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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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
New contributor
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
New contributor
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
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Nodt Greenish
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