User rh - MathOverflow

Can we find a set of elliptic curves over rationals associated with $f$?.

2013-04-27T09:23:14Z

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 question is: Consider a motivic $L$-functions $f$, can we find a set of elliptic curves over rationals associated with $f$?. Simply I ask about the inverse of the first statement in the motivation of this question.

A generalisation of the Birch and Swinnerton-Dyer conjecture

2013-04-18T15:46:17Z

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 and Swinnerton-Dyer conjecture.

How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$?

2013-04-13T14:03:21Z

Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily infinitely many real zeros. Let us consider the functions: $f^{(k)}(1-2\prod_{j=1}^{k}t_{j})$ for $k=1,..,r$ and $(t_1,t_2,...,t_{r})\in (0,1)^{r}$. We know that these functions have also infinitely many real zeros.
My question is: How I can choose $(t_1,t_2,...,t_{k})∈(0,1)^{k}$ such that $f^{(k)}(1-2\prod_{j=1}^{k}t_{j})=0$ and $f^{(k+1)}(1-2 \prod_{j=1}^{k} s_{j})\neq 0$ that is, the real $(1 - 2\prod_{j=1}^{k}s_{j})$ is a simple root of the function $f^{(k)}$?

Does $h$ have infinitely many isolated zeros?

2013-04-10T16:57:35Z

Let $f:ℝ→ℝ$ be a real analytic function with infinitely many isolated zeros. Let us define the function: $$h(s₁,s₂,...,s_{r+1})=\prod_{k=1}^{r+1}f^{(k+1)}(\left(1-2\prod_{j=1}^{k}s_{j}\right)$$ Also, all the $k$-th derivatives of $f$ have infinitely many isolated zeros. Does $h$ have infinitely many isolated zeros? I know that $f′(1-2s₁)=0$ has infinitely many isolated zeros.

The $k^{th}$ derivative of a L-function has necessarily infinitely many zeros

2013-04-06T17:34:42Z

My current question is concerned with a reference (paper or book) containing a proof of this result: The $k^{th}$ derivative of a L-function has necessarily infinitely many zeros.

The Hasse-Weil L-function and some equations

2013-04-02T08:30:19Z

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

$$f(s)=N^{s/2}(2\pi)^{-s}\Gamma(s)\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}$$

where $N$ is an integer and $\Gamma(s)$ is the gamma function.

Let $r$ be an integer. I have a set of equations of the form

$$f^{(k)}\left(1-2\prod_{j=1}^{k}s_{j}\right)=f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)$$

for all $k=1,...,r$. Here $f^{(k)}$ is the $k$-th derivative of $f$.

Can I deduce that

$$(t_1,t_2,\ldots,t_{r})=(s_1,s_2,\ldots,s_{r})$$

under some conditions on the derivatives of $f$?

The injectivity is not possible for this case.

The origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation)

2012-12-11T17:08:02Z

The motivation for this question is the same as in my previous question in MO: http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over

I am just curious to know the origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation $f(s)=w(C/ℚ)f(2-s)=εf(2-s)$) of the curve $C$. I read several papers on this topic, but I cannot find where this root number come from. I wonder if this number has some relation with $a$ and $b$ in the equation of the curve $C$.

Hadamard's product formula for the derivative

2012-12-09T16:02:59Z

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

$$ f(s) =e^{g(s)}s^{r}\prod _ {k=1}^{\infty}\frac{s _ {k}-s}{s _ {k}} e^{s/s _ k} $$

the integer $r$ is the order of vanishing of $f$ at $s=0$, the $s_{k}$ are the other zeros of $f$ listed with multiplicity, $g$ is a polynomial of degree at most $ρ$, and the product converges uniformly in bounded subsets of $ℂ$. My question is how I can deduce directely a Hadamard's product formula for the derivative $f^′$ from the one of the function $f$.

A rapidly-converging series of the Hasse–Weil L-function associated with an elliptic curve over rationals

2012-11-20T16:30:10Z

I know that for some L-series there is still a rapidly-converging series. My question is about the existence of a such a series for the Dirichlet series of the Hasse–Weil L-function associated with an elliptic curve over rationals. A google search do not gives important answers.

Solution of certain forms of equations

2012-11-19T16:15:03Z

I ask about a possible method to find the solution of algebraic equations of the form

$axⁿ+byⁿ+c=0$

where $a,b,c,x,y$ are real constants and $n$ is an integer. Maybe there is a simple method, but I cannot find it.

Riemann Siegel function and gamma function

2012-11-17T14:41:41Z

I ask about an idea to prove this formula:

$Γ(1/2-iβ)=((√π)/(√(coshπβ)))exp(-i(2ϑ(β)+βln2π+arctan(tanh(1/2)πβ)))$

where $ϑ(β)$ is the Riemann Siegel function.

Answer by RH for Whats the application of Hyperchaotic system?

2012-11-18T15:27:40Z

It is well known that if two or more Lyapunov exponents of a dynamical system are positive throughout a range of parameter space, then the resulting attractors are hyperchaotic. The importance of these attractors is that they are less regular and are seemingly "almost full" in space, which explains their importance in fluid mixing [Scheizer & Hasler, 1996, Abel et al., 1997, Ottino,1989; Ottino et al., 1992]. See:

Abel. A, Bauer. A, Kerber. K, Schwarz. W, [1997] " Chaotic codes for CDMA application," Proc. ECCTD '97, 1, 306.

Kapitaniak.T, Chua. L. O, Zhong. Guo-Qun, [1994] " Experimental hyperchaos in coupled Chua's circuits," Circuits,.Syst. I: Fund. Th. Appl. 41 (7), 499 -- 503.

Ottino. J. M, [1989] " The kinematics of mixing: stretching, chaos, and transport," Cambridge: Cambridge University Press.

Ottino. J. M, Muzzion. F. J, Tjahjadi. M, Franjione.J. G, Jana. S. C, Kusch. H. A, [1992] " Chaos, symmetry, and self-similarity: exploring order and disorder in mixing processes," Science. 257, 754--760.

Scheizer. J, Hasler. M, [1996] " Multiple access communication using chaotic signals," Proc. IEEE ISCAS '96. Atlanta, USA, 3, 108.

Thamilmaran. K, Lakshmanan. M, Venkatesan. A, [2004] " Hyperchaos in a Modified Canonical Chua's Circuit," Int. J. Bifurcation and Chaos. 14 (1),221--244.

Functional equation and constant functions

2012-11-04T09:02:40Z

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 of $f$). Then the module of $f(s)/f(bar)(a-s)$ is equal to $1$, implying that the module of $u(s)$ is also $1$. The question is: Does this result implies that in fact the function $u(s)$ is constant. Thank you in advance.

Weierstrass factorization theorem in several variables

2012-11-06T06:11:16Z

Can one indicate to me the Weierstrass factorization theorem in several variables (real or complex). In one complex variable the result is well known. Thank you in advance.

The Dirichlet series of the Hasse–Weil L-function

2012-11-03T16:30:12Z

I have the following question: Is there is a paper claiming that the Dirichlet series of the Hasse–Weil $L$-function (associated with an elliptic curve over rationals) is of finite order. Thank you in advance. I cannot find any result in the net.

functional equation

2012-10-22T13:58:59Z

Hi to all I have the following question: Let f be an analytic function satisfying the functional equation: f(z)=u(z)*f(a-z) where "a" is a real constant. Let g be another function satisfying the same functional equation. In this case I asking if f=g. Thank you in advance.

Functional equation of the alternating zeta function

2012-08-26T16:30:01Z

Can one let me know about the functional equation of the alternating zeta function similar to the well known for the rieman function.

Comment by RH

2013-04-29T16:40:36Z

I reformulate he question.

Comment by RH

2013-04-18T16:46:41Z

@ Arijit: Thank you very much for clarification.

Comment by RH

2013-04-18T16:18:26Z

@ Marc Palm: Does there is a set of special L-functions where this conjecture apply just like the case of the Generalized Riemann hypothesis.

Comment by RH

2013-04-15T15:19:53Z

Yes, thank you. Fixed.

Comment by RH

2013-04-15T13:41:32Z

@Loïc Teyssier: Ok I edit the question. But I notice that this is the first version. Something changes my text!

Comment by RH

2013-04-10T18:56:32Z

@Pietro Majer: I changed the function to be $f:ℝ→ℝ$.

Comment by RH

2013-04-04T19:29:04Z

So, such bijections does not exists.

Comment by RH

2013-04-02T17:31:12Z

Thank you very much for clarification.

Comment by RH

2013-04-02T09:17:44Z

Yes. You make things clearer.

Comment by RH

2013-04-02T09:03:17Z

and what about the case if one side is zero.

Comment by RH

2013-04-02T09:01:19Z

@ Marc Palm: So the functional equation has no role in this analysis.

Comment by RH

2013-03-28T14:58:07Z

My question is about current works on the topic of finiteness of the Mordell-Weil rank r of E/K.

Comment by RH

2012-12-30T21:01:30Z

@ Yemon Choi: Thank you very much for valauable comments and constructive criticism!!!!!

Comment by RH

2012-12-30T13:36:20Z

But it is very hard to obtain a such a solution. The form is closely to a modular form of order $k$ but it is not.

Comment by RH

2012-12-30T13:10:33Z

Thank you for your comment. But Still have trouble with solving it with respect to $f$ not $s$.