{"title":"第二个计算质数测地线的力矩","authors":"I. Kaneko","doi":"10.3792/pjaa.96.002","DOIUrl":null,"url":null,"abstract":": A brighter light has freshly been shed upon the second moment of the Prime Geodesic Theorem. We work with such moments in the two and three dimensional hyperbolic spaces. Letting E (cid:2) ð X Þ be the error term arising from counting prime geodesics associated to (cid:2) ¼ PSL 2 ð Z ½ i (cid:2)Þ , the bound E (cid:2) ð X Þ (cid:3) X 3 = 2 þ (cid:2) is proved in a square mean sense. Our second moment bound is the pure counterpart of the work of Balog et al. for (cid:2) ¼ PSL 2 ð Z Þ , and the main innovation entails the delicate analysis of sums of Kloosterman sums. We also infer pointwise bounds from the standpoint of the second moment. Finally, we announce the pointwise bound E (cid:2) ð X Þ (cid:3) X 67 = 42 þ (cid:2) for (cid:2) ¼ PSL 2 ð Z ½ i (cid:2)Þ by an application of the Weyl-type subconvexity.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"86 18 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The second moment for counting prime geodesics\",\"authors\":\"I. Kaneko\",\"doi\":\"10.3792/pjaa.96.002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\": A brighter light has freshly been shed upon the second moment of the Prime Geodesic Theorem. We work with such moments in the two and three dimensional hyperbolic spaces. Letting E (cid:2) ð X Þ be the error term arising from counting prime geodesics associated to (cid:2) ¼ PSL 2 ð Z ½ i (cid:2)Þ , the bound E (cid:2) ð X Þ (cid:3) X 3 = 2 þ (cid:2) is proved in a square mean sense. Our second moment bound is the pure counterpart of the work of Balog et al. for (cid:2) ¼ PSL 2 ð Z Þ , and the main innovation entails the delicate analysis of sums of Kloosterman sums. We also infer pointwise bounds from the standpoint of the second moment. Finally, we announce the pointwise bound E (cid:2) ð X Þ (cid:3) X 67 = 42 þ (cid:2) for (cid:2) ¼ PSL 2 ð Z ½ i (cid:2)Þ by an application of the Weyl-type subconvexity.\",\"PeriodicalId\":49668,\"journal\":{\"name\":\"Proceedings of the Japan Academy Series A-Mathematical Sciences\",\"volume\":\"86 18 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2020-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Japan Academy Series A-Mathematical Sciences\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3792/pjaa.96.002\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Japan Academy Series A-Mathematical Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3792/pjaa.96.002","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
摘要
一束明亮的光刚刚洒在质数测地线定理的第二矩上。我们在二维和三维双曲空间中处理这样的力矩。设E (cid:2) ð X Þ是由计算与(cid:2)¼PSL 2 ð Z½i (cid:2)Þ相关的素数测地线引起的误差项，在平方平均意义上证明了界限E (cid:2) ð X Þ (cid:3) x3 = 2 Þ (cid:2)。我们的第二个矩界是Balog等人对(cid:2)¼PSL 2 ð Z Þ的工作的纯粹对应，主要的创新需要对Kloosterman和的和进行精细的分析。我们还从第二矩的立场推断出点边界。最后，我们通过应用weyl型子凸性，宣布了(cid:2)¼PSL 2 ð Z½i (cid:2)Þ的点界E (cid:2) ð X Þ (cid:3) X 67 = 42 Þ (cid:2)。
: A brighter light has freshly been shed upon the second moment of the Prime Geodesic Theorem. We work with such moments in the two and three dimensional hyperbolic spaces. Letting E (cid:2) ð X Þ be the error term arising from counting prime geodesics associated to (cid:2) ¼ PSL 2 ð Z ½ i (cid:2)Þ , the bound E (cid:2) ð X Þ (cid:3) X 3 = 2 þ (cid:2) is proved in a square mean sense. Our second moment bound is the pure counterpart of the work of Balog et al. for (cid:2) ¼ PSL 2 ð Z Þ , and the main innovation entails the delicate analysis of sums of Kloosterman sums. We also infer pointwise bounds from the standpoint of the second moment. Finally, we announce the pointwise bound E (cid:2) ð X Þ (cid:3) X 67 = 42 þ (cid:2) for (cid:2) ¼ PSL 2 ð Z ½ i (cid:2)Þ by an application of the Weyl-type subconvexity.
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