answer, Dan Romik answer appeared. Mine is the same but with more detail.
We have
$$\log\Gamma(1-x)=\gamma x+\sum_{n=2}^\infty\zeta(n)\frac{x^n}{n}$$
(this is known and is also a …
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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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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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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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