Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic <i>L</i>‐functions

He, Xiaoguang; Wang, Mengdi

doi:10.1112/mtk.12182

Cited by 1 publication

(2 citation statements)

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“…Theorem 1.1(iv) generalizes (albeit with a slightly weaker logarithmic saving) a result of the first and fourth authors [50,Theorem 1.5] that gave, for 0 < 𝐴 < 1/3, for 𝑋 2/5+𝜀 ≤ 𝐻 ≤ 𝑋. See also [25] for a result with general nilsequences but long intervals. Unfortunately, the methods we use in this paper rely heavily on the convolution structure of the functions involved and do not obviously extend to give results for 𝜆 𝑓 .…”

Section: 𝜇(𝑛)𝑒(−𝛼𝑛)supporting

confidence: 59%

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Higher uniformity of arithmetic functions in short intervals I. All intervals

Matomäki,

Shao,

Tao

et al. 2023

Forum of Mathematics, Pi

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We study higher uniformity properties of the Möbius function $\mu $ , the von Mangoldt function $\Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$ . More precisely, letting $\Lambda ^\sharp $ and $d_k^\sharp $ be suitable approximants of $\Lambda $ and $d_k$ and $\mu ^\sharp = 0$ , we show for instance that, for any nilsequence $F(g(n)\Gamma )$ , we have $$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when $\theta = 5/8$ and $f \in \{\Lambda , \mu , d_k\}$ or $\theta = 1/3$ and $f = d_2$ . As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp \|_{U^s(X,X+H]}$ are also asymptotically small for any fixed s for these choices of $f,\theta $ . As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in $L^2$ . Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type $II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

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Section: 𝜇(𝑛)𝑒(−𝛼𝑛)supporting

confidence: 59%

“…For instance, in [10] it was established that for . See also [25] for a result with general nilsequences but long intervals. Unfortunately, the methods we use in this paper rely heavily on the convolution structure of the functions involved and do not obviously extend to give results for .…”

Section: Introductionmentioning

confidence: 99%