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Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL_2(Z)$ and let $\lambda_f(n)$ be the $n$-th normalized Fourier coefficient of $f.$ Can someone provide me with an effective estimate for this infinite product $$\prod_{\lambda_f(p)=0} \left(1-\frac{1}{p+1}\right).$$

Thanks in advance.

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  • $\begingroup$ I would expect that this infinite product diverges to $0$ (at least this is what happens for $\prod_{p}(1 - 1/(p+1))$. In the CM case, one can show this without too much difficulty, as a positive proportion of primes $p$ satisfy $\lambda_f(p)=0$. In the non-CM case I'm not sure, as the set of primes $p$ for which $\lambda_f(p)=0$ is a lot more sparse. $\endgroup$
    – Daniel Loughran
    Commented Aug 21, 2016 at 14:48
  • $\begingroup$ @DanielLoughran Thanks for your remarks. $\endgroup$
    – Khadija Mbarki
    Commented Aug 21, 2016 at 17:45
  • $\begingroup$ Is there any example known where this product is non-empty? The question asked for forms of level 1, so CM forms do not come up. Lehmer's conjecture definitely predicts that the product should be empty for the weight 12 cusp form $\Delta$. $\endgroup$
    – David Loeffler
    Commented Aug 22, 2016 at 11:20
  • $\begingroup$ I would like to mention that I took this product from Theorem 16 of Serre's paper 'Quelques applications du théorème de densitités de Chebotarev' and it was mentioned in the paper of JEREMY ROUSE AND JESSE THORNER (link: arxiv.org/pdf/1305.5283v3.pdf) that the density of the of the set of non zero eigenvalues for n less than x is asymtotically equal to \alpha_f \times the above product and in case of Hecke eigenforms of level 1 the constant alpha=1 so I was thinking to get effective bound for the product to get effective density for the set of non zero Hecke eigenvalues for $n<x$. $\endgroup$
    – Khadija Mbarki
    Commented Aug 22, 2016 at 13:07

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In general, this product is hard to work with, and good upper/lower bounds require a bit of work. I'll also assume the simplest case of $k=12$, in which case $\lambda_f(n)$ is the famous Ramanujan tau function.

Since the density of primes $p$ for which $\lambda_f(p)=0$ is zero (a result first proved by Serre his paper mentioned above), the product converges absolutely. We expect that $\lambda_f(p)\neq0$ for all $p$ (Lehmer's conjecture), so this product should simply be 1.

For upper bounds, one must know the first many $p$ for which $\lambda_f(p)=0$. Since we expect that no such $p$ exist, this amounts to verifying that $\lambda_f(p)\neq0$ up to some large threshold using SAGE, MAGMA, etc. The arXiv post mentioned above appears to have some discussion on this.

For lower bounds, one needs an upper bound on $\pi_f(x):=\#\{p\leq x\colon \lambda_f(p)=0\}$ for all $x$ (use partial summation on the log of the product). Thus one needs an explicit version of Serre's density zero result, like the one given in Theorem 1.3 of the arXiv post mentioned above.

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