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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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On a certain integral representation for Hurwitz zeta functions
A recent question On a certain integral representation for Dirichlet L-functions referenced an integral representation of $\zeta(s)$ due to Jensen that was new to me:
$$
(s-1)\zeta(s)=\frac{\pi}{2(s-1 …
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Conjecture of Spira on the zeros of $\zeta^\prime(s)$
Let $N(T)$ be the number of complex zeros of $\zeta(s)$ with imaginary part between $0$ and $T$, and let $N_k(T)$ be the analogous counting function for the $k$th derivative $\zeta^{(k)}(s)$. Based o …
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The horizontal distribution of zeros of $\zeta^\prime(s)$
I have a question about a detail in the proof of Proposition 1.6 in "The horizontal distribution of zeros of $\zeta^\prime(s)$", K. Soundararajan, Duke J. Math. vol. 91 1998.
Throughout I will simpli …
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Numerical coincidence for the first Riemann zero
Is there any heuristic explanation for why the imaginary part of the first nontrivial Riemann zero, $\gamma_1\approx 14.134725$, should be so close to $\pi/2$ modulo $2\pi$? Mathematica gives Mod[Im[ …
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Zeros of higher derivatives of $\zeta(s)$
Zeros of successive higher order derivatives of the Riemann zeta function seem to cluster along roughly horizontal lines.
Is there a heuristic explanation of why this happens (especially inside the cr …
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Scattering amplitudes and the Riemann zeta function
I'm reading Amplitudes and the Riemann Zeta Function, which recently appeared in Physical Review Letters. It's received some publicity, including my own campus' PR operation. From the abstract (adap …
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Jensen Polynomials for the Riemann Zeta Function
In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in PNAS) the abstract includes
In the case of the Riemann zeta function, this proves the G …
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Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$
Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example …
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PT Symmetry and the Riemann Hypothesis
Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The …
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Zeros of the derivative of Riemann's $\xi$-function
The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real val …
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The Riemann zeros and the heat equation
The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as
$$
\Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du,
$$
where $\Phi(u)$ is defined as
$$
2\sum_ …
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Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of $\zeta(s) …
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