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  • A proof of Sarnak's golden mean conjecture
    J. Number Theory (IF 0.684) Pub Date : 2020-03-31
    C.J. Mozzochi

    For an irrational number θ, let0=a0

    更新日期:2020-03-31
  • Corrigendum to “A uniqueness property of general Dirichlet series” [J. Number Theory 206 (2020) 123–137]
    J. Number Theory (IF 0.684) Pub Date : 2020-03-31
    Anup B. Dixit

    This corrigendum introduces an extra condition in the statement of Theorem 1.2 of the paper “A uniqueness property of general Dirichlet series”, published in the Journal of Number Theory, vol. 206, January 2020, 123–137. This condition is necessary in the proof of the theorem.

    更新日期:2020-03-31
  • Expanding phenomena over higher dimensional matrix rings
    J. Number Theory (IF 0.684) Pub Date : 2020-03-30
    Nguyen Van The; Le Anh Vinh

    In this paper, we study the expanding phenomena in the setting of higher dimensional matrix rings. More precisely, we obtain a sum-product estimate for large subsets and show that x(y+z), x+yz, xy+z+t are moderate expanders over the matrix ring Mn(Fq). These results generalize recent results of Y. D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author.

    更新日期:2020-03-30
  • New estimates for exponential sums over multiplicative subgroups and intervals in prime fields
    J. Number Theory (IF 0.684) Pub Date : 2020-03-27
    Daniel Di Benedetto; Moubariz Z. Garaev; Victor C. Garcia; Diego Gonzalez-Sanchez; Igor E. Shparlinski; Carlos A. Trujillo

    Let H be a multiplicative subgroup of Fp⁎ of order H>p1/4. We show thatmax(a,p)=1⁡|∑x∈Hep(ax)|≤H1−31/2880+o(1), where ep(z)=exp⁡(2πiz/p), which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product nx with x∈H and n∈N for a short interval N of consecutive integers.

    更新日期:2020-03-27
  • Endomorphism rings of reductions of Drinfeld modules
    J. Number Theory (IF 0.684) Pub Date : 2019-03-22
    Sumita Garai; Mihran Papikian

    Let A=Fq[T] be the polynomial ring over Fq, and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r≥2 over F. For all but finitely many primes p◁A, one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ⊗Fp of rank r over Fp=A/p. The endomorphism ring Ep=EndFp(ϕ⊗Fp) is an order in an imaginary field extension K of F of degree r. Let Op be the integral closure of A in K, and let

    更新日期:2020-03-27
  • A basis for the space of weakly holomorphic Drinfeld modular forms for GL2(A)
    J. Number Theory (IF 0.684) Pub Date : 2019-04-17
    SoYoung Choi

    We construct a canonical basis for the space of weakly holomorphic Drinfeld modular forms. And we find that the basis elements satisfy a generating function and the duality among coefficients of the basis elements. Moreover we obtain the congruence properties of t-expansion coefficients of these functions under some conditions.

    更新日期:2020-03-27
  • Twisted divided powers and applications
    J. Number Theory (IF 0.684) Pub Date : 2019-03-20
    Michel Gros; Bernard Le Stum; Adolfo Quirós

    In order to give a formal treatment of differential equations in positive characteristic p, it is necessary to use divided powers. One runs into an analog problem in the theory of q-difference equations when q is a pth root of unity. We introduce here a notion of twisted divided powers (relative to q) and show that one can recover the twisted Weyl algebra and obtain a twisted p-curvature map that describes

    更新日期:2020-03-27
  • Fitting ideals of class groups in Carlitz–Hayes cyclotomic extensions
    J. Number Theory (IF 0.684) Pub Date : 2019-01-18
    Andrea Bandini; Francesc Bars; Edoardo Coscelli

    We generalize some results of Greither and Popescu to a geometric Galois cover X→Y which appears naturally for example in extensions generated by pn-torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz–Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula

    更新日期:2020-03-27
  • On Drinfeld modular forms of higher rank II
    J. Number Theory (IF 0.684) Pub Date : 2019-01-04
    Ernst-Ulrich Gekeler

    We show that the absolute value |f| of an invertible holomor- phic function f on the Drinfeld symmetric space Ωr (r≥2) is constant on fibers of the building map to the Bruhat–Tits building BT. Its logarithm log⁡|f| is an affine map on the realization of BT. These results are used to study the vanishing loci of modular forms (coefficient forms, Eisenstein series, para-Eisenstein series) and to determine

    更新日期:2020-03-27
  • Equivariant special L-values of abelian t-modules
    J. Number Theory (IF 0.684) Pub Date : 2018-12-17
    Jiangxue Fang

    We prove a special value formula for the Goss L-functions associated to abelian t-modules and Galois representations. We also obtain an equivariant version of this formula.

    更新日期:2020-03-27
  • On multi-poly-Bernoulli–Carlitz numbers
    J. Number Theory (IF 0.684) Pub Date : 2018-12-17
    Ryotaro Harada

    We introduce multi-poly-Bernoulli–Carlitz numbers, function field analogues of multi-poly-Bernoulli numbers of Imatomi–Kaneko–Takeda. We explicitly describe multi-poly-Bernoulli Carlitz numbers in terms of the Carlitz factorial and the Stirling–Carlitz numbers of the second kind and also show their relationships with function field analogues of finite multiple zeta values.

    更新日期:2020-03-27
  • Periods of t-modules as special values
    J. Number Theory (IF 0.684) Pub Date : 2018-12-07
    Andreas Maurischat

    In this article we show that all periods of uniformizable t-modules (resp. their coordinates) can be obtained via specializing a rigid analytic trivialization of a related dual t-motive at t=θ. The proof is even constructive. The central object in the construction is a subset H of the Tate algebra points of E which turns out to be isomorphic to the period lattice of E via kind of generating series

    更新日期:2020-03-27
  • On twisted A-harmonic sums and Carlitz finite zeta values
    J. Number Theory (IF 0.684) Pub Date : 2018-12-07
    F. Pellarin; R. Perkins

    Twisted A-harmonic sums are partial sums of a class of zeta values introduced by the first author. We prove some new identities for such sums and we deduce properties of analogues of finite zeta values in the framework of the Carlitz module. In the theory of finite multiple zeta values as introduced by Kaneko and Zagier, finite zeta values are all zero and there is no known non-zero finite multiple

    更新日期:2020-03-27
  • Discriminant-stability in p-adic Lie towers of number fields
    J. Number Theory (IF 0.684) Pub Date : 2018-09-19
    James Upton

    In this paper we consider a tower of number fields ⋯⊇K(1)⊇K(0)⊇K arising naturally from a continuous p-adic representation of Gal(Q¯/K), referred to as a p-adic Lie tower over K. A recent conjecture of Daqing Wan hypothesizes, for certain p-adic Lie towers of curves over Fp, a stable (polynomial) growth formula for the genus. Here we prove the analogous result in characteristic zero, namely: the p-adic

    更新日期:2020-03-27
  • An oscillation theorem on the additive representative function over N
    J. Number Theory (IF 0.684) Pub Date : 2020-03-25
    Lixia Dai; Hao Pan

    Let A be an infinite non-empty subset of N. For each n∈N, definerA,A(n):=|{(a,b):a,b∈A,a+b=n}| andRA,A(n):=∑j≤nrA,A(j). We show that if the function RA,A(n) is well-distributed in some sense, then it can't be very well-distributed. Explicitly, if for some constant c>0,limsupn→∞|RA,A(n)−cn|n14<+∞ then for some constant δ>0, the set{n∈N:|RA,A(n)−cn|≥δn14} has a positive lower density. This result implies

    更新日期:2020-03-26
  • Hilbert modular polynomials
    J. Number Theory (IF 0.684) Pub Date : 2020-03-18
    Chloe Martindale

    We present an algorithm to compute a higher dimensional analogue of modular polynomials. This higher dimensional analogue, the ‘set of Hilbert modular polynomials’, concerns cyclic isogenies of principally polarised abelian varieties with maximal real multiplication by a fixed totally real number field K0. In the 2-dimensional case with K0=Q(5) we also provide an implementation together with some optimisations

    更新日期:2020-03-19
  • On integers n for which σ(2n + 1)≥σ(2n)
    J. Number Theory (IF 0.684) Pub Date : 2020-03-18
    Mits Kobayashi; Tim Trudgian

    We show that the natural density of positive integers n for which σ(2n+1)≥σ(2n) is between 0.053 and 0.055.

    更新日期:2020-03-19
  • On a generalization of a conjecture of Grosswald
    J. Number Theory (IF 0.684) Pub Date : 2020-03-17
    Pradipto Banerjee; Ranjan Bera

    We generalize a conjecture of Grosswald, now a theorem due to Filaseta and Trifonov, stating that the Bessel polynomials, denoted by yn(x), have the associated Galois group Sn over the rationals for each n. We consider generalized Bessel polynomials yn,β(x) which contain interesting families of polynomials whose discriminants are nonzero rational squares. We show that the Galois group associated with

    更新日期:2020-03-19
  • On the distribution of Salem numbers
    J. Number Theory (IF 0.684) Pub Date : 2020-03-17
    Friedrich Götze; Anna Gusakova

    In this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals I1,…,Ik⊂[0;π], 1≤k≤m let Salm,k(Q,I1,…,Ik) denote the set of ordered (k+1)-tuples (α0,…,αk) of conjugate algebraic integers, such that α0 is a Salem numbers of degree 2m+2 satisfying α≤Q for some positive real number Q and arg⁡αi∈Ii. We derive the following asymptotic approximation#Salm,k(Q

    更新日期:2020-03-19
  • On the mu and lambda invariants of the logarithmic class group
    J. Number Theory (IF 0.684) Pub Date : 2020-03-17
    José-Ibrahim Villanueva-Gutiérrez

    Let ℓ be a rational prime number. Assuming the Gross-Kuz'min conjecture along a Zℓ-extension K∞ of a number field K, we show that there exist integers μ˜, λ˜ and ν˜ such that the exponent e˜n of the order ℓe˜n of the logarithmic class group Cℓ˜n for the n-th layer Kn of K∞ is given by e˜n=μ˜ℓn+λ˜n+ν˜, for n big enough. We show some relations between the classical invariants μ and λ, and their logarithmic

    更新日期:2020-03-19
  • Parity considerations in Rogers–Ramanujan–Gordon type overpartitions
    J. Number Theory (IF 0.684) Pub Date : 2020-03-16
    Doris D.M. Sang; Diane Y.H. Shi; Ae Ja Yee

    In 2010, Andrews investigated a variety of parity questions in the classical partition identities of Euler, Rogers, Ramanujan and Gordon. In particular, he considered the Rogers-Ramanujan-Gordon partitions with some constraints on even and odd parts. At the end of this paper, he left fifteen open questions, of which the eleventh is to extend his parity consideration to overpartitions. The main purpose

    更新日期:2020-03-16
  • On the error term and zeros of the Dedekind zeta function
    J. Number Theory (IF 0.684) Pub Date : 2020-03-16
    Biplab Paul; Ayyadurai Sankaranarayanan

    Let K be a number field of degree k≥8. In this article, we investigate a certain density theorem for zeros of the Dedekind zeta function attached to K. In this context, our theorem strengthens a result of Heath-Brown. Further, we investigate an upper-bound of the error term for an ideal counting function attached to K. When K is a cyclotomic field, we prove a stronger upper-bound for this error term

    更新日期:2020-03-16
  • On quadratic approximation for hyperquadratic continued fractions
    J. Number Theory (IF 0.684) Pub Date : 2020-03-16
    Khalil Ayadi; Tomohiro Ooto

    We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning Diophantine exponents for algebraic Laurent series. As the second application, we determine the degrees of these families in particular case.

    更新日期:2020-03-16
  • Optimal topological generators of U(1)
    J. Number Theory (IF 0.684) Pub Date : 2020-03-16
    Zachary Stier

    Sarnak's golden mean conjecture states that (m+1)dφ(m)⩽1+25 for all integers m⩾1, where φ is the golden mean and dθ is the discrepancy function for m+1 multiples of θ modulo 1. In this paper, we characterize the set S of values θ that share this property, as well as the set T of those with the property for some lower bound m⩾M. Remarkably, Smod1 has only 16 elements, whereas T is the set of GL2(Z)-transformations

    更新日期:2020-03-16
  • Asymptotic lower bound of class numbers along a Galois representation
    J. Number Theory (IF 0.684) Pub Date : 2019-10-15
    Tatsuya Ohshita

    Let T be a free Zp-module of finite rank equipped with a continuous Zp-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a lower bound of the additive p-adic valuation of the class number of Kn, which is the Galois extension field of K fixed by the stabilizer of T/pnT. By applying this result

    更新日期:2020-03-02
  • Arithmetic differential operators with congruence level structures: First results and examples
    J. Number Theory (IF 0.684) Pub Date : 2019-09-23
    Christine Huyghe; Tobias Schmidt; Matthias Strauch

    This paper is a survey on sheaves of arithmetic differential operators with congruence level on formal schemes. We present first results about these sheaves and discuss some examples.

    更新日期:2020-03-02
  • A t-motivic interpretation of shuffle relations for multizeta values
    J. Number Theory (IF 0.684) Pub Date : 2019-06-18
    Wei-Cheng Huang

    Thakur [Tha10] showed that, for r, s∈N, a product of two Carlitz zeta values ζA(r) and ζA(s) can be expressed as an Fp-linear combination of ζA(r+s) and double zeta values of weight r+s. Such an expression is called shuffle relation by Thakur. Fixing r, s∈N, we construct a t-module E′. To determine whether an (r+s)-tuple C in Fq(θ)r+s gives a shuffle relation, we relate it to the Fq[t]-torsion property

    更新日期:2020-03-02
  • On Drinfeld modular forms of higher rank IV: Modular forms with level
    J. Number Theory (IF 0.684) Pub Date : 2019-05-20
    Ernst-Ulrich Gekeler

    We construct and study a natural compactification M‾r(N) of the moduli scheme Mr(N) for rank-r Drinfeld Fq[T]-modules with a structure of level N∈Fq[T]. Namely, M‾r(N)=ProjEis(N), the projective variety associated with the graded ring Eis(N) generated by the Eisenstein series of rank r and level N. We use this to define the ring Mod(N) of all modular forms of rank r and level N. It equals the integral

    更新日期:2020-03-02
  • Tensor powers of rank 1 Drinfeld modules and periods
    J. Number Theory (IF 0.684) Pub Date : 2019-04-18
    Nathan Green

    We study tensor powers of rank 1 sign-normalized Drinfeld A-modules, where A is the coordinate ring of an elliptic curve over a finite field. Using the theory of A-motives, we find explicit formulas for the A-action of these modules. Then, by developing the theory of vector-valued Anderson generating functions, we give formulas for the period lattice of the associated exponential function.

    更新日期:2020-03-02
  • A new proof of a vanishing result due to Berthelot, Esnault, and Rülling
    J. Number Theory (IF 0.684) Pub Date : 2019-04-17
    Veronika Ertl

    The goal of this small note is to give a more concise proof of a result due to Berthelot, Esnault, and Rülling in [4]. For a regular, proper, and flat scheme X over a discrete valuation ring of mixed characteristic (0,p), it relates the vanishing of the cohomology of the structure sheaf of the generic fibre of X with the vanishing of the Witt vector cohomology of its special fibre. We use as a critical

    更新日期:2020-03-02
  • Subspace Theorem for Moving Hypersurfaces and Semi-decomposable Form Inequalities
    J. Number Theory (IF 0.684) Pub Date : 2020-02-27
    Qingchun Ji; Qiming Yan; Guangsheng Yu

    In this paper, a Schmidt's subspace type theorem is given for moving hypersurfaces. As the applications, we give some finiteness criteria for the solutions of the sequence of semi-decomposable form equations and inequalities.

    更新日期:2020-02-28
  • The fourier coefficients of a metaplectic eisenstein distribution on the double cover of SL(3) over Q
    J. Number Theory (IF 0.684) Pub Date : 2020-02-15
    Edmund Karasiewicz

    We compute the Fourier coefficients of a minimal parabolic Eisenstein distribution on the double cover of SL(3) over Q. Two key aspects of the paper are an explicit formula for the constant term, and formulas for the Fourier coefficients at the ramified place p=2. Additionally, the unramified non-degenerate Fourier coefficients of this Eisenstein distribution fit into the combinatorial description

    更新日期:2020-02-20
  • On the Siegel-Weil formula for classical groups over function fields
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Wei Xiong

    We establish a Siegel-Weil formula for classical groups over a function field with odd characteristic, which asserts in many cases that the Siegel Eisenstein series is equal to an integral of a theta function. This is a function-field analogue of the classical result proved by A. Weil in his 1965 Acta Math. paper. We also give a convergence criterion for the theta integral by using Harder's reduction

    更新日期:2020-02-20
  • Recurrence with prescribed number of residues
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Artūras Dubickas; Aivaras Novikas

    In this paper we show that for every positive integer m there exist positive integers x1,x2,M such that the sequence (xn)n=1∞ defined by the Fibonacci recurrence xn+2=xn+1+xn, n=1,2,3,…, has exactly m distinct residues modulo M. As an application we show that for each integer m⩾2 there exists ξ∈R such that the sequence of fractional parts {ξφn}n=1∞, where φ=(1+5)/2, has exactly m limit points. Furthermore

    更新日期:2020-02-20
  • Equidistribution results for sequences of polynomials
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Simon Baker

    Let (fn)n=1∞ be a sequence of polynomials and α>1. In this paper we study the distribution of the sequence (fn(α))n=1∞ modulo one. We give sufficient conditions for a sequence (fn)n=1∞ to ensure that for Lebesgue almost every α>1 the sequence (fn(α))n=1∞ has Poissonian pair correlations. In particular, this result implies that for Lebesgue almost every α>1, for any k≥2 the sequence (αnk)n=1∞ has Poissonian

    更新日期:2020-02-20
  • On the crystal graph description of the stable Weyl group multiple Dirichlet series
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Yuanqing Cai

    For a semisimple Lie algebra admitting a good enumeration, we prove a parametrization for the elements in its Weyl group. As an application, we give a coordinate-free comparison between the crystal graph description (when it is known) and the Lie-theoretic description of the Weyl group multiple Dirichlet series in the stable range.

    更新日期:2020-02-20
  • Variants of Khintchine's theorem in metric Diophantine approximation
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Laima Kaziulytė

    New results towards the Duffin-Schaeffer conjecture, which is a fundamental problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function ψ:N→R we denote by W(ψ) the set of all x∈R such that |nx−a|<ψ(n) for infinitely many a,n. Analogously, denote W′(ψ) if we additionally require a,n to be coprime. Aistleitner et al. [2] proved that W′(ψ) is

    更新日期:2020-02-20
  • Dynamical models for Liouville and obstructions to further progress on sign patterns
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Will Sawin

    We define a class of dynamical systems by modifying a construction due to Tao, which includes certain Furstenburg limits arising from the Liouville function. Most recent progress on the Chowla conjectures and sign patterns of the Möbius and Liouville functions uses methods that apply to any dynamical system in this class. Hence dynamical systems in this class with anomalous local behavior present obstructions

    更新日期:2020-02-20
  • On Andrews' integer partitions with even parts below odd parts
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Chiranjit Ray; Rupam Barman

    Recently, Andrews defined a partition function EO(n) which counts the number of partitions of n in which every even part is less than each odd part. He also defined a partition function EO‾(n) which counts the number of partitions of n enumerated by EO(n) in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties

    更新日期:2020-02-20
  • Romik's conjecture for the Jacobi theta function
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Tanay Wakhare

    Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point i. He then conjectured that the Taylor coefficients d(n) either vanish or are periodic modulo any prime p; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, with the latter trio considering arbitrary half integral weight modular forms. We refine previous

    更新日期:2020-02-20
  • The least primitive root modulo p2
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Bryce Kerr; Kevin J. McGown; Tim Trudgian

    We provide an explicit estimate on the least primitive root mod p2. We show, in particular, that every prime p has a primitive root mod p2 that is less than p0.99.

    更新日期:2020-02-20
  • Generalised Markov numbers
    J. Number Theory (IF 0.684) Pub Date : 2020-02-14
    Oleg Karpenkov; Matty van Son

    In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find corresponding values in the Markov spectrum. Further we construct a counterexample to the generalised Markov uniqueness conjecture. The proposed generalisation is

    更新日期:2020-02-20
  • On a class of Lebesgue-Ljunggren-Nagell type equations
    J. Number Theory (IF 0.684) Pub Date : 2020-02-04
    Andrzej Dąbrowski; Nursena Günhan; Gökhan Soydan

    Text Given odd, coprime integers a, b (a>0), we consider the Diophantine equation ax2+b2l=4yn, x,y∈Z, l∈N, n odd prime, gcd⁡(x,y)=1. We completely solve the above Diophantine equation for a∈{7,11,19,43,67,163}, and b a power of an odd prime, under the conditions 2n−1bl≢±1(mod a) and gcd⁡(n,b)=1. For other square-free integers a>3 and b a power of an odd prime, we prove that the above Diophantine equation

    更新日期:2020-02-04
  • Hilbert modularity of some double octic Calabi–Yau threefolds
    J. Number Theory (IF 0.684) Pub Date : 2019-10-15
    Sławomir Cynk; Matthias Schütt; Duco van Straten

    We exhibit three double octic Calabi–Yau threefolds, a non-rigid threefold defined over Q and two rigid threefolds over the quadratic fields Q[5],Q[−3], and prove their modularity. The non-rigid threefold has two conjugate Hilbert modular forms for the field Q[2] of weight [4,2] and [2,4] attached while the two rigid threefolds correspond to a Hilbert modular form of weight [4,4] and to the twist of

    更新日期:2020-01-29
  • Distribution of real algebraic integers
    J. Number Theory (IF 0.684) Pub Date : 2019-10-15
    Denis V. Koleda

    In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their naïve height tends to infinity. For an arbitrary interval I⊂R and sufficiently large Q>0, we obtain an asymptotic formula for the number of algebraic integers α∈I of fixed degree n and naïve height H(α)≤Q. In particular, we show that the real algebraic integers of degree n, with their height growing

    更新日期:2020-01-29
  • Twin-prime and Goldbach theorems for Z[[x]]
    J. Number Theory (IF 0.684) Pub Date : 2020-01-27
    Elad Paran

    We show that an element f in the ring Z[[x]] of formal power series over the integers is a sum of two irreducible elements in Z[[x]] if and only if the constant term of f is of the form ±pk±ql or of the form ±pk, where p,q are prime numbers and k,l are positive integers. Moreover, if f0 is of such form, then there exist 2ℵ0 pairwise coprime elements g∈Z[[x]] such that both g and g+f are irreducible

    更新日期:2020-01-27
  • Galois theoretic study on simultaneous representation of primes by binary quadratic forms
    J. Number Theory (IF 0.684) Pub Date : 2020-01-24
    Hiroto Horiba; Masanari Kida; Genki Koda

    We study intrinsic Galois structure behind theorems by Kaplansky, Brink, and Mortenson on simultaneous representation of primes by binary quadratic forms with different discriminants. This study brings new theorems like theirs also in indefinite quadratic forms.

    更新日期:2020-01-24
  • Algebraic differential independence regarding the Riemann ζ-function and the Euler Γ-function
    J. Number Theory (IF 0.684) Pub Date : 2020-01-24
    Qi Han; Jingbo Liu

    In this paper, we prove that ζ cannot be a solution to any nontrivial algebraic differential equation whose coefficients are polynomials in Γ,Γ(n) and Γ(ℓn) over the ring of polynomials in C, where ℓ,n≥1 are positive integers.

    更新日期:2020-01-24
  • On completely multiplicative automatic sequences
    J. Number Theory (IF 0.684) Pub Date : 2020-01-21
    Shuo Li

    In this article, we prove that all completely multiplicative automatic sequences (an)n∈N defined on C, vanishing or not, can be written in the form an=bnχn, where (bn)n∈N is an almost constant sequence, and (χn)n∈N is a Dirichlet character.

    更新日期:2020-01-22
  • The distribution of multiples of real points on an elliptic curve
    J. Number Theory (IF 0.684) Pub Date : 2020-01-21
    Alex Cowan

    Given an elliptic curve E and a point P in E(R), we investigate the distribution of the points nP as n varies over the integers, giving bounds on the x and y coordinates of nP and determining the natural density of integers n for which nP lies in an arbitrary open subset of R2. Our proofs rely on a connection to classical topics in the theory of Diophantine approximation.

    更新日期:2020-01-22
  • Heron Triangles and their Elliptic Curves
    J. Number Theory (IF 0.684) Pub Date : 2020-01-21
    Lorenz Halbeisen; Norbert Hungerbühler

    In geometry, a Heron triangle is a triangle with rational side lengths and integral area. We investigate Heron triangles and their elliptic curve. In particular, we provide some new results concerning Heron triangles and give elementary proofs for some results concerning Heronian elliptic curves.

    更新日期:2020-01-22
  • Non-vanishing of special L-values of cusp forms on GL(2) with totally split prime power twists
    J. Number Theory (IF 0.684) Pub Date : 2020-01-21
    Jaesung Kwon; Hae-Sang Sun

    We prove the non-vanishing of special L-values of cuspidal automorphic forms on GL(2) twisted by Hecke characters of prime power orders and totally split prime power conductors. Main ingredients of the proof are estimating the Galois averages of fast convergent series expressions of special L-values and considering Shintani cone decomposition.

    更新日期:2020-01-21
  • The local zeta function in enumerating quartic fields
    J. Number Theory (IF 0.684) Pub Date : 2020-01-21
    Robert D. Hough

    An exact formula is obtained for the Fourier transform of the local condition of maximality modulo primes p>3 in the prehomogeneous vector space 2⊗Sym2(Zp3) parametrizing quartic fields, thus solving the local ‘quartic case’ in enumerating quartic fields.

    更新日期:2020-01-21
  • Explicit Reciprocity Laws for Higher Local Fields
    J. Number Theory (IF 0.684) Pub Date : 2020-01-21
    Jorge Flórez

    Using previously constructed reciprocity laws for the generalized Kummer pairing of an arbitrary (one-dimensional) formal group, in this article a special consideration is given to Lubin-Tate formal groups. In particular, this allows for a completely explicit description of the Kummer pairing in terms of multidimensional p-adic differentiation. The results obtained here constitute a generalization

    更新日期:2020-01-21
  • Primitive divisors of elliptic divisibility sequences over function fields with constant j-invariant
    J. Number Theory (IF 0.684) Pub Date : 2020-01-20
    Bartosz Naskręcki; Marco Streng

    We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the j-invariant of the elliptic curve is constant. In more detail, given an elliptic curve E with a point P of infinite order over a global field, the sequence D1, D2,… of denominators of multiples P, 2P,… of P is a strong divisibility sequence in the sense that gcd⁡(Dm,Dn)=Dgcd⁡(m,n). This is the genus-one

    更新日期:2020-01-21
  • On the l.c.m. of random terms of binary recurrence sequences
    J. Number Theory (IF 0.684) Pub Date : 2020-01-20
    Carlo Sanna

    For every positive integer n and every δ∈[0,1], let B(n,δ) denote the probabilistic model in which a random set A⊆{1,…,n} is constructed by choosing independently every element of {1,…,n} with probability δ. Moreover, let (uk)k≥0 be an integer sequence satisfying uk=a1uk−1+a2uk−2, for every integer k≥2, where u0=0, u1≠0, and a1,a2 are fixed nonzero integers; and let α and β, with |α|≥|β|, be the two

    更新日期:2020-01-21
  • On extending Artin's conjecture to composite moduli in function fields
    J. Number Theory (IF 0.684) Pub Date : 2020-01-20
    Eugene Eisenstein; Lalit K. Jain; Wentang Kuo

    In 1927, Artin hypothesized that for any given non-zero integer a other than 1, −1, or a perfect square, there exists infinitely many primes p for which a is a primitive root modulo p. In 1967, Hooley proved it under the assumption of the generalized Riemann hypothesis. Since then, there are many analogues and generalization of this conjecture. In this paper, we work on its generalization to composite

    更新日期:2020-01-21
  • Effective bounds for traces of singular moduli
    J. Number Theory (IF 0.684) Pub Date : 2020-01-20
    Havi Ellers; Meagan Kenney; Riad Masri; Wei-Lun Tsai

    We give an asymptotic formula for traces of weak Maass forms at CM points with an effective bound on the error term. Upon specializing to the modular j-function, we deduce such a result for traces of singular moduli. Due to work of Zagier, and Bringmann and Ono, these traces of weak Maass forms at CM points appear as Fourier coefficients of half-integral weight weakly holomorphic modular forms. Hence

    更新日期:2020-01-21
  • On quadratic progression sequences on smooth plane curves
    J. Number Theory (IF 0.684) Pub Date : 2020-01-20
    Eslam Badr; Mohammad Sadek

    We study the arithmetic (geometric) progressions in the x-coordinates of quadratic points on smooth planar curves defined over a number field k. Unless the curve is hyperelliptic, we prove that these progressions must be finite. We, moreover, show that the arithmetic gonality of the curve determines the infinitude of these progressions in the set of k‾-points with field of definition of degree at most

    更新日期:2020-01-21
  • Weighted greatest common divisors and weighted heights
    J. Number Theory (IF 0.684) Pub Date : 2020-01-20
    L. Beshaj; J. Gutierrez; T. Shaska

    Text We introduce the weighted greatest common divisor of a tuple of integers and explore some of its basic properties. Furthermore, for a set of weights w=(q0,…,qn), we use the concept of the weighted greatest common divisor to define a height h(p) on weighted projective spaces WPwn(k). We prove some of the basic properties of this weighted height, including an analogue of the Northcott's theorem

    更新日期:2020-01-21
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