{"title":"An explicit evaluation of $10^{text{th}}$-power moment of quadratic Gauss sums and some applications","authors":"Nilanjan Bag, Antonio Rojas-León, Zhang Wenpeng","doi":"10.7169/facm/1995","DOIUrl":"https://doi.org/10.7169/facm/1995","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46082326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Galois realization of Schur covers of dihedral groups of $2$-power order","authors":"Ryosuke Amano, Akira Ishimaru, Masanari Kida","doi":"10.7169/facm/1975","DOIUrl":"https://doi.org/10.7169/facm/1975","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48025337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hadamard product of series with special numbers","authors":"Khristo N. Boyadzhiev, R. Frontczak","doi":"10.7169/facm/2050","DOIUrl":"https://doi.org/10.7169/facm/2050","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47604231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We prove Haynes’ version of the Duffin–Schaeffer conjecture for the p -adic numbers. In addition, we prove several results about an associated related but false conjecture, related to p -adic approximation in the spirit of Jarn´ık and Lutz.
{"title":"The $p$-adic Duffin-Schaeffer conjecture","authors":"S. Kristensen, M. Laursen","doi":"10.7169/facm/2042","DOIUrl":"https://doi.org/10.7169/facm/2042","url":null,"abstract":". We prove Haynes’ version of the Duffin–Schaeffer conjecture for the p -adic numbers. In addition, we prove several results about an associated related but false conjecture, related to p -adic approximation in the spirit of Jarn´ık and Lutz.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48928543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let K = Q(θ) be a number field generated by a complex root θ of a monic irreducible trinomial F (x) = x + ax + b ∈ Z[x]. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of Z[θ]. But if Z[θ] is not integrally closed, then Jhorar and Khanduja’s results cannot answer on the monogenity of K. In this paper, based on Newton polygon techniques, we deal with the problem of monogenity of K. More precisely, when ZK 6= Z[θ], we give sufficient conditions on n, a and b for K to be not monogenic. For n ∈ {5, 6, 3, 2 · 3, 2 · 3 + 1}, we give explicitly some infinite families of these number fields that are not monogenic. Finally, we illustrate our results by some computational examples.
{"title":"On monogenity of certain number fields defined by trinomials","authors":"H. B. Yakkou, L. E. Fadil","doi":"10.7169/facm/1987","DOIUrl":"https://doi.org/10.7169/facm/1987","url":null,"abstract":"Let K = Q(θ) be a number field generated by a complex root θ of a monic irreducible trinomial F (x) = x + ax + b ∈ Z[x]. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of Z[θ]. But if Z[θ] is not integrally closed, then Jhorar and Khanduja’s results cannot answer on the monogenity of K. In this paper, based on Newton polygon techniques, we deal with the problem of monogenity of K. More precisely, when ZK 6= Z[θ], we give sufficient conditions on n, a and b for K to be not monogenic. For n ∈ {5, 6, 3, 2 · 3, 2 · 3 + 1}, we give explicitly some infinite families of these number fields that are not monogenic. Finally, we illustrate our results by some computational examples.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49452001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rankin--Cohen brackets on Hermitian Jacobi forms and the adjoint of some linear maps","authors":"S. Sumukha, Singh Sujeet Kumar","doi":"10.7169/facm/1890","DOIUrl":"https://doi.org/10.7169/facm/1890","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43488492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Accurate computations of Euler products over primes in arithmetic progressions","authors":"Ramaré Olivier","doi":"10.7169/facm/1853","DOIUrl":"https://doi.org/10.7169/facm/1853","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44200830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we give the order of magnitude for the summatory function of the number of divisors of the least common multiple of $p^i-1$ for $i=1,2,ldots,k$ when $ple x$ is prime.
{"title":"On the number of divisors of the least common multiples of shifted prime powers","authors":"F. Luca, F. Pappalardi","doi":"10.7169/FACM/1866","DOIUrl":"https://doi.org/10.7169/FACM/1866","url":null,"abstract":"In this paper, we give the order of magnitude for the summatory function of the number of divisors of the least common multiple of $p^i-1$ for $i=1,2,ldots,k$ when $ple x$ is prime.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48784409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Abstract intersection theory for zeta-functions: geometric aspects","authors":"Grzegorz Banaszak, Y. Uetake","doi":"10.7169/FACM/1916","DOIUrl":"https://doi.org/10.7169/FACM/1916","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48980004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let k be a number field and let H(k) denote the Hilbert class field of k, that is the maximal abelian unramified extension of k. It is known by class field theory that the Galois group of the extension H(k)/k, i.e., G := Gal(H(k)/k), is isomorphic to Cl(k), the class group of k (cf. [13, p. 228]). The Hilbert genus field of k, denoted by E(k), is the invariant field of G. Thus, by Galois theory, we have: Cl(k)/Cl(k) ≃ G/G ≃ Gal(E(k)/k),
设k为数域，H(k)表示k的Hilbert类域，即k的最大阿贝尔无分支扩展。由类场论可知，扩展H(k)/k的伽罗瓦群，即G:= Gal(H(k)/k)，与k的类群Cl(k)同构(cf. [13, p. 228])。k的Hilbert格场，用E(k)表示，是G的不变场。因此，根据伽罗瓦理论，我们得到:Cl(k)/Cl(k)≃G/G≃Gal(E(k)/k);
{"title":"The construction of the Hilbert genus fields of real cyclic quartic fields","authors":"M. M. Chems-Eddin, Moulay Ahmed Hajjami, M. Taous","doi":"10.7169/facm/2014","DOIUrl":"https://doi.org/10.7169/facm/2014","url":null,"abstract":"Let k be a number field and let H(k) denote the Hilbert class field of k, that is the maximal abelian unramified extension of k. It is known by class field theory that the Galois group of the extension H(k)/k, i.e., G := Gal(H(k)/k), is isomorphic to Cl(k), the class group of k (cf. [13, p. 228]). The Hilbert genus field of k, denoted by E(k), is the invariant field of G. Thus, by Galois theory, we have: Cl(k)/Cl(k) ≃ G/G ≃ Gal(E(k)/k),","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44343428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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