Let GP (q, d) be the d-Paley graph defined on the finite field Fq . It is notoriously difficult to improve the trivial upper bound √ q on the clique number of GP (q, d). In this paper, we investigate the connection between Gauss sums over a finite field and the maximum cliques of their corresponding generalized Paley graphs. We show that the trivial upper bound on the clique number of GP (q, d) is tight if and only if d | (√q + 1), which strengthens the previous related results by Broere-Döman-Ridley and Schneider-Silva. We also obtain a new simple proof of Stickelberger’s theorem on evaluating semi-primitive Gauss sums.
{"title":"Gauss sums and the maximum cliquesin generalized Paley graphs of square order","authors":"Chi Hoi Yip","doi":"10.7169/facm/1981","DOIUrl":"https://doi.org/10.7169/facm/1981","url":null,"abstract":"Let GP (q, d) be the d-Paley graph defined on the finite field Fq . It is notoriously difficult to improve the trivial upper bound √ q on the clique number of GP (q, d). In this paper, we investigate the connection between Gauss sums over a finite field and the maximum cliques of their corresponding generalized Paley graphs. We show that the trivial upper bound on the clique number of GP (q, d) is tight if and only if d | (√q + 1), which strengthens the previous related results by Broere-Döman-Ridley and Schneider-Silva. We also obtain a new simple proof of Stickelberger’s theorem on evaluating semi-primitive Gauss sums.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44025836","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":"Asymptotic behavior of Bernoulli-Dunkland Euler-Dunkl polynomials and their zeros","authors":"J. M. Ceniceros, J. Varona","doi":"10.7169/facm/1968","DOIUrl":"https://doi.org/10.7169/facm/1968","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44295523","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":"Analytic continuation of multi-variable Arakawa-Kaneko zeta function for positive indices and its values at positive integers","authors":"K. Ito","doi":"10.7169/facm/1974","DOIUrl":"https://doi.org/10.7169/facm/1974","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49466735","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":"On a generalization of the Euler totient function","authors":"W. Zhai","doi":"10.7169/FACM/1917","DOIUrl":"https://doi.org/10.7169/FACM/1917","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42036478","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":"Heat operators on modular and quasimodular polynomials","authors":"Min Ho Lee","doi":"10.7169/facm/1978","DOIUrl":"https://doi.org/10.7169/facm/1978","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47411051","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}
Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the"small"solutions of these equations. Under"small"solutions we mean the solutions, say, with absolute values or sizes $leq 10^{100}$. Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain"totally real"relative Thue equations can be reduced to absolute Thue equations (equations over $Bbb Z$). As a common generalization of the above results, in our paper we develop a fast algorithm for calculating"small"solutions (say with sizes $leq 10^{100}$) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the"totally real"case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.
{"title":"Calculating “small” solutions of inhomogeneous relative Thue inequalities","authors":"Istv'an Ga'al","doi":"10.7169/facm/1876","DOIUrl":"https://doi.org/10.7169/facm/1876","url":null,"abstract":"Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the\"small\"solutions of these equations. Under\"small\"solutions we mean the solutions, say, with absolute values or sizes $leq 10^{100}$. Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain\"totally real\"relative Thue equations can be reduced to absolute Thue equations (equations over $Bbb Z$). As a common generalization of the above results, in our paper we develop a fast algorithm for calculating\"small\"solutions (say with sizes $leq 10^{100}$) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the\"totally real\"case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48913708","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}
Paul Erd˝os, Janos Galambos and others have studied the relative size of the consecutive prime divisors of an integer. Here, we further extend this study by examining the distribution of the consecutive neighbour spacings between the prime divisors p 1 ( n ) < p 2 ( n ) < · · · < p r ( n ) of a typical integer n ≥ 2. In particular, setting γ j ( n ) := log p j ( n ) / log p j +1 ( n ) for j = 1 , 2 , . . . , r − 1 and, for any λ ∈ (0 , 1], introducing U λ ( n ) := # { j ∈ { 1 , 2 , . . . , r − 1 } : γ j ( n ) < λ } , we establish the mean value of U λ ( n ) and prove that U λ ( n ) /r ∼ λ for almost all integers n ≥ 2. We also examine the shifted prime version of these two results and study other related functions.
Paul Erdõos、Janos Galambos等人研究了整数的连续素数的相对大小。在这里，我们通过检验典型整数n≥2的素数p1（n）
{"title":"On the consecutive prime divisors of an integer","authors":"J. Koninck, Imre Kátai Imre Kátai","doi":"10.7169/facm/1922","DOIUrl":"https://doi.org/10.7169/facm/1922","url":null,"abstract":"Paul Erd˝os, Janos Galambos and others have studied the relative size of the consecutive prime divisors of an integer. Here, we further extend this study by examining the distribution of the consecutive neighbour spacings between the prime divisors p 1 ( n ) < p 2 ( n ) < · · · < p r ( n ) of a typical integer n ≥ 2. In particular, setting γ j ( n ) := log p j ( n ) / log p j +1 ( n ) for j = 1 , 2 , . . . , r − 1 and, for any λ ∈ (0 , 1], introducing U λ ( n ) := # { j ∈ { 1 , 2 , . . . , r − 1 } : γ j ( n ) < λ } , we establish the mean value of U λ ( n ) and prove that U λ ( n ) /r ∼ λ for almost all integers n ≥ 2. We also examine the shifted prime version of these two results and study other related functions.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42847513","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":"${P}$-adic approximation of Dedekind sumsin function fields","authors":"Y. Hamahata","doi":"10.7169/facm/1961","DOIUrl":"https://doi.org/10.7169/facm/1961","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41449406","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}
By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $geq 4$, and show the existence of infinitely many elliptic curves of rank $geq 5$, parameterized by the points of an elliptic curve of positive rank.
{"title":"Integral triangles and perpendicular quadrilateral pairs with a common area and a common perimeter","authors":"A. S. Zargar, Yong Zhang","doi":"10.7169/facm/1842","DOIUrl":"https://doi.org/10.7169/facm/1842","url":null,"abstract":"By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $geq 4$, and show the existence of infinitely many elliptic curves of rank $geq 5$, parameterized by the points of an elliptic curve of positive rank.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46205215","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 discuss evaluating fractional Stieltjes constants γα(a), arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions ζ(α)(s, a). We give an upper bound for the absolute value of Cα(a) = γα(a) − log(a)/a and an asymptotic formula C̃α(a) for Cα(a) that yields a good approximation even for most small values of α. We bound |C̃α(a)| and based on this conjecture a tighter bound for |Cα(a)|
{"title":"Approximating and bounding fractional Stieltjes constants","authors":"Ricky E. Farr, S. Pauli, F. Saidak","doi":"10.7169/facm/1868","DOIUrl":"https://doi.org/10.7169/facm/1868","url":null,"abstract":"We discuss evaluating fractional Stieltjes constants γα(a), arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions ζ(α)(s, a). We give an upper bound for the absolute value of Cα(a) = γα(a) − log(a)/a and an asymptotic formula C̃α(a) for Cα(a) that yields a good approximation even for most small values of α. We bound |C̃α(a)| and based on this conjecture a tighter bound for |Cα(a)|","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47499838","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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