arXiv subjects

Ned Carmichael

Publications and source records attributed to Ned Carmichael.

3 recordsLinked to original sources

arXiv2025

Non-vanishing of Poincar\'e Series on Average

We study when Poincar\'e series for congruence subgroups do not vanish identically. We show that almost all Poincar\'e series with suitable parameters do not vanish when either the weight $k$ or the index $m$ varies in a dyadic interval. Crucially, analyzing the problem `on average' over these weights or indices allows us to prove non-vanishing in ranges where the index $m$ is significantly larger than $k^2$ - a range in which proving non-vanishing for individual Poincar\'e series remains out of reach of current methods.

math.NT↗

arXiv2025

The Second Moment of Sums of Hecke Eigenvalues II

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq 1}$ denote its sequence of normalised Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)=\sum_{x\leq n\leq 2x} \lambda_f(n)$, on average over forms $f$ of large weight $k$. In the range $k^2/(8\pi^2)\leq x\leq k^{12/5-\epsilon}$, the size of the second moment lies between $x^{1/2-o(1)}$ and $x^{1/2}$. This is in sharp contrast to the regime $x\leq k^{2-o(1)}$, where the second moment was shown in preceding work (part I) to be of size $\asymp x$.

math.NT↗

arXiv2025

The Second Moment of Sums of Hecke Eigenvalues I

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq1}$ denote its sequence of Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)=\sum_{x\leq n\leq 2x}\lambda_f(n)$, on average over forms $f$ of large weight $k$, in the regime where the length of the sums $x$ is smaller than $k^2$. We observe transitions in the size of the sums when $x\approx k$ and $x\approx k^2$. In subsequent work (part II), it will be shown that once $x$ is larger than $k^2$ (where the latter transition occurs), the average size of the sums $S(x,f)$ becomes dramatically smaller.

math.NT↗