Adrian Hauffe-Waschbusch, A. Krieg, Brandon Williams
We consider the Hermitian Eisenstein series $E^{(mathbb{K})}_k$ of degree $2$ and weight $k$ associated with an imaginary-quadratic number field $mathbb{K}$ and determine the influence of $mathbb{K}$ on the arithmetic and the growth of its Fourier coefficients. We find that they satisfy the identity $E^{{(mathbb{K})}^2}_4 = E^{{(mathbb{K})}}_8$, which is well-known for Siegel modular forms of degree $2$, if and only if $mathbb{K} = mathbb{Q} (sqrt{-3})$. As an application, we show that the Eisenstein series $E^{(mathbb{K})}_k$, $k=4,6,8,10,12$ are algebraically independent whenever $mathbb{K}neq mathbb{Q}(sqrt{-3})$. The difference between the Siegel and the restriction of the Hermitian to the Siegel half-space is a cusp form in the Maass space that does not vanish identically for sufficiently large weight; however, when the weight is fixed, we will see that it tends to $0$ as the discriminant tends to $-infty$. Finally, we show that these forms generate the space of cusp forms in the Maass Spezialschar as a module over the Hecke algebra as $mathbb{K}$ varies over imaginary-quadratic number fields.
{"title":"On Hermitian Eisenstein series of degree $2$","authors":"Adrian Hauffe-Waschbusch, A. Krieg, Brandon Williams","doi":"10.7169/facm/2047","DOIUrl":"https://doi.org/10.7169/facm/2047","url":null,"abstract":"We consider the Hermitian Eisenstein series $E^{(mathbb{K})}_k$ of degree $2$ and weight $k$ associated with an imaginary-quadratic number field $mathbb{K}$ and determine the influence of $mathbb{K}$ on the arithmetic and the growth of its Fourier coefficients. We find that they satisfy the identity $E^{{(mathbb{K})}^2}_4 = E^{{(mathbb{K})}}_8$, which is well-known for Siegel modular forms of degree $2$, if and only if $mathbb{K} = mathbb{Q} (sqrt{-3})$. As an application, we show that the Eisenstein series $E^{(mathbb{K})}_k$, $k=4,6,8,10,12$ are algebraically independent whenever $mathbb{K}neq mathbb{Q}(sqrt{-3})$. The difference between the Siegel and the restriction of the Hermitian to the Siegel half-space is a cusp form in the Maass space that does not vanish identically for sufficiently large weight; however, when the weight is fixed, we will see that it tends to $0$ as the discriminant tends to $-infty$. Finally, we show that these forms generate the space of cusp forms in the Maass Spezialschar as a module over the Hecke algebra as $mathbb{K}$ varies over imaginary-quadratic number fields.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43307070","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":"Ulam stability of a quadratic-type functional equation in 3 variables in the quasi-Banach space","authors":"Ravi Sharma, S. Chandok","doi":"10.7169/facm/1934","DOIUrl":"https://doi.org/10.7169/facm/1934","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48603871","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":"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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