First we reprove two results in additive number theorydue to Dombi and Chen & Wang, respectively, on the number ofrepresentations of $n$ as the sum of two odious or evil numbers, using techniques from automata theory and logic. We also use this technique to prove a new result aboutthe numbers represented by five summands. Furthermore, we prove some new results on the tenfold sums of the evil and odious numbers, as well as $k$-fold sums of similar sequences of integers, by using techniques of analytic number theory involving trigonometric sums associated with the $pm 1$ characteristic sequences of these integers.
{"title":"Additive properties of the evil and odious numbers and similar sequences","authors":"Jean-Paul Allouche, Jeffrey Shallit","doi":"10.7169/facm/2108","DOIUrl":"https://doi.org/10.7169/facm/2108","url":null,"abstract":"First we reprove two results in additive number theorydue to Dombi and Chen & Wang, respectively, on the number ofrepresentations of $n$ as the sum of two odious or evil numbers, using techniques from automata theory and logic. We also use this technique to prove a new result aboutthe numbers represented by five summands. Furthermore, we prove some new results on the tenfold sums of the evil and odious numbers, as well as $k$-fold sums of similar sequences of integers, by using techniques of analytic number theory involving trigonometric sums associated with the $pm 1$ characteristic sequences of these integers.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135107188","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 study the number of integer pair solutions to the equation $|F(x,y)| = 1$ where $F(x,y) in Z[x,y]$ is an irreducible (over $Z$) binary form with degree $n geq 3$ and exactly three nonzero summands. In particular, we improve Thomas' explicit upper bounds on the number of solutions to this equation (see [13]). For instance, when $n geq 219$, we show that there are no more than 32 integer pair solutions to this equation when $n$ is odd and no more than 40 integer pair solutions to this equation when $n$ is even, an improvement on Thomas' work in [13], where he shows that there are no more than 38 such solutions when $n$ is odd and no more than 48 such solutions when $n$ is even.
本文研究了方程$|F(x,y)| = 1$的整数对解的个数，其中$F(x,y) in Z[x,y]$是次为$n geq 3$的不可约(在$Z$上)二进制形式，且恰好有三个非零和。特别地，我们改进了Thomas关于该方程解个数的显式上界(见[13])。例如，当$n geq 219$时，我们表明，当$n$为奇数时，该方程的整数对解不超过32个，当$n$为偶数时，该方程的整数对解不超过40个，这是对Thomas在[13]中的工作的改进，他表明，当$n$为奇数时，该方程的整数对解不超过38个，当$n$为偶数时，该方程的整数对解不超过48个。
{"title":"The number of solutionsto the trinomial Thue equation","authors":"Greg Knapp","doi":"10.7169/facm/2093","DOIUrl":"https://doi.org/10.7169/facm/2093","url":null,"abstract":"In this paper, we study the number of integer pair solutions to the equation $|F(x,y)| = 1$ where $F(x,y) in Z[x,y]$ is an irreducible (over $Z$) binary form with degree $n geq 3$ and exactly three nonzero summands. In particular, we improve Thomas' explicit upper bounds on the number of solutions to this equation (see [13]). For instance, when $n geq 219$, we show that there are no more than 32 integer pair solutions to this equation when $n$ is odd and no more than 40 integer pair solutions to this equation when $n$ is even, an improvement on Thomas' work in [13], where he shows that there are no more than 38 such solutions when $n$ is odd and no more than 48 such solutions when $n$ is even.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135106313","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 introduce the study of triple convolution Ramanujan sums and apply it to give a~heuristic derivation of the Hardy-Littlewood conjecture on prime 3-tuples without using the circle method. We also estimate the triple convolution of the Jordan totient function using Ramanujan sums.
{"title":"On the Hardy-Littlewood prime tuples conjecture and higher convolutions of Ramanujan sums","authors":"Sneha Chaubey, Shivani Goel, M. Ram Murty","doi":"10.7169/facm/2048","DOIUrl":"https://doi.org/10.7169/facm/2048","url":null,"abstract":"We introduce the study of triple convolution Ramanujan sums and apply it to give a~heuristic derivation of the Hardy-Littlewood conjecture on prime 3-tuples without using the circle method. We also estimate the triple convolution of the Jordan totient function using Ramanujan sums.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135495011","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 $N$ be a square-free integer. Let $fin mathcal{B}^ast_k(N)$ (or $mathcal{B}_lambda^ast(N)$) be a primitive (either holomorphic or Maaß) cusp form of level $N$, with $lambda_f(n)$ denoting the $n$-th Hecke eigenvalue. In this paper, we explicitly determine the dependence on the level aspect for the sum [sum_{nle X}lambda_f(n) e{left(n^2alpha+nbeta right)},] which is uniform in any $alpha,betain R$ and $Xge 2$. In addition, we also investigate the analog at the prime arguments.
{"title":"The exponential sums related to cusp formsin the level aspect","authors":"Fei Hou","doi":"10.7169/facm/2079","DOIUrl":"https://doi.org/10.7169/facm/2079","url":null,"abstract":"Let $N$ be a square-free integer. Let $fin mathcal{B}^ast_k(N)$ (or $mathcal{B}_lambda^ast(N)$) be a primitive (either holomorphic or Maaß) cusp form of level $N$, with $lambda_f(n)$ denoting the $n$-th Hecke eigenvalue. In this paper, we explicitly determine the dependence on the level aspect for the sum [sum_{nle X}lambda_f(n) e{left(n^2alpha+nbeta right)},] which is uniform in any $alpha,betain R$ and $Xge 2$. In addition, we also investigate the analog at the prime arguments.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135495012","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":"Rational right triangle tripleswith special linear relationship of areas and perimeters","authors":"Yangcheng Li, Y. Zhang","doi":"10.7169/facm/2031","DOIUrl":"https://doi.org/10.7169/facm/2031","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44259606","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":"Kronecker's first limit formula for Kleinian groups","authors":"Zihan Miao, A. Nguyen, T. Wong","doi":"10.7169/facm/1997","DOIUrl":"https://doi.org/10.7169/facm/1997","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41950775","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 the product of translated division polynomials and Somos sequences","authors":"B. Gezer, O. Bizim","doi":"10.7169/facm/2038","DOIUrl":"https://doi.org/10.7169/facm/2038","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43491280","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 $q$ be a prime. We give an elementary proof of the fact that for any $kinmathbb{N}$, the proportion of $k$-element subsets of $mathbb{Z}$ that contain a $q^{text{th}}$ power modulo almost every prime, is zero. This result holds regardless of whether the proportion is measured additively or multiplicatively. More specifically, the number of $k$-element subsets of $[-N, N]capmathbb{Z}$ that contain a $q^{text{th}}$ power modulo almost every prime is no larger than $a_{q,k} N^{k-(1-frac{1}{q})}$, for some positive constant $a_{q,k}$. Furthermore, the number of $k$-element subsets of ${pm p_{1}^{e_{1}} p_{2}^{e_{2}} cdots p_N^{e_N} : 0 leq e_{1}, e_{2}, ldots, e_Nleq N}$ that contain a $q^{text{th}}$ power modulo almost every prime is no larger than $m_{q,k} frac{N^{Nk}}{q^N}$ for some positive constant $m_{q,k}$.
{"title":"Almost no finite subset of integers containsa $q^{text{th}}$ power modulo almost every prime","authors":"Bhawesh Mishra","doi":"10.7169/facm/2122","DOIUrl":"https://doi.org/10.7169/facm/2122","url":null,"abstract":"Let $q$ be a prime. We give an elementary proof of the fact that for any $kinmathbb{N}$, the proportion of $k$-element subsets of $mathbb{Z}$ that contain a $q^{text{th}}$ power modulo almost every prime, is zero. This result holds regardless of whether the proportion is measured additively or multiplicatively. More specifically, the number of $k$-element subsets of $[-N, N]capmathbb{Z}$ that contain a $q^{text{th}}$ power modulo almost every prime is no larger than $a_{q,k} N^{k-(1-frac{1}{q})}$, for some positive constant $a_{q,k}$. Furthermore, the number of $k$-element subsets of ${pm p_{1}^{e_{1}} p_{2}^{e_{2}} cdots p_N^{e_N} : 0 leq e_{1}, e_{2}, ldots, e_Nleq N}$ that contain a $q^{text{th}}$ power modulo almost every prime is no larger than $m_{q,k} frac{N^{Nk}}{q^N}$ for some positive constant $m_{q,k}$.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134891449","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":"Remarks on rank one Drinfeld modules and their torsion elements","authors":"El Kati Mohamed, Oukhaba Hassan","doi":"10.7169/facm/1956","DOIUrl":"https://doi.org/10.7169/facm/1956","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43836457","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 study the average behaviour of the coefficients of triple product L-functions and some related L-functions corresponding to normalized primitive holomorphic cusp form f ( z ) of weight k for the full modular group SL (2 , Z ) . Here we call f ( z ) a primitive cusp form if it is an eighenfunction of all Hecke operators simultane-ously.
{"title":"On the average behavior of coefficients related to triple product $L$-functions","authors":"K. Venkatasubbareddy, S. Ayyadurai","doi":"10.7169/facm/2046","DOIUrl":"https://doi.org/10.7169/facm/2046","url":null,"abstract":"In this paper, we study the average behaviour of the coefficients of triple product L-functions and some related L-functions corresponding to normalized primitive holomorphic cusp form f ( z ) of weight k for the full modular group SL (2 , Z ) . Here we call f ( z ) a primitive cusp form if it is an eighenfunction of all Hecke operators simultane-ously.","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":"48131466","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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