REVIEW
Large zeta sums and zeros of the Riemann zeta function
Number Theory · Submitted 31 Aug 2026 · 11 references
Pith's verdict: A careful, unconditional inverse theorem linking large zeta sums and zeros—solid work that deserves publication.
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Referee report
The paper
The paper claims
Large zeta sums force zero clusters near the line Re(s)=1
The reading
What carries the argument
The core machinery is a Gaussian transorm identity (Lemma3.3) that expresses a weighted integral of S(e^y,τ) as an integral of ζ(1-λ+ i(ξ-τ)) over ξ plus a residue from the pole at s=1. This is combinoed with Halász-type mean value estimates (Lemma2.1) for multiplicative functions specialized to $n^{{it}}$, Lipschitz estimates for the normilized sums, and a bound from the Hadamard product of ξ(s) (Lemma3.2) that converts a lower bound on |ζ| into a lower bound on a weighted sum over zeros. The zero-forcing argument then shows that if the weighted sum is large, many zeros must lie in a small disk.
Extended reading notes
Core claim
The central discovery of the paper is Theorem 1.1, which estabilshes an unconditional quantitative relation between a single large value of S(x,t) and the local density of zeta zeros near the line Re(s)=1. The theorem says that if |S(x,t)|=x/N with N small relative to log x, then for any L up to half log x, a small disk centered near 1+it (radius proportional to L logT/(log x)^2) contains at least L/360 zeros of ζ(s). This is the first such inverse theorem for zeta sums, modeloed on the analogous result for Dirichlet character sums but with the novel feature that the pole of ζ introduces a residue term in the Fourier transform, which is shown to be negligible in the chosen parameter range.
Load-bearing premise
The theorem relies on the range condition exp(√log T) ≤ x ≤ √T; without it the error terms in the Gaussian transform and the mean-value estimates would be too large for the zero-forcing argument to work.
Editorial extensions
If this is right
- If the zeros of ζ(s) are sparse in every thin left-of-1 strip, then S(x,t) must be as small as x/(log x)^{1/100} for x up to any polynomial of T (Theorem 1.2).
- Combining Theorem 1.2 with zero-density estimates (e.g., Guth–Maynard) shows that the local sparsity hypothhesis fails only on a set of t of small measure (Remark 1.3).
- The method provides a quantitative way to bound zeta sums from above if one knows local zero distributions near the line Re(s)=1.
Reading between the lines
- The exponent 6 in the condition L≥ cN^6 likely arises from the specific estimates used; improving this exponent would sharpen the theorem and could connect to the Lindelöf hypothesis.
- The technique could potentialy be extended to other Dirichlet series with a pole, such as Dedekind zeta functions, to relate large partial sums to zeros near the pole.
- A numerical test for moderate T might check the zero-forcing prediction by computing S(x,t) and counting zeros in small disks, though such computations remain challenging for the required large parameters.
Original 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 $cN^6\le L\le (\log x)/2$ a disk centered at $1+\ii\phi$, where $|\phi-t|\ll N$, 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.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
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 $cN^6\le L\le (\log x)/2$ a disk centered at $1+\ii\phi$, where $|\phi-t|\ll N$, 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For real $t$ and $x\ge 1$, set \[ S(x,t)=\sum_{n\le x} n^{\ii t}. | background 0.92 | 0.36 | 0.02 | unclear 0.59 |
| s1 | \] 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 $t$. | contribution 0.66 | 0.96 | 0.87 | asserted 1.00 |
| s2 | 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 $cN^6\le L\le (\log x)/2$ a disk centered at $1+\ii\phi$, where $|\phi-t|\ll N$, contains at least $L/360$ zeros of $\zeta(s)$. | result 0.99 | 0.93 | 0.63 | asserted 1.00 |
| s3 | 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 $x$. | interpretation 0.60 | 0.86 | 0.23 | asserted 0.99 |
| s4 | The proof adapts the zero-forcing mechanism of Granville and Soundararajan for large character sums. | method 0.99 | 0.64 | 0.03 | asserted 1.00 |
| s5 | 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. | method 0.96 | 0.72 | 0.03 | asserted 1.00 |
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Cites as background 1 paper
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The typical and averaged sizes of zeta sums have also been studied by Harper [ref] , while Gao [ref] obtained conditional upper bounds for their moments.
The typical size of character and zeta sums is $o(\sqrt{x})$ · Introduction · 0.99
Discussion
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