The following question came up while I was working through an example:
Does there exist an $\ell^1$ sequence of complex numbers $a_n$, not all zero, such that $\sum_n a_n n^{p} = 0$ for all $p \in [0,1]$?
The following question came up while I was working through an example:



Let $U\subset \Bbb C$ be the set of complex number with positive real part and $f\colon U\to\Bbb C$ given by $f(z):=\sum_{n=1}^{+\infty}a_n\exp(z\log n)$. Since there is uniform convergence on compact subsets of $\Bbb C$, $f$ is holomorphic. We have $f(z)=0$ if $z\in (0,1]$, hence $f$ vanishes identically on the connected set $U$. In particular, the initial assumption is valid for any $p>0$. Then we can prove by induction that $a_n=0$ for all $n$. We have $a_1=\left\sum_{n\geqslant 2}a_n\exp(p\log n)\right\leqslant \exp(p\log 2)\lVert a\rVert_{\ell^1}$ for each $p$, hence $a_1=0$. Assume that for $n\geqslant 2$, $a_0=\dots=a_{n1}=0$, then $$a_n \leqslant \exp\left(p\log \frac{n+1}{n}\right)\lVert a\rVert_{\ell^1},$$ giving what we want. Note that we can relax the initial assumption "for all $p\in [0,1]$" by for "all $p$ in an nondiscrete subset of the unit interval". 

