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Results tagged with zeta-functions
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user 74668
Zeta functions are typically analogues or generalizations of the Riemann zeta function. Examples include Dedekind zeta functions of number fields, and zeta functions of varieties over finite fields. They are typically initially defined as formal generating functions, but often admit analytic continuations.
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Does there exist a known Dirichlet series verifying all these conditions and have non trivia...
Let $s=α+iβ$ be a complex number. Consider the Dirichlet series of the form $$f(s)=∑_{n=1}^{∞}(a_{n})/n^{s}$$
where $(a_{n})_{n≥1}$ is a real sequence.
We consider the class of Dirichlet series sati …
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2
votes
1
answer
266
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The Hasse-Weil L-function and some equations
Let $f$ be an analytic function verfifying
$f(s)=\epsilon f(2-s)$
where $\epsilon=\pm 1$. The expression of Hasse-Weil L-function $f$ is
$$f(s)=N^{s/2}(2\pi)^{-s}\Gamma(s)\sum_{n=1}^{\infty}\frac …
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2
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3
answers
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A generalisation of the Birch and Swinnerton-Dyer conjecture
We know that the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis. My question is about the existence of a similar generalisation of the Birch …
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1
answer
465
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Functional equation of the alternating zeta function
Can one let me know about the functional equation of the alternating zeta function similar to the well known for the rieman function.
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1
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2
answers
858
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The origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation)
The motivation for this question is the same as in my previous question in MO: https://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over
I am just curious to know t …
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1
vote
0
answers
265
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Does the property (P) holds true for the derivatives of $L$?
Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros …
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1
vote
1
answer
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Real points $a∈ℝ$ such that the equation $f^{(k)}(s)=a$ have a finite number of real solutio...
Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the $L$-functi …
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0
votes
1
answer
259
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Functional equation and constant functions
I ask about this claim:
let $f$ be an entire function satisfying $f(s)=u(s)f(a-s)$. Assume that $s$ and $a-s$ are not zeroes of $f$ and $f (bar)(a-s)=f(s)$ in a region $D$ ($f(bar)$ is the conjugate o …
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0
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2
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Hadamard's product formula for the derivative
Let $f$ be an entire function of order $ρ<\infty$. Assume that $f$ does not vanish identically on $\mathbb{C}$. Then, we know that $f$ has a Hadamard's product formula
$$ f(s) =e^{g(s)}s^{r}\prod _ { …
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0
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1
answer
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Can we find a set of elliptic curves over rationals associated with $f$?.
We know that for each elliptic curve over rationals, we can define the Dirichlet series of the Hasse–Weil $L$-function, i.e., the function associated with an elliptic curve over rationals.
Then my qu …
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2
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Riemann Siegel function and gamma function
I ask about an idea to prove this formula:
$Γ(1/2-iβ)=((\sqrt{π})/(\sqrt{\coshπβ}))\exp(-i(2ϑ(β)+βln2π+\arctan(\tanh(1/2)πβ)))$
where $ϑ(β)$ is the Riemann Siegel function.
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