Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract For a prime number p, let \(A_3(p)= \{ m \in \mathbb {N}: \exists m_1,m_2,m_3 \in \mathbb {N}, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \} \) . In 2019 Luca and Pappalardi proved that \(x (\log x)^3 \ll \sum _{p \le x} A_{3}(p) \ll x (\log x)^5\) . We improve the upper bound, showing \(\sum _{p \le x} A_{3}(p) \ll x (\log x)^3 (\log \log x)^2\) . PubDate: 2022-06-26
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract Let \(f(x,y)=1+\sum \limits _{\begin{array}{c} p=1\\ m+n=p \end{array}}^{\infty }a_{m,n}x^my^n\) be a formal power series. We convert f(x, y) into the formal product \(\prod \limits _{\begin{array}{c} p=1\\ m+n=p \end{array}}^{\infty }(1+g_{m,n}x^m y^n)\) , namely the power product expansion in two independent variables. By developing new machinery involving the majorizing infinite product, we provide estimates on the domain of absolute convergence of the infinite product via the Taylor series coefficients of f(x, y). This machinery introduces a myriad of “mixed expansions”, uncovers various algebraic connections between the \((a_{m,n})\) and the \((g_{m,n})\) , and leads to the identification of the domain of absolute convergence of the power product as the Cartesian product of polydiscs associated with a quadratic equation. This makes it possible to use the truncated power product expansions \(\prod \limits _{\begin{array}{c} p=1\\ m+n=p \end{array}}^{P}(1+g_{m,n}x^my^n)\) as approximations to the analytic function f(x, y). We derive an asymptotic formula for the \(g_{m,n}\) , with m fixed, associated with the majorizing infinite product. We also discuss various combinatorial interpretations provided by these power product expansions and derive an additional theorem which shows that with \(g_{m,n}\ge 0\) , the polydisc of absolute convergence for the power product is identical to that of its Taylor series. PubDate: 2022-06-23
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract Fueter and Polya proved that the only quadratic polynomials giving a bijection between N and N \(^{2}\) are the two Cantor polynomials. It is conjectured that there is no bijection from N \(^2\) onto N given by a polynomial of degree at least 3. A similar problem arises when the domain of the map is replaced by the set of integral points in some sector in R \(^2\) . Rational sectors were considered by Nathanson and Stanton. Here, we study and solve the case of general irrational sectors. In fact, our method enables us also to recover the results on rational sectors and also answer a question posed by Nathanson. PubDate: 2022-06-19
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract Let \(\varvec{F}_q\) be the finite field of q elements, where \(q=p^r\) is a power of the prime p, and \(\left( \beta _1, \beta _2, \dots , \beta _r \right) \) be an ordered basis of \(\varvec{F}_q\) over \(\varvec{F}_p\) . For $$\begin{aligned} \xi =\sum _{i=1}^rx_i\beta _i, \quad x_i\in \varvec{F}_p, \end{aligned}$$ we define the Thue–Morse or sum-of-digits function \(T(\xi )\) on \(\varvec{F}_q\) by $$\begin{aligned} T(\xi )=\sum _{i=1}^{r}x_i. \end{aligned}$$ For a given pattern length s with \(1\le s\le q\) , a vector \(\varvec{\alpha }=(\alpha _1,\ldots ,\alpha _s)\in \varvec{F}_q^s\) with different coordinates \(\alpha _{j_1}\not = \alpha _{j_2}\) , \(1\le j_1<j_2\le s\) , a polynomial \(f(X)\in \varvec{F}_q[X]\) of degree d and a vector \(\mathbf{c} =(c_1,\ldots ,c_s)\in \varvec{F}_p^s\) we put $$\begin{aligned} \mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)=\{\xi \in \varvec{F}_q : T(f(\xi +\alpha _i))=c_i,~i=1,\ldots ,s\}. \end{aligned}$$ In this paper we will see that under some natural conditions, the size of \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)\) is asymptotically the same for all \(\mathbf{c} \) and \(\varvec{\alpha }\) in both cases, \(p\rightarrow \infty \) and \(r\rightarrow \infty \) , respectively. More precisely, we have $$\begin{aligned} \left \mathcal{T}(\mathbf{c} , \varvec{\alpha }, f) - p^{r-s} \right \le (d-1)q^{1/2} \end{aligned}$$ under certain conditions on d, q and s. For monomials of large degree we improve this bound as well as we find conditions on d, q and s for which this bound is not true. In particular, if \(1\le d<p\) we have the dichotomy that the bound is valid if \(s\le d\) and for \(s\ge d+1\) there are vectors \(\mathbf{c} \) and \(\varvec{\alpha }\) with PubDate: 2022-06-12
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract The q-rational numbers and the q-irrational numbers were introduced by Morier-Genoud and Ovsienko. In this paper, we focus on q-real quadratic irrational numbers, especially q-metallic numbers and q-rational sequences which converge to q-metallic numbers, and consider the radiuses of convergence of them when we assume that q is a complex number. We construct two sequences given by recurrence formula as a generalization of the q-deformation of the Fibonacci and Pell numbers which are introduced by Morier-Genoud and Ovsienko. For these two sequences, we prove a conjecture of Leclere, Morier-Genoud, Ovsienko and Veselov concerning the expected lower bound of the radiuses of convergence. In addition, we obtain a relationship between the radius of convergence of these two sequences in two special cases. PubDate: 2022-06-09
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract We construct a family of harmonic Maass forms of polynomial growth of any level corresponding to any cusp whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil’s converse theorem. PubDate: 2022-05-25
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract We investigate a special sequence of random variables A(N) defined by an exponential power series with independent standard complex Gaussians \((X(k))_{k \ge 1}\) . Introduced by Hughes, Keating, and O’Connell in the study of random matrix theory, this sequence relates to Gaussian multiplicative chaos (in particular “holomorphic multiplicative chaos” per Najnudel, Paquette, and Simm) and random multiplicative functions. Soundararajan and Zaman recently determined the order of \(\mathbb {E}[ A(N) ]\) . By constructing an algorithm to calculate A(N) in \(O(N^2 \log N)\) steps, we produce computational evidence that their result can likely be strengthened to an asymptotic result with a numerical estimate for the asymptotic constant. We also obtain similar conclusions when A(N) is defined using standard real Gaussians or uniform \(\pm 1\) random variables. However, our evidence suggests that the asymptotic constants do not possess a natural product structure. PubDate: 2022-05-23
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract L. Moret-Bailly constructed families \({\mathfrak {C}}\rightarrow {\mathbb {P}}^1\) of genus 2 curves with supersingular Jacobian. In this paper we first classify the reducible fibers of a Moret-Bailly family using linear algebra over a quaternion algebra. The main result is an algorithm that exploits properties of two reducible fibers to compute a hyperelliptic model for any irreducible fiber of a Moret-Bailly family. PubDate: 2022-05-18
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract We study the structure of combinatorial Burnside groups, which receive equivariant birational invariants of actions of finite groups on algebraic varieties. PubDate: 2022-05-18
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract We classify the coefficients \((a_1, \ldots , a_n) \in {\mathbb {F}}_q[t]^n\) that appear in a linear relation \(\sum _{i=1}^n a_i \gamma _i =0\) among Galois conjugates \(\gamma _i \in \overline{{\mathbb {F}}_q(t)}\) . We call such an n-tuple a Smyth tuple. Our main theorem gives an affirmative answer to a function field analogue of a 1986 conjecture of Smyth (J Numb Theory 23, 243–254, 1986) over \({\mathbb {Q}}\) . Smyth showed that certain local conditions on the \(a_i\) are necessary and conjectured that they are sufficient. Our main result is that the analogous conditions are necessary and sufficient over \({\mathbb {F}}_q(t)\) , which we show using a combinatorial characterization of Smyth tuples from Smyth (J Numb Theory 23, 243–254, 1986). We also formulate a generalization of Smyth’s Conjecture in an arbitrary number field that is not a straightforward generalization of the conjecture over \({\mathbb {Q}}\) due to a subtlety occurring at the archimedean places. PubDate: 2022-05-18
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract The aim of this paper is to give a criterion of algebraic independence for the values at the same point of two modular functions under certain conditions. As an application, we show that any two infinite products in $$\begin{aligned} \prod _{n=1}^{\infty }\left( 1+\frac{1}{F_n} \right) ,\quad \prod _{n=3}^{\infty }\left( 1-\frac{1}{F_n} \right) ,\quad \prod _{n=1}^{\infty }\left( 1+\frac{1}{L_n} \right) ,\quad \prod _{n=2}^{\infty }\left( 1-\frac{1}{L_n} \right) \end{aligned}$$ are algebraically independent over \(\mathbb {Q}\) , where \(\{F_n\}\) and \( \{L_n\} \) are the Fibonacci and Lucas sequences, respectively. The proof of our main theorem is based on the properties of the field of all modular functions for the principal congruence subgroup, together with a deep result of Yu. V. Nesterenko on algebraic independence of the values of the Eisenstein series. PubDate: 2022-05-10
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract We study the zeta Mahler function (ZMF, also zeta Mahler measure), which is closely related to the Mahler measure. Here we discuss a family of ZMFs attached to the Laurent polynomials \(k + (x_1 + x_1^{-1}) \cdots \left( x_r + x_r^{-1}\right) \) , where k is real. We give explicit formulae, present examples and establish properties for these ZMFs, such as an RH-type phenomenon. Further, we explore connections with the Mahler measure. PubDate: 2022-04-24
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract For an algebraic number field K with ring of integers \(\mathcal {O}_{K}\) , an important subgroup of the ideal class group \(Cl_{K}\) is the Pólya group, denoted by Po(K), which measures the failure of the \(\mathcal {O}_{K}\) -module \(Int(\mathcal {O}_{K})\) of integer-valued polynomials on \(\mathcal {O}_{K}\) from admitting a regular basis. In this paper, we prove that for any integer \(n \ge 2\) , there are infinitely many totally real bi-quadratic fields K with \(Po(K) \simeq ({\mathbb {Z}}/2{\mathbb {Z}})^{n}\) . In fact, we explicitly construct such an infinite family of number fields. This also provides an infinite family of bi-quadratic fields with ideal class groups of 2-ranks at least n. PubDate: 2022-04-24
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract We study the splitting of primes in number fields generated by points on modular curves. Momose (Nagoya Math J 96:139–165, 1984) was the first to notice that quadratic points on \(X_1(N)\) generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm (Quadratic Points on Modular Curves. PhD thesis, University of Georgia, 2013). We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves \(X_0(N)\) which are hyperelliptic (except for \(N=37\) ) and in cubic fields generated by points on \(X_1(2,14)\) . PubDate: 2022-04-21
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract Let K be a discretely valued field with ring of integers \(\mathcal {O}_K\) with perfect residue field. Let K(x) be the rational function field in one variable. Let \({\mathbb {P}}^1_{\mathcal {O}_K}\) be the standard smooth model of \({\mathbb {P}}^1_K\) with coordinate x. Let \(f(x) \in \mathcal {O}_K[x]\) be a squarefree polynomial with corresponding divisor of zeroes \({{\,\mathrm{div}\,}}_0(f)\) on \({\mathbb {P}}^1_{\mathcal {O}_K}\) . We give an explicit description of the minimal embedded resolution \(\mathcal {Y}\) of the pair \(({\mathbb {P}}^1_{\mathcal {O}_K}, {{\,\mathrm{div}\,}}_0(f))\) by using Mac Lane’s theory to write down the discrete valuations on K(x) corresponding to the irreducible components of the special fiber of \(\mathcal {Y}\) . PubDate: 2022-04-13
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract Shifted convolution sums play a prominent rôle in analytic number theory. Here these sums are considered in the context of holomorphic Hecke eigenforms. We investigate pointwise bounds, mean-square bounds consistent with the optimal conjectural bound, and find asymptotics on average for their variance. PubDate: 2022-04-08
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of the Boyd–Lawton formula, and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker’s lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer’s question is true. PubDate: 2022-03-31
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract We construct families of Bailey pairs \((\alpha _n,\beta _n)\) where the exponent of q in \(\alpha _n\) is an indefinite quadratic form, but where the usual \((-1)^j\) is replaced by a sign function. This leads to identities involving “false” indefinite binary theta series. These closely resemble q-identities for mock theta functions or Maass waveforms, but the sign function prevents them from having the usual modular properties. PubDate: 2022-03-23
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: Abstract We study the elliptic curves given by \(y^2 = x^3 + b x +t^{3^n+1}\) over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve \(u^3 + b u = v^{3^n + 1}\) . In this way, using the Néron–Tate height on the Mordell–Weil group, we obtain lattices in dimension \(2 \cdot 3^n\) for every \(n \ge 1\) , which improve on the currently best known sphere packing densities in dimensions 162 (case \(n=4\) ) and 486 (case \(n=5\) ). For \(n=3\) , the construction has the same packing density as the best currently known sphere packing in dimension 54, and for \(n=1\) it has the same density as the lattice \(E_6\) in dimension 6. PubDate: 2022-03-17
Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.
Abstract: A Correction to this paper has been published: https://doi.org/10.1007/s40993-021-00307-4 PubDate: 2022-03-07
Hybrid journal (It can contain Open Access articles)
[228 journals]
