Lower bound for sum of Hecke eigenvalues
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Let $lambda$ be weakly multiplicative, $lambda(n)geq0$, $p$ prime and $S(x)=sum_{nleq x}lambda(n)log(frac{x}{n})$ for real $x$.
How can I show $S(x)gg left(sum_{pleq sqrt{x/3}}lambda(p)right)^2-left(sum_{pleq sqrt{x/3}}1right)$?
Here is the background:
The question is coming from IKS, section 3. $lambda(n)$ are the eigenvalues of a newform $f$ of level $N$. All above sums are chosen such that $(n,N)=1$ or $pnotmid N$, thus giving multiplicativity.
In Xu, section 3.1 something similar happens. Here the hints $lambda(p)geq1$ and $|lambda(n)|leqsigma_0(n)$ (divisor function) are given.
My approach so far starts estimating $log(x/n)geq1$ but I could not figure out the strange bound of $sqrt{x/3}$...
number-theory eigenvalues-eigenvectors estimation modular-forms upper-lower-bounds
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Nodt Greenish
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up vote
1
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Let $lambda$ be weakly multiplicative, $lambda(n)geq0$, $p$ prime and $S(x)=sum_{nleq x}lambda(n)log(frac{x}{n})$ for real $x$.
How can I show $S(x)gg left(sum_{pleq sqrt{x/3}}lambda(p)right)^2-left(sum_{pleq sqrt{x/3}}1right)$?
Here is the background:
The question is coming from IKS, section 3. $lambda(n)$ are the eigenvalues of a newform $f$ of level $N$. All above sums are chosen such that $(n,N)=1$ or $pnotmid N$, thus giving multiplicativity.
In Xu, section 3.1 something similar happens. Here the hints $lambda(p)geq1$ and $|lambda(n)|leqsigma_0(n)$ (divisor function) are given.
My approach so far starts estimating $log(x/n)geq1$ but I could not figure out the strange bound of $sqrt{x/3}$...
number-theory eigenvalues-eigenvectors estimation modular-forms upper-lower-bounds
asked yesterday
Nodt Greenish
266
New contributor
Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $lambda$ be weakly multiplicative, $lambda(n)geq0$, $p$ prime and $S(x)=sum_{nleq x}lambda(n)log(frac{x}{n})$ for real $x$.
How can I show $S(x)gg left(sum_{pleq sqrt{x/3}}lambda(p)right)^2-left(sum_{pleq sqrt{x/3}}1right)$?
Here is the background:
The question is coming from IKS, section 3. $lambda(n)$ are the eigenvalues of a newform $f$ of level $N$. All above sums are chosen such that $(n,N)=1$ or $pnotmid N$, thus giving multiplicativity.
In Xu, section 3.1 something similar happens. Here the hints $lambda(p)geq1$ and $|lambda(n)|leqsigma_0(n)$ (divisor function) are given.
My approach so far starts estimating $log(x/n)geq1$ but I could not figure out the strange bound of $sqrt{x/3}$...
number-theory eigenvalues-eigenvectors estimation modular-forms upper-lower-bounds
asked yesterday
Nodt Greenish
266
New contributor
Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Let $lambda$ be weakly multiplicative, $lambda(n)geq0$, $p$ prime and $S(x)=sum_{nleq x}lambda(n)log(frac{x}{n})$ for real $x$.
How can I show $S(x)gg left(sum_{pleq sqrt{x/3}}lambda(p)right)^2-left(sum_{pleq sqrt{x/3}}1right)$?
Here is the background:
The question is coming from IKS, section 3. $lambda(n)$ are the eigenvalues of a newform $f$ of level $N$. All above sums are chosen such that $(n,N)=1$ or $pnotmid N$, thus giving multiplicativity.
In Xu, section 3.1 something similar happens. Here the hints $lambda(p)geq1$ and $|lambda(n)|leqsigma_0(n)$ (divisor function) are given.
My approach so far starts estimating $log(x/n)geq1$ but I could not figure out the strange bound of $sqrt{x/3}$...
number-theory eigenvalues-eigenvectors estimation modular-forms upper-lower-bounds
number-theory eigenvalues-eigenvectors estimation modular-forms upper-lower-bounds
asked yesterday
Nodt Greenish
266
New contributor
Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Nodt Greenish
266
New contributor
Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
asked yesterday
Nodt Greenish
266
New contributor
Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Nodt Greenish
266
asked yesterday
Nodt Greenish
266
266
New contributor
Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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Nodt Greenish is a new contributor. Be nice, and check out our Code of Conduct.
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