Mathematika最新文献

{"title":"Coloring unions of nearly disjoint hypergraph cliques","authors":"Dhruv Mubayi,&nbsp;Jacques Verstraete","doi":"10.1112/mtk.12234","DOIUrl":"10.1112/mtk.12234","url":null,"abstract":"<p>We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are generally known to exist only when the number of cliques is exponential in the clique size (Glock, Kühn, Lo, and Osthus, <i>Mem. Amer. Math. Soc</i>. <b>284</b> (2023) v+131 pp; Keevash, Preprint; Rödl, <i>Eur. J. Combin</i>. 6 (1985) 69–78). We construct near designs where the number of cliques is polynomial in the clique size, and show that they have large chromatic number. The case when the cliques have pairwise intersections of size at most one seems particularly challenging. Here, we give lower bounds by analyzing a random greedy hypergraph process. We also consider the related question of determining the maximum number of caps in a finite projective/affine plane and obtain nontrivial upper and lower bounds.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135243024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Upper bounds for the constants of Bennett's inequality and the Gale–Berlekamp switching game","authors":"Daniel Pellegrino,&nbsp;Anselmo Raposo Jr.","doi":"10.1112/mtk.12229","DOIUrl":"10.1112/mtk.12229","url":null,"abstract":"<p>In 1977, G. Bennett proved, by means of nondeterministic methods, an inequality that plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>p</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>p</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$p_{1},p_{2} in [1,infty ]$</annotation>\u0000 </semantics></math> and all positive integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$n_{1},n_{2}$</annotation>\u0000 </semantics></math>, there exists a bilinear form <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mrow>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>:</mo>\u0000 <mo>(</mo>\u0000 </mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msup>\u0000 <msub>\u0000 <mrow>\u0000 <mo>,</mo>\u0000 <mo>∥</mo>\u0000 <mo>·</mo>\u0000 <mo>∥</mo>\u0000 </mrow>\u0000 <msub>\u0000 <mi>p</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msub>\u0000 <mrow>\u0000 <mo>)</mo>\u0000 <mo>×</mo>\u0000 <mo>(</mo>\u0000 </mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </msup>\u0000 <msub>\u0000 <mrow>\u0000 <mo>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Correlation of multiplicative functions over \u0000 \u0000 \u0000 \u0000 F\u0000 q\u0000 \u0000 \u0000 [\u0000 x\u0000 ]\u0000 \u0000 \u0000 $mathbb {F}_q[x]$\u0000 : A pretentious approach","authors":"Pranendu Darbar,&nbsp;Anirban Mukhopadhyay","doi":"10.1112/mtk.12227","DOIUrl":"https://doi.org/10.1112/mtk.12227","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$mathcal {M}_n$</annotation>\u0000 </semantics></math> denote the set of monic polynomials of degree <i>n</i> over a finite field <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>q</mi>\u0000 </msub>\u0000 <annotation>$mathbb {F}_q$</annotation>\u0000 </semantics></math> of <i>q</i> elements. For multiplicative functions <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ψ</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>ψ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$psi _1,psi _2$</annotation>\u0000 </semantics></math>, using the recently developed “pretentious method,” we establish a “local-global” principle for correlation functions of the form\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12227","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68181295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Intrinsic Diophantine approximation on circles and spheres","authors":"Byungchul Cha,&nbsp;Dong Han Kim","doi":"10.1112/mtk.12228","DOIUrl":"https://doi.org/10.1112/mtk.12228","url":null,"abstract":"<p>We study Lagrange spectra arising from intrinsic Diophantine approximation of circles and spheres. More precisely, we consider three circles embedded in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^2$</annotation>\u0000 </semantics></math> or <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^3$</annotation>\u0000 </semantics></math> and three spheres embedded in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^3$</annotation>\u0000 </semantics></math> or <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>4</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^4$</annotation>\u0000 </semantics></math>. We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$mathbb {C}$</annotation>\u0000 </semantics></math>. Thanks to prior work of Asmus L. Schmidt on the spectra of <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$mathbb {C}$</annotation>\u0000 </semantics></math>, we obtain as a corollary, for each of the six spectra, the smallest accumulation point and the initial discrete part leading up to it completely.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68181297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the \u0000 \u0000 \u0000 a\u0000 b\u0000 c\u0000 \u0000 $abc$\u0000 conjecture in algebraic number fields","authors":"Andrew Scoones","doi":"10.1112/mtk.12230","DOIUrl":"https://doi.org/10.1112/mtk.12230","url":null,"abstract":"<p>In this paper, we prove a weak form of the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$abc$</annotation>\u0000 </semantics></math> conjecture generalised to algebraic number fields. Given integers satisfying <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>+</mo>\u0000 <mi>b</mi>\u0000 <mo>=</mo>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$a+b=c$</annotation>\u0000 </semantics></math>, Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. <b>291</b> (1991), 225–230], Stewart and Yu [Duke Math. J. <b>108</b> (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>K</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>b</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>c</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$H_{K}(a,,b,,c)$</annotation>\u0000 </semantics></math> in terms of the radical for algebraic numbers (Győry [Acta Arith. <b>133</b> (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field <i>K</i>. Given algebraic integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>b</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$a,,b,,c$</annotation>\u0000 </semantics></math> in a number field <i>K</i> satisfying <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>+</mo>\u0000 <mi>b</mi>\u0000 <mo>=</mo>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$a+b=c$</annotation>\u0000 </semantics></math>, we give an upper bound for the logarithm of the projective height <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>H</","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12230","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68181298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the number of tiles visited by a line segment on a rectangular grid","authors":"Alex Arkhipov,&nbsp;Luis Mendo","doi":"10.1112/mtk.12223","DOIUrl":"https://doi.org/10.1112/mtk.12223","url":null,"abstract":"<p>Consider a line segment placed on a two-dimensional grid of rectangular tiles. This paper addresses the relationship between the length of the segment and the number of tiles it visits (i.e., has intersection with). The square grid is also considered explicitly, as some of the specific problems studied are more tractable in that particular case. The segment position and orientation can be modeled as either deterministic or random. In the deterministic setting, the maximum possible number of visited tiles is characterized for a given length, and conversely, the infimum segment length needed to visit a desired number of tiles is analyzed. In the random setting, the average number of visited tiles and the probability of visiting the maximum number of tiles on a square grid are studied as a function of segment length. These questions are related to Buffon's needle problem and its extension by Laplace.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12223","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50149073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal Hardy-weights for elliptic operators with mixed boundary conditions","authors":"Yehuda Pinchover,&nbsp;Idan Versano","doi":"10.1112/mtk.12226","DOIUrl":"https://doi.org/10.1112/mtk.12226","url":null,"abstract":"<p>We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>P</mi>\u0000 <mo>,</mo>\u0000 <mi>B</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(P,B)$</annotation>\u0000 </semantics></math> with degenerate mixed boundary conditions. By an optimal Hardy-weight for a subcritical operator we mean a nonzero nonnegative weight function <i>W</i> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>P</mi>\u0000 <mo>−</mo>\u0000 <mi>W</mi>\u0000 <mo>,</mo>\u0000 <mi>B</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(P-W,B)$</annotation>\u0000 </semantics></math> is critical, and null-critical with respect to <i>W</i>. Our results rely on a recently developed criticality theory for positive solutions of the corresponding mixed boundary value problem.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12226","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability of polydisc slicing","authors":"Nathaniel Glover,&nbsp;Tomasz Tkocz,&nbsp;Katarzyna Wyczesany","doi":"10.1112/mtk.12225","DOIUrl":"https://doi.org/10.1112/mtk.12225","url":null,"abstract":"<p>We prove a dimension-free stability result for polydisc slicing due to Oleszkiewicz and Pełczyński. Intriguingly, compared to the real case, there is an additional asymptotic maximizer. In addition to Fourier-analytic bounds, we crucially rely on a self-improving feature of polydisc slicing, established via probabilistic arguments.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12225","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50145819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two problems on the distribution of Carmichael's lambda function","authors":"Paul Pollack","doi":"10.1112/mtk.12222","DOIUrl":"https://doi.org/10.1112/mtk.12222","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$lambda (n)$</annotation>\u0000 </semantics></math> denote the exponent of the multiplicative group modulo <i>n</i>. We show that when <i>q</i> is odd, each coprime residue class modulo <i>q</i> is hit equally often by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$lambda (n)$</annotation>\u0000 </semantics></math> as <i>n</i> varies. Under the stronger assumption that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>gcd</mo>\u0000 <mo>(</mo>\u0000 <mi>q</mi>\u0000 <mo>,</mo>\u0000 <mn>6</mn>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$gcd (q,6)=1$</annotation>\u0000 </semantics></math>, we prove that equidistribution persists throughout a Siegel–Walfisz-type range of uniformity. By similar methods we show that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$lambda (n)$</annotation>\u0000 </semantics></math> obeys Benford's leading digit law with respect to natural density. Moreover, if we assume Generalized Riemann Hypothesis, then Benford's law holds for the order of <i>a</i> mod <i>n</i>, for any fixed integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>∉</mo>\u0000 <mo>{</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mo>±</mo>\u0000 <mn>1</mn>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$anotin lbrace 0,pm 1rbrace$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12222","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50145833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the distribution of modular inverses from short intervals","authors":"Moubariz Z. Garaev,&nbsp;Igor E. Shparlinski","doi":"10.1112/mtk.12224","DOIUrl":"https://doi.org/10.1112/mtk.12224","url":null,"abstract":"<p>For a prime number <i>p</i> and integer <i>x</i> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>gcd</mo>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$gcd (x,p)=1$</annotation>\u0000 </semantics></math>, let <math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>x</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <annotation>$overline{x}$</annotation>\u0000 </semantics></math> denote the multiplicative inverse of <i>x</i> modulo <i>p</i>. In this paper, we are interested in the problem of distribution modulo <i>p</i> of the sequence\u0000\u0000 </p><p>For any fixed <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$A &gt; 1$</annotation>\u0000 </semantics></math> and for any sufficiently large integer <i>N</i>, there exists a prime number <i>p</i> with\u0000\u0000 </p><p>For any fixed positive <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>&lt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$gamma &lt; 1$</annotation>\u0000 </semantics></math>, there exists a positive constant <i>c</i> such that the following holds: for any sufficiently large integer <i>N</i> there is a prime number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>&gt;</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$p &gt; N$</annotation>\u0000 </semantics></math> such that\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12224","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50145829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}