Let ${A'_n}$ be the Ap'ery numbers given by $A'_n=sum_{k=0}^nbinom nk^2binom{n+k}k.$ For any prime $pequiv 3pmod 4$ we show that $A'_{frac{p-1}2}equiv frac{p^2}3binom{frac{p-3}2}{frac{p-3}4}^{-2}pmod {p^3}$. Let ${t_n}$ be given by $$t_0=1, t_1=5quadhbox{and}quad t_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1)^2t_{n-1} (nge 1).$$ We also obtain the congruences for $t_ppmod {p^3}, t_{p-1}pmod {p^2}$ and $t_{frac{p-1}2}pmod {p^2}$, where $p$ is an odd prime.
{"title":"Congruences for the Apéry numbers modulo $p^3$","authors":"Zhi-Hong Sun","doi":"arxiv-2409.06544","DOIUrl":"https://doi.org/arxiv-2409.06544","url":null,"abstract":"Let ${A'_n}$ be the Ap'ery numbers given by $A'_n=sum_{k=0}^nbinom\u0000nk^2binom{n+k}k.$ For any prime $pequiv 3pmod 4$ we show that\u0000$A'_{frac{p-1}2}equiv frac{p^2}3binom{frac{p-3}2}{frac{p-3}4}^{-2}pmod\u0000{p^3}$. Let ${t_n}$ be given by $$t_0=1, t_1=5quadhbox{and}quad\u0000t_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1)^2t_{n-1} (nge 1).$$ We also obtain the\u0000congruences for $t_ppmod {p^3}, t_{p-1}pmod {p^2}$ and $t_{frac{p-1}2}pmod\u0000{p^2}$, where $p$ is an odd prime.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203830","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 obtain a mean value theorem for a general Dirichlet series $f(s)= sum_{j=1}^infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = sum_{n_{j}le x}a_{j}$ satisfies $A(x)=rho x + O(x^beta)$ for some $rho>0$ and $beta<1$. We prove that $frac1Tint_0^T |f(sigma+it)|^2, dt to sum_{j=1}^infty a_j^2n_j^{-2sigma}$ for $sigma>frac{1+beta}{2}$ and obtain an upper bound for this moment for $beta
{"title":"A Mean Value Theorem for general Dirichlet Series","authors":"Frederik Broucke, Titus Hilberdink","doi":"arxiv-2409.06301","DOIUrl":"https://doi.org/arxiv-2409.06301","url":null,"abstract":"In this paper we obtain a mean value theorem for a general Dirichlet series\u0000$f(s)= sum_{j=1}^infty a_j n_j^{-s}$ with positive coefficients for which the\u0000counting function $A(x) = sum_{n_{j}le x}a_{j}$ satisfies $A(x)=rho x +\u0000O(x^beta)$ for some $rho>0$ and $beta<1$. We prove that $frac1Tint_0^T\u0000|f(sigma+it)|^2, dt to sum_{j=1}^infty a_j^2n_j^{-2sigma}$ for\u0000$sigma>frac{1+beta}{2}$ and obtain an upper bound for this moment for\u0000$beta<sigmale frac{1+beta}{2}$. We provide a number of examples indicating\u0000the sharpness of our results.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203828","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}
Rahul Kumar, Paul Levrie, Jean-Christophe Pain, Victor Scharaschkin
We propose a relation between values of the Riemann zeta function $zeta$ and a family of integrals. This results in an integral representation for $zeta(2p)$, where $p$ is a positive integer, and an expression of $zeta(2p+1)$ involving one of the above mentioned integrals together with a harmonic-number sum. Simplification of the latter eventually leads to an integral representation of $zeta(2p + 1)$.
{"title":"A family of integrals related to values of the Riemann zeta function","authors":"Rahul Kumar, Paul Levrie, Jean-Christophe Pain, Victor Scharaschkin","doi":"arxiv-2409.06546","DOIUrl":"https://doi.org/arxiv-2409.06546","url":null,"abstract":"We propose a relation between values of the Riemann zeta function $zeta$ and\u0000a family of integrals. This results in an integral representation for\u0000$zeta(2p)$, where $p$ is a positive integer, and an expression of\u0000$zeta(2p+1)$ involving one of the above mentioned integrals together with a\u0000harmonic-number sum. Simplification of the latter eventually leads to an\u0000integral representation of $zeta(2p + 1)$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203826","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 give rigorous proofs and generalizations of partial fraction expansions for string amplitudes that were recently discovered by Saha and Sinha.
我们给出了萨哈和辛哈最近发现的弦振幅部分分数展开的严格证明和概括。
{"title":"String theory amplitudes and partial fractions","authors":"Hjalmar Rosengren","doi":"arxiv-2409.06658","DOIUrl":"https://doi.org/arxiv-2409.06658","url":null,"abstract":"We give rigorous proofs and generalizations of partial fraction expansions\u0000for string amplitudes that were recently discovered by Saha and Sinha.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203834","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 that for a dominant rational self-map $f$ on a quasi-projective variety defined over $overline{mathbb{Q}}$, there is a point whose $f$-orbit is well-defined and its arithmetic degree is arbitrary close to the first dynamical degree of $f$. As an application, we prove that Zariski dense orbit conjecture holds for a birational map defined over $overline{mathbb{Q}}$ such that the first dynamical degree is strictly larger than the third dynamical degree. In particular, the conjecture holds for birational maps on threefolds with first dynamical degree larger than $1$.
{"title":"Arithmetic degree and its application to Zariski dense orbit conjecture","authors":"Yohsuke Matsuzawa, Junyi Xie","doi":"arxiv-2409.06160","DOIUrl":"https://doi.org/arxiv-2409.06160","url":null,"abstract":"We prove that for a dominant rational self-map $f$ on a quasi-projective\u0000variety defined over $overline{mathbb{Q}}$, there is a point whose $f$-orbit\u0000is well-defined and its arithmetic degree is arbitrary close to the first\u0000dynamical degree of $f$. As an application, we prove that Zariski dense orbit\u0000conjecture holds for a birational map defined over $overline{mathbb{Q}}$ such\u0000that the first dynamical degree is strictly larger than the third dynamical\u0000degree. In particular, the conjecture holds for birational maps on threefolds\u0000with first dynamical degree larger than $1$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"109 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203836","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 a formula for the proper class number of a binary quadratic polynomial assuming that the conductor ideal is sufficiently divisible at dyadic places. This allows us to study the growth of the proper class numbers of totally positive binary quadratic polynomials. As an application, we prove finiteness results on totally positive binary quadratic polynomials with a fixed quadratic part and a fixed proper class number.
{"title":"Class numbers of binary quadratic polynomials","authors":"Zichen Yang","doi":"arxiv-2409.06244","DOIUrl":"https://doi.org/arxiv-2409.06244","url":null,"abstract":"In this paper, we give a formula for the proper class number of a binary\u0000quadratic polynomial assuming that the conductor ideal is sufficiently\u0000divisible at dyadic places. This allows us to study the growth of the proper\u0000class numbers of totally positive binary quadratic polynomials. As an\u0000application, we prove finiteness results on totally positive binary quadratic\u0000polynomials with a fixed quadratic part and a fixed proper class number.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203829","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 know that any finite abelian group $G$ appears as a subgroup of infinitely many multiplicative groups $mathbb{Z}_n^times$ (the abelian groups of size $phi(n)$ that are the multiplicative groups of units in the rings $mathbb{Z}/nmathbb{Z}$). It seems to be less well appeciated that $G$ appears as a subgroup of almost all multiplicative groups $mathbb{Z}_n^times$. We exhibit an asymptotic formula for the counting function of those integers whose multiplicative group fails to contain a copy of $G$, for all finite abelian groups $G$ (other than the trivial one-element group).
{"title":"Multiplicative groups avoiding a fixed group","authors":"Matthias Hannesson, Greg Martin","doi":"arxiv-2409.06869","DOIUrl":"https://doi.org/arxiv-2409.06869","url":null,"abstract":"We know that any finite abelian group $G$ appears as a subgroup of infinitely\u0000many multiplicative groups $mathbb{Z}_n^times$ (the abelian groups of size\u0000$phi(n)$ that are the multiplicative groups of units in the rings\u0000$mathbb{Z}/nmathbb{Z}$). It seems to be less well appeciated that $G$ appears\u0000as a subgroup of almost all multiplicative groups $mathbb{Z}_n^times$. We\u0000exhibit an asymptotic formula for the counting function of those integers whose\u0000multiplicative group fails to contain a copy of $G$, for all finite abelian\u0000groups $G$ (other than the trivial one-element group).","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203822","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 $g$ be sufficiently large, $bin{0,ldots,g-1}$, and $mathcal{S}_b$ be the set of integers with no digit equal to $b$ in their base $g$ expansion. We prove that every sufficiently large odd integer $N$ can be written as $p_1 + p_2 + p_3$ where $p_i$ are prime and $p_iin mathcal{S}_b$.
{"title":"Vinogradov's theorem for primes with restricted digits","authors":"James Leng, Mehtaab Sawhney","doi":"arxiv-2409.06894","DOIUrl":"https://doi.org/arxiv-2409.06894","url":null,"abstract":"Let $g$ be sufficiently large, $bin{0,ldots,g-1}$, and $mathcal{S}_b$ be\u0000the set of integers with no digit equal to $b$ in their base $g$ expansion. We\u0000prove that every sufficiently large odd integer $N$ can be written as $p_1 +\u0000p_2 + p_3$ where $p_i$ are prime and $p_iin mathcal{S}_b$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203821","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 provide an effective upper bound for positive integers with bounded Hamming weights with respect to both a linear recurrence numeration system and an Ostrowski-$alpha$ numeration system, where $alpha$ is a quadratic irrational. We prove a similar result for the representation of an integer in two textit{different} Ostrowski numeration systems.
{"title":"On the representation of an integer in Ostrowski and recurrence numeration systems","authors":"Mohit Mittal, Divyum Sharma","doi":"arxiv-2409.06232","DOIUrl":"https://doi.org/arxiv-2409.06232","url":null,"abstract":"We provide an effective upper bound for positive integers with bounded\u0000Hamming weights with respect to both a linear recurrence numeration system and\u0000an Ostrowski-$alpha$ numeration system, where $alpha$ is a quadratic\u0000irrational. We prove a similar result for the representation of an integer in\u0000two textit{different} Ostrowski numeration systems.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203831","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}
Analogous to Andrews' and Newman's discovery and work on the minimal excludant or "mex" of partitions, we define four new classes of minimal excludants for overpartitions and unearth relations to certain functions due to Ramanujan.
{"title":"On new minimal excludants of overpartitions related to some $q$-series of Ramanujan","authors":"Aritram Dhar, Avi Mukhopadhyay, Rishabh Sarma","doi":"arxiv-2409.06121","DOIUrl":"https://doi.org/arxiv-2409.06121","url":null,"abstract":"Analogous to Andrews' and Newman's discovery and work on the minimal\u0000excludant or \"mex\" of partitions, we define four new classes of minimal\u0000excludants for overpartitions and unearth relations to certain functions due to\u0000Ramanujan.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203832","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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