American Journal of Mathematics最新文献

On the Liouville function at polynomial arguments 论多项式参数下的柳维尔函数

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-07-17 DOI: 10.1353/ajm.2024.a932436

Joni Teräväinen

{"title":"On the Liouville function at polynomial arguments","authors":"Joni Teräväinen","doi":"10.1353/ajm.2024.a932436","DOIUrl":"https://doi.org/10.1353/ajm.2024.a932436","url":null,"abstract":"abstract:Let $lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne--Ferenczi--Mauduit--Rivat--S'ark\"ozy asks to show that if $P(x)inmathbb{Z}[x]$, then the sequence $lambda(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial.We show that the sequence $lambda(P(n))$ indeed changes sign infinitely often, provided that either (i) $P$ factorizes into linear factors over the rationals; or (ii) $P$ is a reducible cubic polynomial; or (iii) $P$ factorizes into a product of any number of quadratics of a certain type; or (iv) $P$ is any polynomial not belonging to an exceptional set of density zero.Concerning (i), we prove more generally that the partial sums of $g(P(n))$ for $g$ a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on $g$. This establishes a ``99% version'' of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of $g(P(n))$ and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"25 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Planar minimal surfaces with polynomial growth in the Sp(4,ℝ)-symmetric space Sp(4,ℝ)- 对称空间中多项式增长的平面极小曲面

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-07-17 DOI: 10.1353/ajm.2024.a932432

Andrea Tamburelli, Michael Wolf

引用次数: 0

Fourier transform and expanding maps on Cantor sets 康托尔集合上的傅立叶变换和展开图

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-07-17 DOI: 10.1353/ajm.2024.a932433

Tuomas Sahlsten, Connor Stevens

引用次数: 0

Large Fourier coefficients of half-integer weight modular forms 半整数权模块形式的大傅里叶系数

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-07-17 DOI: 10.1353/ajm.2024.a932437

S. Gun, W. Kohnen, K. Soundararajan

引用次数: 0

Wave equation on general noncompact symmetric spaces 一般非紧凑对称空间上的波方程

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-07-17 DOI: 10.1353/ajm.2024.a932434

Jean-Philippe Anker, Hong-Wei Zhang

引用次数: 0

The spectrum of an operator associated with G2-instantons with 1-dimensional singularities and Hermitian Yang–Mills connections with isolated singularities 与具有一维奇点的 G2-不等子和具有孤立奇点的赫尔墨斯杨-米尔斯连接相关的算子谱

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-07-17 DOI: 10.1353/ajm.2024.a932435

Yuanqi Wang

{"title":"The spectrum of an operator associated with G2-instantons with 1-dimensional singularities and Hermitian Yang–Mills connections with isolated singularities","authors":"Yuanqi Wang","doi":"10.1353/ajm.2024.a932435","DOIUrl":"https://doi.org/10.1353/ajm.2024.a932435","url":null,"abstract":"abstract:This is the first step in an attempt at a deformation theory for $G_{2}$-instantons with $1$-dimensional conic singularities. Under a set of model data, the linearization yields a Dirac operator $P$ on a certain bundle over $mathbb{S}^{5}$, called the textit{link operator}. As a dimension reduction, the link operator also arises from Hermitian Yang--Mills connections with isolated conic singularities on a Calabi--Yau $3$-fold.Using the quaternion structure in the Sasakian geometry of $mathbb{S}^{5}$, we describe the set of all eigenvalues of $P$, denoted by $Spec P$. We show that $Spec P$ consists of finitely many integers induced by certain sheaf cohomologies on $mathbb{P}^{2}$, and infinitely many real numbers induced by the spectrum of the rough Laplacian on the pullback endomorphism bundle over $mathbb{S}^{5}$. The multiplicities and the form of an eigensection can be described fairly explicitly.In particular, there is a relation between the spectrum on $mathbb{S}^{5}$ to certain sheaf cohomologies on~$mathbb{P}^{2}$.Moreover, on a Calabi--Yau $3$-fold, the index of the linearized operator for admissible singular Hermitian Yang--Mills connections is also calculated, in terms of these sheaf cohomologies.Using the representation theory of $SU(3)$ and the subgroup $S[U(1)times U(2)]$, we show an example in which $Spec P$ and the multiplicities can be completely determined.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"13 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three 三维规定标量曲率超曲面的内部曲率估算

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-05-24 DOI: 10.1353/ajm.2024.a928319

Guohuan Qiu

引用次数: 0

Prime and Möbius correlations for very short intervals in $fq[x]$ $fq[x]$ 中极短区间的质点和莫比乌斯相关性

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-05-24 DOI: 10.1353/ajm.2024.a928320

Pär Kurlberg, Lior Rosenzweig

{"title":"Prime and Möbius correlations for very short intervals in $fq[x]$","authors":"Pär Kurlberg, Lior Rosenzweig","doi":"10.1353/ajm.2024.a928320","DOIUrl":"https://doi.org/10.1353/ajm.2024.a928320","url":null,"abstract":"abstract:We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in ``very short intervals'' of the form $I(f):={f(x) + a : a infp}$ for $f(x)infp[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the M\"{o}bius $mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for ``generic'' set of shifts. We can also exhibit examples for which there is no cancellation at all in M\"{o}bius/Chowla type sums (in fact, it turns out that (square root) cancellation in M\"{o}bius sums is {em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic ``primes are independent'' fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the M\"{o}bius $mu$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x)infq[x]$ and intervals of the form $f(x)+a$ for $ainfq$, where $p$ is fixed and $q=p^{l}$ grows.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"37 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

New characterization of plurisubharmonic functions and positivity of direct image sheaves 多次谐函数的新表征和直映剪切的实在性

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-05-24 DOI: 10.1353/ajm.2024.a928324

Fusheng Deng, Zhiwei Wang, Liyou Zhang, Xiangyu Zhou

引用次数: 0

Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups 外特殊 2 群的色度定点理论和巴尔默谱

IF 1.7 1区数学

American Journal of Mathematics Pub Date : 2024-05-24 DOI: 10.1353/ajm.2024.a928325

Nicholas J. Kuhn, Christopher J. R. Lloyd

{"title":"Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups","authors":"Nicholas J. Kuhn, Christopher J. R. Lloyd","doi":"10.1353/ajm.2024.a928325","DOIUrl":"https://doi.org/10.1353/ajm.2024.a928325","url":null,"abstract":"abstract:In the early 1940s, P. A. Smith showed that if a finite $p$-group $G$ acts on a finite dimensional complex $X$ that is mod $p$ acyclic, then its space of fixed points, $X^G$, will also be mod $p$ acyclic.In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $G$-space $X$ is a equivariant homotopy retract of the $p$-localization of a based finite $G$-C.W. complex, given $H<G$ and $n$, what is the smallest $r$ such that if $X^H$ is acyclic in the $(n+r)$th Morava $K$-theory, then $X^G$ must be acyclic in the $n$th Morava $K$-theory? Barthel et.~al. then answered this when $G$ is abelian, by finding general lower and upper bounds for these ``blue shift'' numbers which agree in the abelian case.In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod $p$ homology, and thus applies to all finite dimensional $G$-spaces. This unlocks new techniques and applications in chromatic fixed point theory.Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups.Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for $K(n)_*$-homology disks and spheres.Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava $K$-theories of all real Grassmanians.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"125 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0