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On the jth smallest modulus of a covering system with distinct moduli Int. J. Number Theory (IF 0.7) Pub Date : 2024-01-11 Jonah Klein, Dimitris Koukoulopoulos, Simon Lemieux
Covering systems were introduced by Erdős in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounded. In 2015, Hough answered affirmatively this long standing question. In 2022, Balister et al. gave a simpler and more versatile proof of Hough’s result. Building upon their work, we show that there exists some absolute constant
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Congruence properties modulo powers of 2 for overpartitions and overpartition pairs Int. J. Number Theory (IF 0.7) Pub Date : 2023-12-14 Dazhao Tang
In 2004, Corteel and Lovejoy introduced the notion of overpartitions in order to give a combinatorial proof of several celebrated q-series identities. Let p¯(n) denote the number of overpartitions of n. Many scholars have been investigated subsequently congruence properties modulo powers of 2 satisfied by p¯(n). Congruence properties modulo powers of 2 for pp¯(n) were also considered by several scholars
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Primes in denominators of algebraic numbers Int. J. Number Theory (IF 0.7) Pub Date : 2023-12-08 Deepesh Singhal, Yuxin Lin
Denote the set of algebraic numbers as ℚ¯ and the set of algebraic integers as ℤ¯. For γ∈ℚ¯, consider its irreducible polynomial in ℤ[x], Fγ(x)=anxn+⋯+a0. Denote e(γ)=gcd(an,an−1,…,a1). Drungilas, Dubickas and Jankauskas show in a recent paper that ℤ[γ]∩ℚ={α∈ℚ|{p|vp(α)<0}⊆{p:p|e(γ)}}. Given a number field K and γ∈ℚ¯, we show that there is a subset X(K,γ)⊆Spec(𝒪K), for which 𝒪K[γ]∩K={α∈K|{𝔭|v𝔭(α)<0}⊆X(K
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On the X-coordinates of Pell equations X2 − dY2 = ±1 as difference of two Fibonacci numbers Int. J. Number Theory (IF 0.7) Pub Date : 2023-12-06 Carlos A. Gómez, Salah Eddine Rihane, Alain Togbé
In this paper, we show that there is at most one value of the positive integer X participating in the Pell equation X2−dY2=±1, which is a difference of two Fibonacci numbers.
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Hybrid subconvexity bounds for twists of GL(3) L-functions Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-23 Xin Wang, Tengyou Zhu
Let F be a Hecke–Maass cusp form on SL(3,ℤ) and χ a primitive Dirichlet character of prime power conductor 𝔮=pk with p prime. In this paper, we will prove the following subconvexity bound L(12+it,F×χ)≪π,𝜀p3/4(𝔮(1+|t|))3/4−3/40+𝜀, for any 𝜀>0 and t∈ℝ.
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Interlacing properties for zeros of a family of modular forms Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-23 William Frendreiss, Jennifer Gao, Austin Lei, Amy Woodall, Hui Xue, Daozhou Zhu
Getz presented a family of level one modular forms fk for which all zeros lie on the unit circle in the fundamental domain. Expanding on work from Nozaki, Griffin et al., and Saha and Saradha, we show that the non-elliptic zeros of these fk satisfy two interlacing properties: standard interlacing, where the zeros of fk and fk+a alternate if and only if a∈{2,4,6,8,12} for sufficiently large k; and Stieltjes
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On 12-congruences of elliptic curves Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-23 Sam Frengley
We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over ℚ with 12-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of 12-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient
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Computing Shintani domains Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-21 Alex Capuñay
For a number field k having at least one embedding into the real numbers we give an algorithm to obtain a Shintani domain for the action of the totally positive units of k under its geometric embedding. Our algorithm modifies a known signed, or virtual, fundamental domain until it becomes a true one. We examine the results of extensive runs of our algorithm for cubic, quartic and quintic number fields
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Log-concavity of infinite product and infinite sum generating functions Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-16 Bernhard Heim, Markus Neuhauser
In this paper, we expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let {gd(n)}d≥0,n≥1 be the double sequences σd(n)=∑ℓ|nℓd or ψd(n)=nd. We associate double sequences {pgd(n)} and {qgd(n)}, defined as the coefficients of ∑n=0∞pgd(n)tn:=∏n=1∞(1−tn)−∑ℓ|nμ(ℓ)gd(n/ℓ)n,∑n=0∞qgd(n)tn:=11−∑n=1∞gd(n)tn. These coefficients are related to the number of partitions
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On 5ψ5 identities of Bailey Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-16 Aritram Dhar
In this paper, we provide proofs of two 5ψ5 summation formulas of Bailey using a 5ϕ4 identity of Carlitz. We show that in the limiting case, the two 5ψ5 identities give rise to two 3ψ3 summation formulas of Bailey. Finally, we prove the two 3ψ3 identities using a technique initially used by Ismail to prove Ramanujan’s 1ψ1 summation formula and later by Ismail and Askey to prove Bailey’s very-well-poised
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Metrical properties for functions of consecutive multiple partial quotients in continued fractions Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-16 Yuqing Zhang
Recently, the growth of the products of consecutive partial quotients ai(x) in the continued fraction expansion of a real number x was studied in connections with improvements to Dirichlet’s theorem. In this paper, for a non-decreasing positive measurable function F(x1,…,xm) and a function ϕ:ℕ→ℝ>0, we consider the set ℰF(ϕ)={x∈[0,1]:F(an(x),…,an+m−1(x))≥ϕ(n) for infinitely many n∈ℕ}, and obtain its
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Non-vanishing of theta components of Jacobi forms with level and an application Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-16 Pramath Anamby
We prove that a nonzero Jacobi form of level N (an odd integer) and square-free index m1m2 with m1|N and (N,m2)=1 has a nonzero theta component hμ with either (μ,2m1m2)=1 or (μ,2m1m2)∤2m2. As an application, we prove that a nonzero Siegel cusp form F of degree 2 and an odd level N in the Atkin–Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of N.
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On additive decompositions of primitive elements in finite fields Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-16 Yue-Feng She, Hai-Liang Wu
In this paper, we study several topics on additive decompositions of primitive elements in finite fields. Also we refine some bounds obtained by Dartyge and Sárközy as well as Shparlinski.
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On bounded basis with prescribed representation functions Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-02 Fang-Gang Xue
Let ℤ be the set of integers and ℕ the set of positive integers. For a nonempty set A of integers and any integers n, h with h≥2, define rA,h(n) as the number of solutions of n=a1+⋯+ah, where a1≤⋯≤ah and ai∈A for i=1,…,h. A set A of integers is defined as a basis of order h for ℤ if rA,h(n)≥1 for every integer n. In 2004, Nešetřil and Serra considered the Erdős–Turán conjecture for a class of bounded
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A lower bound on the proportion of modular elliptic curves over Galois CM fields Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-02 Zachary Feng
We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing ζ5. Applied to imaginary quadratic fields, this proportion is at least 2/5. Applied to cyclotomic fields ℚ(ζn) with 5∤n, this proportion is at least 1−ε with only finitely many exceptions of n, for any choice of ε>0.
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Infinite families of solutions for A3 + B3 = C3 + D3 and A4 + B4 + C4 + D4 + E4 = F4 Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-02 Archit Agarwal, Meghali Garg
Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler’s Diophantine equation A3+B3=C3+D3. In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation A4+B4+C4+D4+E4=F4 in the spirit of Ramanujan.
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Convergence of improper Iwasawa continued fractions Int. J. Number Theory (IF 0.7) Pub Date : 2023-10-07 Anton Lukyanenko, Joseph Vandehey
We prove the convergence of a wide class of continued fractions, including generalized continued fractions over quaternions and octonions. Fractional points in these systems are not bounded away from the unit sphere, so that the iteration map is not uniformly expanding. We bypass this problem by analyzing digit sequences for points that converge to the unit sphere under iteration, expanding on previous
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Large algebraic integers Int. J. Number Theory (IF 0.7) Pub Date : 2023-08-07 Denis Simon, Lea Terracini
An algebraic integer is said large if all its real or complex embeddings have absolute value larger than 1. An integral ideal is said large if it admits a large generator. We investigate the notion of largeness, relating it to some arithmetic invariants of the field involved, such as the regulator and the covering radius of the lattice of units. We also study its connection with the Weil height and
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A global invariant and Hasse invariants at finite or real primes Int. J. Number Theory (IF 0.7) Pub Date : 2023-08-07 Maozhou Huang
In their 2019 paper, Lee and Park presented a formula for the arithmetic Chern–Simons invariant. This formula gives a relation between this invariant and the local Hasse invariants at certain finite primes. Given a number field having real embeddings, we present alternative formulas to give relations between the arithmetic Chern–Simons invariant and the local Hasse invariants at certain primes including
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On a family of Siegel Poincaré series Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-29 Sonja Žunar
Let Γ be a congruence subgroup of Sp2n(ℤ). Using Poincaré series of K-finite matrix coefficients of integrable discrete series representations of Sp2n(ℝ), we construct a spanning set for the space Sm(Γ) of Siegel cusp forms of weight m∈ℤ>2n. We prove the non-vanishing of certain elements of this spanning set using Muić’s integral non-vanishing criterion for Poincaré series on locally compact Hausdorff
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S-unit equation in two variables and Padé approximations Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-22 Noriko Hirata-Kohno, Makoto Kawashima, Anthony Poëls, Yukiko Washio
In this paper, we use Padé approximations, constructed in [Kawashima and Poëls, Padé approximations for shifted functions and parametric geometry of numbers, J. Number Theory243 (2023) 646–687] for binomial functions, to give a new upper bound for the number of the solutions of the S-unit equation in two variables. Combining explicit Padé approximants with a simple argument relying on Mahler measure
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On ∞-adic and v-adic multiple zeta functions in positive characteristic Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-21 Daichi Matsuzuki
In this paper, we pursue positive characteristic analogs of the results of Furusho, Komori, Matsumoto, and Tsumura on p-adic multiple L-functions. We consider ∞-adic and v-adic multiple zeta functions concerned by Anglès, Ngo Dac, and Tavares Ribeiro. Our main results in this paper consist of: (1) integral expressions of special values of ∞-adic multiple zeta functions, (2) integral expressions of
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Detailed asymptotic expansions for partitions into powers Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-19 Cormac O’Sullivan
In this paper, we examine the number of ways to partition an integer n into kth powers when n is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion coefficients. The convexity and log-concavity of these partitions is shown for large n, and the stronger conjectures of Ulas are proved. The asymptotics of Wright’s
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Duality for finite/symmetric multiple zeta values of fixed weight, depth, and height Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-19 Kosuke Sakurada
In this paper, we prove a duality formula for the sum of the finite/symmetric multiple zeta-star values of fixed weight, depth, and height, which was conjectured by Kaneko. This result is a generalization of “height-one duality” proved by Hoffman and an analogue of the duality formula for the sum of the multiple zeta-star values proved by Li.
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On the parity of the number of (a,b,m)-copartitions of n Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-17 Hannah E. Burson, Dennis Eichhorn
We continue the study of the (a,b,m)-copartition function cpa,b,m(n), which arose as a combinatorial generalization of Andrews’ partitions with even parts below odd parts. The generating function of cpa,b,m(n) has a nice representation as an infinite product. In this paper, we focus on the parity of cpa,b,m(n). We find specific cases of a,b,m such that cpa,b,m(n) is even with density 1. Additionally
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The Shimura–Shintani correspondence via singular theta lifts and currents Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-17 Jonathan Crawford, Jens Funke
We describe the construction and properties of a singular theta lift for the orthogonal group SO(2,1). We obtain locally harmonic Maass forms in the sense of Bringmann, Kane and Kohnen with singular sets along geodesics in the upper half plane. We consider these forms as currents and derive properties of the Shimura–Shintani correspondence. This work provides extensions of the theta lifts considered
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A Freiman-type Theorem for restricted sumsets Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-14 David Daza, Mario Huicochea, Carlos Martos, Carlos Trujillo
Let A and B be nonempty finite subsets of ℤ. Freiman’s 3k−4 Theorem states that if |A+A|≤3|A|−4, then A is contained in a short arithmetic progression. Freiman generalized his theorem establishing that if |A+B|≤|A|+|B|+min{|A|,|B|}−4, then A and B are contained in short arithmetic progressions with common difference. Take S⊆A×B and write A+SB={a+b:(a,b)∈S}. There have been several attempts to generalize
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Diversity in rationally parameterized number fields Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-12 Benjamin Klahn
Let X be a curve defined over ℚ and let t∈ℚ(X) be a non-constant rational function on X of degree v≥2. For every rational number a/b pick a point Pa/b∈X(ℚ¯) such that t(Pa/b)=a/b. In this paper, we obtain lower bounds on the number of distinct fields among ℚ(Pa/b) with 1≤a,b≤N under some assumptions on t. We show that if t has a pole of order at least 2 or if there is a rational number α such that
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On the analytic strong multiplicity one for SL(2) Int. J. Number Theory (IF 0.7) Pub Date : 2023-07-11 Bin Guan
With the results on GL(3) and the adjoint lift, a restricted version of analytic strong multiplicity one is proved which holds for SL(2), the restriction being that the representations are required to be generic with respect to the same additive character. In this case there exists a positive constant c depending on 𝜀>0 and the number field F only, such that π can be decided completely by its local
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On the number of even values of an eta-quotient Int. J. Number Theory (IF 0.7) Pub Date : 2023-06-30 Fabrizio Zanello
The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient F, over any arithmetic progression. Namely, if ga,b(x) denotes the number of even coefficients of F in degrees n≡b (mod a) such that n≤x, then we show that ga,b(x)/x is unbounded for x large. Note that our result is very close to the best bound currently known
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The mean number of divisors for rough, dense and practical numbers Int. J. Number Theory (IF 0.7) Pub Date : 2023-06-30 Andreas Weingartner
We give asymptotic estimates for the mean number of divisors of integers without small prime factors, integers with bounded ratios of consecutive divisors, and for practical numbers. In the last case, this confirms a conjecture of Margenstern.
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The pro-p-Iwahori invariants of the universal quotient representation for GL2(F) Int. J. Number Theory (IF 0.7) Pub Date : 2023-06-15 Arindam Jana
Let F be a finite extension of ℚp. Every supersingular mod p representation of GL2(F) is a quotient of a certain universal module, say πr. The space of pro-p-Iwahori invariants of πr is instrumental in classifying the supersingular mod p representations of GL2(ℚp). For F≠ℚp, the space of pro-p-Iwahori invariants of πr has been determined by Schein and Hendel using the spherical Hecke algebra. An alternative
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Some remarks on the Erdős Distinct subset sums problem Int. J. Number Theory (IF 0.7) Pub Date : 2023-06-07 Stefan Steinerberger
Let {a1,…,an}⊂ℕ be a set of positive integers, an denoting the largest element, so that for any two of the 2n subsets the sum of all elements is distinct. Erdős asked whether this implies an≥c⋅2n for some universal c>0. We prove, slightly extending a result of Elkies, that for any a1,…,an∈ℝ>0, ∫ℝsinxx2∏i=1ncos(aix)2dx≥π2n with equality if and only if all subset sums are 1-separated. This leads to a
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On the superspecial loci of orthogonal type Shimura varieties Int. J. Number Theory (IF 0.7) Pub Date : 2023-06-07 Haining Wang
In this note, we study the superspecial loci of orthogonal type Shimura varieties of signature (n−2,2) with n≥3. We prove a conjecture of Gross that the superspecial loci and their lift in the integral models can be parametrized by certain homogeneous spaces. As applications, we provide mass formulas for the superspecial loci. We also indicate how Gross’s conjectures can be proved in the case of Coxeter
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Evaluation of iterated log-sine integrals in terms of multiple polylogarithms Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-31 Ryota Umezawa
It is known that multiple zeta values can be written in terms of certain iterated log-sine integrals. Conversely, we evaluate iterated log-sine integrals in terms of multiple zeta values and multiple polylogarithms in this paper. We also suggest some conjectures on multiple zeta values, multiple Clausen values, multiple Glaisher values and iterated log-sine integrals.
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Modulo d extension of parity results in Rogers–Ramanujan–Gordon type overpartition identities Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-31 Kağan Kurşungöz, Mohammad Zadehdabbagh
Sang, Shi and Yee, in 2020, found overpartition analogs of Andrews’ results involving parity in Rogers–Ramanujan–Gordon identities. Their result partially answered an open question of Andrews’. The open question was to involve parity in overpartition identities. We extend Sang, Shi and Yee’s work to arbitrary moduli, and also provide a missing case in their identities. We also unify proofs of Rogers–Ramanujan–Gordon
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Weighted badly approximable complex vectors and bounded orbits of certain diagonalizable flows Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-31 Gaurav Sawant
We show an analog of a theorem of [J. An, A. Ghosh, L. Guan and T. Ly, Bounded orbits of diagonalizable flows on finite volume quotients of products of SL2(ℝ), Adv. Math. 354 (2019) 106743] on weighted badly approximable vectors for totally imaginary number fields. We show that for G=SL2(ℂ)×⋯×SL2(ℂ) and Γ
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Zero-free regions for spectral averages of Hecke L-functions Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-29 Satadal Ganguly, E. M. Sandeep
We obtain an explicit zero-free region for a weighted sum of L-functions over the orthogonal basis of Hecke eigen cusp forms of a large integral weight for the full modular group. We also estimate the number of such forms whose L value does not vanish at a given point inside this zero-free region.
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The average of the exponential sums related to cusp forms Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-25 Fei Hou
In this paper, we investigate the average ∑f∈𝒮κ∗(P)ωf−1ℒf(α,β;N), where ℒf(α,β;N)=∑n∼Nλf(n)e(n2α+nβ) for any α,β∈ℝ and N≥2, with f∈𝒮κ∗(P) running over primitive holomorphic cusp forms of weight κ and prime level P. As a result, we prove pointwise uniform bounds with respect to α, β for the frequency of the P-values, and present that there exist strong oscillations in the exponential sum ℒf(α,β;N)
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An explicit comparison of anticyclotomic p-adic L-functions for Hida families Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-22 Chan-Ho Kim, Matteo Longo
We provide comparison results for anticyclotomic p-adic L-functions attached to Hida families of modular forms. The main result is a comparison between the anticyclotomic restriction of the three variable p-adic L-function introduced by Skinner and Urban, and the anticyclotomic L-function constructed by means of p-adic families of Gross points in the setting of definite quaternion algebras. Several
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Natural homomorphism of Witt rings of a certain cubic order Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-20 Paweł Gładki, Mateusz Pulikowski
In this paper, we provide an example of a nonmaximal order in a cubic number field K whose Witt ring is embedded into the Witt ring of K.
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Hasse principle violations in twist families of superelliptic curves Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-20 Lori D. Watson
Conditionally on the abc conjecture, we generalize the previous work of Clark and the author to show that a superelliptic curve C:yn=f(x) of sufficiently high genus has infinitely many twists violating the Hasse Principle if and only if f(x) has no ℚ-rational roots. We also show unconditionally that a curve defined by C:ypN=f(x) (for p prime and N>1) has infinitely many twists violating the Hasse Principle
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Analogue of Ramanujan’s function k(τ) for the cubic continued fraction Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-20 Yoon Kyung Park
We study the modularity of the function u(τ)=C(τ)C(2τ), where C(τ) is Ramanujan’s cubic continued fraction. It is an analogue of Ramanujan’s function k(τ)=r(τ)r(2τ)2, where r(τ) is the Rogers–Ramanujan continued fraction. We first prove the modularity of u(τ) and express C(τ) and C(2τ) in terms of u(τ). Subsequently, we find modular equations of u(τ) of level n for every positive integer n by using
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Congruences for k-elongated plane partition diamonds Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-20 Nayandeep Deka Baruah, Hirakjyoti Das, Pranjal Talukdar
In the eleventh paper in the series on MacMahon’s partition analysis, Andrews and Paule introduced the k-elongated partition diamonds. Recently, they revisited the topic. Let dk(n) count the partitions obtained by adding the links of the k-elongated plane partition diamonds of length n. Andrews and Paule obtained several generating functions and congruences for d1(n), d2(n), and d3(n). Da Silva et al
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On k-measures and Durfee squares of partitions Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-20 Damanvir Singh Binner
Recently, Andrews, Bhattacharjee and Dastidar introduced the concept of k-measure of an integer partition, and proved a surprising identity that the number of partitions of n which have 2-measure m is equal to the number of partitions of n with a Durfee square of side m. The authors asked for a bijective proof of this result and also suggested a further exploration of the properties of the number of
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Eigenform product identities of genus two Siegel modular forms of general congruence level Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-22 Jim Brown, Justine Dell, Hanna Noelle Griesbach, Amanda Hernandez
Given two eigenforms, it is a natural question to ask if the product of the eigenforms is again an eigenform. In the case of elliptic modular forms, this was answered in the full level case by Duke and Ghate and in the general level case by Johnson. In this paper, we consider the case of genus two Siegel modular forms with general congruence level.
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On the 2-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of ℚ(−7) Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-20 Takumi Yoshida
For a modular elliptic curve A=X0(49) and its quadratic twists A(M), we give equivalent conditions such that the 2-Selmer group S2(A(M)/ℚ) is minimal, namely, it is of order 2. One of these conditions is described by the L-value L(A(M)/ℚ,1). The other conditions are described by quadratic and biquadratic residue symbol, so explicit and computable (and one can compute the density of M). Also we prove
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On squares of Fourier coefficients twist exponential functions with applications Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-20 Wei Zhang
Let f be a Hecke–Maass cusp form of weight zero for SL2(ℤ) and λf(n) be the nth Fourier coefficient. For almost all 𝜗, we have ∑n≤xλf2(n)e(𝜗n)≪x0.8125+𝜀, which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than x0.8297. For all α>1 of type τ<∞, we also show that
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An octic diophantine equation and related families of elliptic curves Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-20 Ajai Choudhry, Arman Shamsi Zargar
We obtain two parametric solutions of the diophantine equation ϕ(x1,x2,x3)=ϕ(y1,y2,y3) where ϕ(x1,x2,x3) is the octic form defined by ϕ(x1,x2,x3)=x18+x28+x38−2x14x24−2x14x34−2x24x34. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank
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On the error term in the explicit formula of Riemann–von Mangoldt Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-16 Michaela Cully-Hugill, Daniel R. Johnston
We provide an explicit O(xlogx/T) error term for the Riemann–von Mangoldt formula by making results of Wolke [On the explicit formula of Riemann–von Mangoldt, II, J. London Math. Soc.2(3) (1983) 406–416] and Ramaré [Modified truncated Perron formulae, Ann. Math. Blaise Pascal23(1) (2016) 109–128] explicit. We also include applications to primes between consecutive powers, the error term in the prime
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Iterated integrals, multiple zeta values and Selberg integrals Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-11 Jiangtao Li
Classical multiple zeta values can be viewed as iterated integrals of the differentials dtt,dt1−t from 0 to 1. In this paper, we reprove Brown’s theorem: For ai,bi,cij∈ℤ, the iterated integral of the form ∫⋯∫0
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On length of the period of the continued fraction of nd Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-09 Filip Gawron, Tomasz Kobos
For a given quadratic irrational α, let us denote by D(α) the length of the periodic part of the continued fraction expansion of α. We prove that for every positive integer d, which is not a perfect square, there are infinitely many even integers k≥1, for which the equality D(nd)=k holds for infinitely many integers n≥1.
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On classification of sequences containing arbitrarily long arithmetic progressions Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-09 Şermin Çam Çelik, Sadık Eyidoğan, Haydar Göral, Doğa Can Sertbaş
In this paper, we study the classification of sequences containing arbitrarily long arithmetic progressions. First, we deal with the question how the polynomial map ns can be extended so that it contains arbitrarily long arithmetic progressions. Under some growth conditions, we construct sequences which contain arbitrarily long arithmetic progressions. Also, we give a uniform and explicit arithmetic
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Rational solutions to the variants of Erdős–Selfridge superelliptic curves Int. J. Number Theory (IF 0.7) Pub Date : 2023-05-04 Pranabesh Das, Shanta Laishram, N. Saradha, Divyum Sharma
For the superelliptic curves of the form (x+1)⋯(x+i−1)(x+i+1)⋯(x+k)=yℓ with x,y∈ℚ, y≠0, k≥3, 1≤i≤k, ℓ≥2, a prime, Das, Laishram, Saradha and Edis showed that the superelliptic curve has no rational points for ℓ≥e3k. In fact, the double exponential bound, obtained in these papers is far from the reality. In this paper, we study the superelliptic curves for small values of k. In particular, we explicitly
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Expressing q-series in terms of building blocks of Hecke-type double-sums Int. J. Number Theory (IF 0.7) Pub Date : 2023-04-27 Eric T. Mortenson, Ankit Sahu
We express recent double-sums studied by Wang, Yee, and Liu in terms of two types of Hecke-type double-sum building blocks. When possible we determine the (mock) modularity. We also express a recent q-hypergeometric function of Andrews as a mixed mock modular form.
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Torsion groups of elliptic curves over ℚ(μp∞) Int. J. Number Theory (IF 0.7) Pub Date : 2023-04-12 Tomislav Gužvić, Borna Vukorepa
Let E/ℚ be an elliptic curve and p∈{5,7,11} be a prime. We determine the possibilities for E(ℚ)(ζp))tors. Additionally, we determine all the possibilities for E(ℚ(ζ16))tors and E(ℚ(ζ27))tors. Using these results, we can determine the possibilities for E(ℚ(μp∞))tors for p∈{2,3,5,7,11}.
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Rational 𝜃-parallelogram envelopes via 𝜃-congruent elliptic curves Int. J. Number Theory (IF 0.7) Pub Date : 2023-03-30 Sajad Salami, Arman Shamsi Zargar
We introduce a new generalization of 𝜃-congruent numbers by defining a rational 𝜃-parallelogram envelope. It turns out that the existence of a rational 𝜃-parallelogram envelope associated to an n is connected to the property of n being a 𝜃-congruent number. Then, we study some problems related to the rational 𝜃-parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize
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Identities of general Kloosterman sums Int. J. Number Theory (IF 0.7) Pub Date : 2023-03-29 Xiaoge Liu, Tianping Zhang
Let q,m,n be any integers with q≥3, and χ be a Dirichlet character modulo q. The general Kloosterman sums K(m,n,χ;q) are defined as follows: K(m,n,χ;q)=∑a=1qχ(a)ema+nāq, where e(y)=e2πiy, and ā denotes the multiplicative inverse of a modulo q. By combining elementary methods and analytic methods, along with Montgomery and Vaughan’s clever trick on primitive characters, we derive some new identities
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On a variant of Pillai’s problem with binary recurrences and S-units Int. J. Number Theory (IF 0.7) Pub Date : 2023-03-29 Volker Ziegler
Let U=(Un)n∈ℕ be a fixed binary recurrence with real characteristic roots α,β satisfying |α|>|β| and let p1,…,ps be fixed distinct prime numbers. In this paper, we show that there exist effectively computable, positive constants C+ and C− such that the Diophantine equation Un−p1x1⋯psxs=c has at most s solutions (n,x1,…,xs)∈ℕs+1 if c>C+ and at most s+1 solutions (n,x1,…,xs)∈ℕs+1 if c<−C−. In order to
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On affinoids in quotients of Fermat varieties and explicit formula for Jacobi sum Hecke characters Int. J. Number Theory (IF 0.7) Pub Date : 2023-03-29 Takahiro Tsushima
By regarding the Fermat variety over a local field as a rigid analytic variety, we construct a family of affinoids in quotients of the Fermat varieties and compute the reduction of them. As a result, we explicitly compute the middle cohomology of the Fermat variety as a Galois representation. As a byproduct, we give an explicit formula for the ramified components of Jacobi sum Hecke characters in many