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":"20 1","pages":"0"},"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":"119 1","pages":"0"},"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":" ","pages":""},"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":"1 1","pages":""},"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":" ","pages":""},"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":"46 1","pages":"0"},"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":" ","pages":""},"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":" ","pages":""},"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}
In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function $pi(x)$ and the Chebyshev $vartheta$-function. Some of these estimates depend on the correctness of the Riemann hypothesis on the nontrivial zeros of the Riemann zeta function $zeta(s)$.
{"title":"New estimates for some integrals of functions defined over primes","authors":"Christian Axler","doi":"10.7169/facm/2049","DOIUrl":"https://doi.org/10.7169/facm/2049","url":null,"abstract":"In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function $pi(x)$ and the Chebyshev $vartheta$-function. Some of these estimates depend on the correctness of the Riemann hypothesis on the nontrivial zeros of the Riemann zeta function $zeta(s)$.","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":"47401634","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":"Multiplication on Besov and Triebel-Lizorkin spaces of power weights","authors":"Hamza Brahim Boulares, D. Drihem","doi":"10.7169/facm/1991","DOIUrl":"https://doi.org/10.7169/facm/1991","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":"45707060","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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