Large zeta sums and zeros of the Riemann zeta function

Abstract

For real $t$ and $x\ge 1$, set \[ S(x,t)=\sum_{n\le x} n^{\ii t}. \] We prove an unconditional inverse theorem relating large values of $S(x,t)$, with $|t|$ large, to zeros of the Riemann zeta function near height $t$. More precisely, if $T\le |t|\le 2T$, $\exp (\sqrt{\log T})\le x\le \sqrt{T}$, and $|S(x,t)|=x/N$ with $N\le (\log x{)}^{1/100}$, then for every $c{N}^6\le L\le (\log x)/2$ a disk centered at $1+\iiϕ$, where $|\varphi -t|\ll N$, contains at least $L/360$ zeros of $\zeta (s)$. As a consequence, a quantitative restriction on zeros in a short family of arbitrarily thin fixed windows to the left of the line $\Ree s=1$ yields $S(x,t)\ll x/(\log x{)}^{1/100}$ in polynomial ranges of $x$. The proof adapts the zero-forcing mechanism of Granville and Soundararajan for large character sums. In the zeta setting the spectral height is shifted by $t$, and an additional residue from the pole of $\zeta (s)$ appears in the Gaussian transform; in the range considered here that residue is exponentially small.

19 pages