The following question came up while I was working through an example:



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> 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 $p \in [0,1]$?
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> 



Does there exist a sequence of complex numbers such that...
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 $n\geqslant 2$, $a\_0=\dots=a\_{n-1}=0$, 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 non-discrete subset of the unit interval".


