Zeros of special Dirichlet series.
$begingroup$
There is given Dirichlet series:
$$f:mathbb{C}rightarrow mathbb{R}_{ge 0}$$
$$f(x+iy)=eta(x+iy)eta(x-iy)=sum_{n=1}^{infty} frac{a(n)}{n^{x+iy}}$$
Where $a(n)=sum_{d|n}(-1)^{d+frac{n}{d}}d^{2yi}$ And of course $x>0$.
Let's consider $g(x,y)=frac{d^{2}f(x+iy)}{dx^{2}}=sum_{n=1}^{infty} frac{(ln n)^{2}a(n)}{n^{x+iy}}$.
My question: Is that function always non negative?
I made some research and i deduced that for $x>0$ small enough function $g(x,y)$ is positive hence for constant $y$ if there would be negative value, then it would exist some $x$ that:
$$g(x,y)=0$$
Regards.
complex-analysis dirichlet-series
asked Dec 22 '18 at 18:15
mkultramkultra
1038
$endgroup$
add a comment |
$begingroup$
There is given Dirichlet series:
$$f:mathbb{C}rightarrow mathbb{R}_{ge 0}$$
$$f(x+iy)=eta(x+iy)eta(x-iy)=sum_{n=1}^{infty} frac{a(n)}{n^{x+iy}}$$
Where $a(n)=sum_{d|n}(-1)^{d+frac{n}{d}}d^{2yi}$ And of course $x>0$.
Let's consider $g(x,y)=frac{d^{2}f(x+iy)}{dx^{2}}=sum_{n=1}^{infty} frac{(ln n)^{2}a(n)}{n^{x+iy}}$.
My question: Is that function always non negative?
I made some research and i deduced that for $x>0$ small enough function $g(x,y)$ is positive hence for constant $y$ if there would be negative value, then it would exist some $x$ that:
$$g(x,y)=0$$
Regards.
complex-analysis dirichlet-series
asked Dec 22 '18 at 18:15
mkultramkultra
1038
$endgroup$
$begingroup$
What is meant by "it must not have Dirichlet character"?
$endgroup$
– coffeemath
Dec 22 '18 at 18:37
$begingroup$
It means that $$a(n)$$ may be Dirichlet character or not.
$endgroup$
– mkultra
Dec 22 '18 at 19:49
1
$begingroup$
Dirichlet Series are with $;{a_n};$ any complex sequence. Why would you stress that it could or could not be a Dirichlet character?
$endgroup$
– DonAntonio
Dec 22 '18 at 20:28
add a comment |
$begingroup$
There is given Dirichlet series:
$$f:mathbb{C}rightarrow mathbb{R}_{ge 0}$$
$$f(x+iy)=eta(x+iy)eta(x-iy)=sum_{n=1}^{infty} frac{a(n)}{n^{x+iy}}$$
Where $a(n)=sum_{d|n}(-1)^{d+frac{n}{d}}d^{2yi}$ And of course $x>0$.
Let's consider $g(x,y)=frac{d^{2}f(x+iy)}{dx^{2}}=sum_{n=1}^{infty} frac{(ln n)^{2}a(n)}{n^{x+iy}}$.
My question: Is that function always non negative?
I made some research and i deduced that for $x>0$ small enough function $g(x,y)$ is positive hence for constant $y$ if there would be negative value, then it would exist some $x$ that:
$$g(x,y)=0$$
Regards.
complex-analysis dirichlet-series
asked Dec 22 '18 at 18:15
mkultramkultra
1038
$endgroup$
There is given Dirichlet series:
$$f:mathbb{C}rightarrow mathbb{R}_{ge 0}$$
$$f(x+iy)=eta(x+iy)eta(x-iy)=sum_{n=1}^{infty} frac{a(n)}{n^{x+iy}}$$
Where $a(n)=sum_{d|n}(-1)^{d+frac{n}{d}}d^{2yi}$ And of course $x>0$.
Let's consider $g(x,y)=frac{d^{2}f(x+iy)}{dx^{2}}=sum_{n=1}^{infty} frac{(ln n)^{2}a(n)}{n^{x+iy}}$.
My question: Is that function always non negative?
I made some research and i deduced that for $x>0$ small enough function $g(x,y)$ is positive hence for constant $y$ if there would be negative value, then it would exist some $x$ that:
$$g(x,y)=0$$
Regards.
complex-analysis dirichlet-series
complex-analysis dirichlet-series
asked Dec 22 '18 at 18:15
mkultramkultra
1038
asked Dec 22 '18 at 18:15
mkultramkultra
1038
edited Jan 12 at 12:03
mkultra
asked Dec 22 '18 at 18:15
mkultramkultra
1038
asked Dec 22 '18 at 18:15
mkultramkultra
1038
asked Dec 22 '18 at 18:15
mkultramkultra
1038
1038
$begingroup$
What is meant by "it must not have Dirichlet character"?
$endgroup$
– coffeemath
Dec 22 '18 at 18:37
$begingroup$
It means that $$a(n)$$ may be Dirichlet character or not.
$endgroup$
– mkultra
Dec 22 '18 at 19:49
1
$begingroup$
Dirichlet Series are with $;{a_n};$ any complex sequence. Why would you stress that it could or could not be a Dirichlet character?
$endgroup$
– DonAntonio
Dec 22 '18 at 20:28
add a comment |
$begingroup$
What is meant by "it must not have Dirichlet character"?
$endgroup$
– coffeemath
Dec 22 '18 at 18:37
$begingroup$
It means that $$a(n)$$ may be Dirichlet character or not.
$endgroup$
– mkultra
Dec 22 '18 at 19:49
1
$begingroup$
Dirichlet Series are with $;{a_n};$ any complex sequence. Why would you stress that it could or could not be a Dirichlet character?
$endgroup$
– DonAntonio
Dec 22 '18 at 20:28
$begingroup$
What is meant by "it must not have Dirichlet character"?
$endgroup$
– coffeemath
Dec 22 '18 at 18:37
$begingroup$
What is meant by "it must not have Dirichlet character"?
$endgroup$
– coffeemath
Dec 22 '18 at 18:37
$begingroup$
It means that $$a(n)$$ may be Dirichlet character or not.
$endgroup$
– mkultra
Dec 22 '18 at 19:49
$begingroup$
It means that $$a(n)$$ may be Dirichlet character or not.
$endgroup$
– mkultra
Dec 22 '18 at 19:49
1
1
$begingroup$
Dirichlet Series are with $;{a_n};$ any complex sequence. Why would you stress that it could or could not be a Dirichlet character?
$endgroup$
– DonAntonio
Dec 22 '18 at 20:28
$begingroup$
Dirichlet Series are with $;{a_n};$ any complex sequence. Why would you stress that it could or could not be a Dirichlet character?
$endgroup$
– DonAntonio
Dec 22 '18 at 20:28
add a comment |
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$begingroup$
What is meant by "it must not have Dirichlet character"?
$endgroup$
– coffeemath
Dec 22 '18 at 18:37
$begingroup$
It means that $$a(n)$$ may be Dirichlet character or not.
$endgroup$
– mkultra
Dec 22 '18 at 19:49
1
$begingroup$
Dirichlet Series are with $;{a_n};$ any complex sequence. Why would you stress that it could or could not be a Dirichlet character?
$endgroup$
– DonAntonio
Dec 22 '18 at 20:28