bibkey: liuwuyang2021variant authors: Kui Liu; Jie Wu; Zhishan Yang year: 2021 title: A variant of the prime number theorem doi: null url: https://arxiv.org/abs/2105.10844v1 claim: Proposition 3.1 bounds triple monomial exponential sums with arbitrary bounded coefficient arrays; it supplies a high-harmonic rectangular estimate under its stated size condition, without controlling a complete drifted Robin positive part. strata_touched: [] license: citation-only triage: anchor

A variant of the prime number theorem

The inspected primary is arXiv:2105.10844v1, submitted 23 May 2021. Proposition 3.1 and equation (3.2), PDF p.3, have been read in both the PDF and source TeX. This note records that classical statement and its application boundary, without an independent audit of the whole proof or Lean certification.

Triple sums with arbitrary bounded coefficients

Write $e(t)=e^{2\pi it}$ and $m\sim M$ for $M<m\le 2M$. For fixed $\alpha ,\beta ,\gamma >0$, $\delta \in \mathbb{R}$, and $X>0$, $H,M,N\ge 1$, the source defines

$$S_{\delta }=\sum _{h\sim H}\sum _{m\sim M}\sum _{n\sim N}{a}_{h,m}{b}_{n}e\left(X\frac{M^{\beta }{N}^{\gamma }}{H^{\alpha }}\frac{h^{\alpha }}{m^{\beta }{n}^{\gamma }+\delta }\right),|a_{h,m}|,|b_n|\le 1.$$

The first estimate of Proposition 3.1 states, for every $\epsilon >0$,

$$S_{\delta }{\ll }_{\alpha ,\beta ,\gamma ,\epsilon }\left((XHMN{)}^{1/2}+(HM{)}^{1/2}N+HM{N}^{1/2}+X^{-1/2}HMN\right)X^{\epsilon },$$

uniformly under $H\le M^{\beta -1}{N}^{\gamma }$ and $|\delta |\le 1/\epsilon$. The application uses $\delta =0$. The paper also gives an exponent-pair variant; that variant is not needed for the application recorded here.

Robin sampling correspondence and limits

The actual sieve-response application uses $(\alpha ,\beta ,\gamma )=(1,k,1)$ and $(M,N,X)=(P,D,Hx/(P^{k}D))$. Its prime and harmonic test coefficients fit $a_{h,m}$, while the normalized actual LCM coefficient fits $b_n$. The size hypothesis becomes $H\le P^{k-1}D$; arbitrary bounded sample-dependent signs are permitted by the coefficient quantifiers. All four terms and coefficient normalization costs must be retained. The resulting high-harmonic estimate does not bound the low harmonics, mean, complete drifted positive part, prime layer or signed Robin tail.