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Abstract: Abstract We give asymptotics for the number of isomorphism classes of Drinfeld \({\mathbb {F}}_q[T]\) -modules over \({\mathbb {F}}_q(T)\) of a given height, which satisfy prescribed sets of local conditions. This is done by relating our problem to a problem about counting points on weighted projective stacks. Our results for counting points of bounded height on weighted projective stacks over global function fields may be of independent interest. PubDate: 2024-02-13

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Abstract: Abstract Let E be an elliptic curve over \({{\mathbb {Q}}}\) . We conjecture asymptotic estimates for the number of vanishings of \(L(E,1,\chi )\) as \(\chi \) varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on E. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order \(\ell \) characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order \(\ell \) twisted elliptic curve L-functions, with \(\ell \) even and composite, is a mixed family with both unitary and orthogonal aspects. PubDate: 2024-02-01

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Abstract: Abstract We show that the values of elliptic Dedekind sums, after normalization, are equidistributed mod 1. The key ingredient is a non-trivial bound on generalized Selberg–Kloosterman sums for discrete subgroups of \(\textrm{PSL}_2({\mathbb {C}})\) using Poincaré series. PubDate: 2024-01-31

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Abstract: Abstract Let \(\beta \) be a Salem number with \(\beta >1\) and x be an element in \({\mathbb {Q}}(\beta ) \cap [0,1)\) . K. Schmidt conjectured that any element in \({\mathbb {Q}}(\beta ) \cap [0,1)\) has a periodic greedy expansion in base \(\beta \) . In this note, we prove that if x has a non-periodic greedy expansion in base \(\beta \) then the sequence of the greedy expansion of x has higher complexity more than polynomial complexity. PubDate: 2024-01-29

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Abstract: Abstract A divisibility sequence is a sequence of integers \(\{d_n\}\) such that \(d_m\) divides \(d_n\) if m divides n. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca–Hsia–Tucker and Silverman for elliptic curves and prove an analogue of Silverman’s conjecture over function fields for abelian and split semiabelian varieties and some generalizations of this result. We employ tools coming from the theory of unlikely intersections as well as properties of the so-called Betti map associated to a section of an abelian scheme. PubDate: 2024-01-23

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Abstract: Abstract Let \(A_k (n)\) be the number of \(a < n\) coprime to n for which the equation \(a/n = 1/m_1 + \cdots + 1/m_k\) has a solution. We show that for a fixed \(k \ge 3\) , the sum of \(A_k (n)\) over all \(n \le x\) is \(\gg x(\log x)^{2k - 1}\) . In the \(k = 3\) case, we find a new upper bound for this sum as well as a new upper bound for individual values of \(A_3 (n)\) . PubDate: 2024-01-22

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Abstract: Abstract We study a sequence of constants known as the Bendersky-Adamchik constants which appear quite naturally in number theory and generalize the classical Glaisher-Kinkelin constant. Our main initial purpose is to elucidate the close relation between the logarithm of these constants and the Ramanujan summation of certain divergent series. In addition, we also present a remarkable, and previously unknown, expansion of the logarithm of these constants in convergent series involving the Bernoulli numbers of the second kind. PubDate: 2024-01-21

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Abstract: Abstract We state and prove a Gras-type equality relating a certain module generated by Rubin–Stark units and the ray class number. We use for this purpose the notion of generalized index. PubDate: 2024-01-19

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Abstract: Abstract We show that the Generalized Riemann Hypothesis for Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of \(\zeta (s)\) satisfy those properties under GRH. PubDate: 2024-01-18

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Abstract: Abstract We investigate the compatibility of Arthur’s conjecture for R-groups in the restriction of discrete series representations from Levi subgroups of a p-adic group to those of its closed subgroup having the same derived group. The relation is conditional as it holds under the conjectural local Langlands correspondence in general. This work provides a uniform way to verify Arthur’s conjecture for R-groups in the setting. It can be also applied to several cases where those assumptions are already established. PubDate: 2024-01-18

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Abstract: Abstract We prove that the Riemann zeta-function \(\zeta (\sigma + it)\) has no zeros in the region \(\sigma \ge 1 - 1/(55.241(\log t )^{2/3} (\log \log t )^{1/3})\) for \( t \ge 3\) . In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region \(\sigma \ge 1 - 1/(5.558691\log t )\) for \( t \ge 2\) . We also provide new bounds that are useful for intermediate values of \( t \) . Combined, our results improve the largest known zero-free region within the critical strip for \(3\cdot 10^{12} \le t \le \exp (64.1)\) and \( t \ge \exp (1000)\) . PubDate: 2024-01-12

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Abstract: Abstract Andrews introduced the partition function \({\overline{C}}_{k, i}(n)\) , called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i\pmod {k}\) may be overlined. We study the parity and distribution results for \({\overline{C}}_{k,i}(n),\) where \(k>3\) and \(1\le i \le \left\lfloor \frac{k}{2}\right\rfloor \) . More particularly, we prove that for each integer \(\ell \ge 2\) depending on k and i, the interval \(\left[ \ell , \frac{\ell (3\ell +1)}{2}\right] \) \(\Big (\) resp. \(\left[ 2\ell -1, \frac{\ell (3\ell -1)}{2}\right] \Big )\) contains an integer n such that \({\overline{C}}_{k,i}(n)\) is even (resp. odd). Finally we study the distribution for \({\overline{C}}_{p,1}(n)\) where \(p\ge 5\) is a prime number. PubDate: 2024-01-04

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Abstract: Abstract Let \(z\in \mathbb {C}\) . We consider the following product-to-sum representation: $$\begin{aligned} \prod _{m=1}^{\infty }(1-q^m)^{-z}=\sum _{n=0}^{\infty }P_n(z)q^n. \end{aligned}$$ The term \(P_n(z)\) introduced by D’Arcais was found to be a polynomial in z of degree n. In this article, we revisit the Kostant’s representation for the coefficients of the polynomial \(P_n(z)\) . This polynomial representation of \(P_n(z)\) helps in obtaining some congruence properties when \(z\in \mathbb Z\) . Moreover, for any given n, we obtain a set of integers (possibly of small size) that contains the integral roots of \(P_n(z)\) . PubDate: 2023-12-29

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Abstract: Abstract We use Rankin–Cohen brackets on \(\textrm{O}(n, 2)\) to prove that the Fourier coefficients of reflective Borcherds products often satisfy congruences modulo certain primes. PubDate: 2023-12-20

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Abstract: Abstract In this paper, we prove that the number of unimodal sequences of size n is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on false theta functions. Log-concavity and higher Turán inequalities have been well-studied for (restricted) partitions and coefficients of weakly holomorphic modular forms, and analytic proofs generally require precise asymptotic series with error term. In this paper, we proceed from the exact formula for unimodal sequences to carry out this calculation. We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects. PubDate: 2023-12-19

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Abstract: Abstract Let C be a curve defined over a number field k. We say a closed point \(x\in C\) of degree d is isolated if it does not belong to an infinite family of degree d points parametrized by the projective line or a positive rank abelian subvariety of the curve’s Jacobian. Building on work of Bourdon et al. (Adv Math 357(33):106824, 2019), we characterize elliptic curves with rational j-invariant which give rise to an isolated point of odd degree on \(X_1(N)/{\mathbb {Q}}\) for some positive integer N. PubDate: 2023-12-17

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Abstract: Abstract We complete the recent research by Akiyama and Kaneko on the higher-order derivative values \(\Phi _n^{(k)}(1)\) of the cyclotomic polynomials. This article focuses on Lehmer’s explicit formula of \(\Phi _n^{(k)}(1)/\Phi _n(1)\) as a polynomial of the Euler and Jordan totient functions over \(\mathbb {Q}\) . Then we prove Akiyama–Kaneko’s conjecture that the polynomials have a specific simple factor. PubDate: 2023-12-13

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Abstract: Abstract Let p be a prime number. Continuing and extending our previous paper with the same title, we prove explicit rates of overconvergence for modular functions of the form \(\frac{E^*_k}{V(E^*_k)}\) where \(E^*_k\) is a classical, normalized Eisenstein series on \(\Gamma _0(p)\) and V the p-adic Frobenius operator. In particular, we extend our previous paper to the primes 2 and 3. For these primes our main theorem improves somewhat upon earlier results by Emerton, Buzzard and Kilford, and Roe. We include a detailed discussion of those earlier results as seen from our perspective. We also give some improvements to our earlier paper for primes \(p\ge 5\) . Apart from establishing these improvements, our main purpose here is also to show that all of these results can be obtained by a uniform method, i.e., a method where the main points in the argumentation is the same for all primes. We illustrate the results by some numerical examples. PubDate: 2023-12-13

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Abstract: Abstract In this paper, we provide a new framework for studying continued fractions (CFs) by means of the backwards continued fraction (BCF). We develop an approximation theory for BCFs based on taking expansions of a fixed length, show the correspondence between continued fractions and their BCFs counterpart, and illustrate a rich approximation theory for continued fractions based off the methods of the approximation theory for the backwards case. In particular, we construct explicit functions that are sharp bounds for the BCF or CF error infinitely often over any BCF or CF cylinder set, and work out the details to pass seamlessly between the BCF and CF expansion of any real number. PubDate: 2023-11-22