样式: 排序: IF: - GO 导出 标记为已读
-
Statistics for Iwasawa invariants of elliptic curves, II Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-18 Debanjana Kundu, Anwesh Ray
We study the average behavior of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.
-
Double and triple character sums and gaps between the elements of subgroups of finite fields Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-10 Jiankang Wang, Zhefeng Xu
For an odd prime p, let 𝔽p be the finite field of p elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any a,b,c∈𝔽p∗, we also obtain new upper bounds of the following double character sum Ta,b,c(χ,ℋ1,ℋ2)=∑h1∈ℋ1∑h2∈ℋ2χ(a+bh1+ch2) and a triple character sum Sχ(a,b,ℋ1,ℋ2,𝒩)=∑x∈𝒩∑h1∈ℋ1∑h2∈ℋ2χ(x+ah1+bh2)
-
Oscillations of Fourier coefficients of product of L-functions at integers in a sparse set Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-10 Babita, Mohit Tripathi, Lalit Vaishya
Let f be a normalized Hecke eigenform of weight k for the full modular group SL2(ℤ). In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg L-functions R(s,f×f), supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant D<0. We improve
-
A note on Schmidt’s subspace type theorems for hypersurfaces in subgeneral position Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-06 Lei Shi, Qiming Yan
In this paper, motivated by Nochka weights and the replacing hypersurfaces technique, we give an improvement of Schmidt’s subspace type theorem for hypersurfaces which are located in subgeneral position.
-
On the solutions of some Lebesgue–Ramanujan–Nagell type equations Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-06 Elif Kızıldere Mutlu, Gökhan Soydan
Denote by h=h(−p) the class number of the imaginary quadratic field ℚ(−p) with p prime. It is well known that h=1 for p∈{3,7,11,19,43,67,163}. Recently, all the solutions of the Diophantine equation x2+ps=4yn with h=1 were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen97(3–4) (2020) 339–352]. In this paper, we study the Diophantine
-
The transcendence of growth constants associated with polynomial recursions Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-06 Veekesh Kumar
Let P(x):=adxd+⋯+a0∈ℚ[x], ad>0, be a polynomial of degree d≥2. Let (xn) be a sequence of integers satisfying xn+1=P(xn)for all n=0,1,2,…andxn→∞as n→∞. Set α:=limn→∞xnd−n. Then, under the assumption ad1/(d−1)∈ℚ, in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J.57 (2022) 569–581], either α is transcendental or α can
-
Moments of Dirichlet L-functions to a fixed modulus over function fields Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-06 Peng Gao, Liangyi Zhao
In this paper, we establish the expected order of magnitude of the kth-moment of central values of the family of Dirichlet L-functions to a fixed prime modulus over function fields for all real k≥0.
-
A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-06 Yuchen Ding, Lilu Zhao
In this paper, we show that the Ruzsa number Rm is bounded by 192 for any positive integer m, which improves the prior bound Rm≤288 given by Chen in 2008.
-
Congruence classes for modular forms over small sets Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-06 Subham Bhakta, Srilakshmi Krishnamoorthy, R. Muneeswaran
Serre showed that for any integer m,a(n)≡0(modm) for almost all n, where a(n) is the nth Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study #{a(n)(modm)}n≤x over the set of integers with O(1) many prime factors. Moreover, we show that any residue class a∈ℤ/mℤ can be written as the sum of at most 13 Fourier coefficients
-
On the non-vanishing of Fourier coefficients of half-integral weight cuspforms Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-06 Jun-Hwi Min
We prove the best possible upper bounds of the gaps between non-vanishing Fourier coefficients of half-integral weight cuspforms. This improves the works of Balog–Ono and Thorner. We also show an asymptotic formula of central modular L-values for short intervals.
-
Relations of multiple t-values of general level Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-05 Zhonghua Li, Zhenlu Wang
We study the relations of multiple t-values of general level. The generating function of sums of multiple t-(star) values of level N with fixed weight, depth and height is represented by the generalized hypergeometric function 3F2, which generalizes the results for multiple zeta(-star) values and multiple t-(star) values. As applications, we obtain formulas for the generating functions of sums of multiple
-
Near-squares in binary recurrence sequences Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-05 Nikos Tzanakis, Paul Voutier
We call an integer a near-square if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers a≥3 by u0(a)=0, u1(a)=1 and un+2(a)=aun+1(a)−un(a) for n≥0. We show that for a given a≥3, there is at most one n≥5 such that un(a) is a near-square. With the exceptions of u6(3)=122 and u7(6)=239⋅132, any such un(a) can
-
Irreducibility and galois groups of truncated binomial polynomials Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-05 Shanta Laishram, Prabhakar Yadav
For positive integers n≥m, let Pn,m(x):=∑j=0mnjxj=n0+n1x+…+nmxm be the truncated binomial expansion of (1+x)n consisting of all terms of degree ≤m. It is conjectured that for n>m+1, the polynomial Pn,m(x) is irreducible. We confirm this conjecture when 2m≤n<(m+1)10. Also we show for any r≥10 and 2m≤n<(m+1)r+1, the polynomial Pn,m(x) is irreducible when m≥max{106,2r3}. Under the explicit abc-conjecture
-
The cuspidal cohomology of GL3/ℚ and cubic fields Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-05 Avner Ash, Dan Yasaki
We investigate the subspace of the homology of a congruence subgroup Γ of SL3(ℤ) with coefficients in the Steinberg module St(ℚ3) which is spanned by certain modular symbols formed using the units of a totally real cubic field E. By Borel–Serre duality, H0(Γ,St(ℚ3)) is isomorphic to H3(Γ,ℚ). The Borel–Serre duals of the modular symbols in question necessarily lie in the cuspidal cohomology Hcusp3(Γ
-
Dynatomic Galois groups for a family of quadratic rational maps Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-05 David Krumm, Allan Lacy
For every nonconstant rational function ϕ∈ℚ(x), the Galois groups of the dynatomic polynomials of ϕ encode various properties of ϕ are of interest in the subject of arithmetic dynamics. We study here the structure of these Galois groups as ϕ varies in a particular one-parameter family of maps, namely, the quadratic rational maps having a critical point of period 2. In particular, we provide explicit
-
Finite sequences of integers expressible as sums of two squares Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-05 Ajai Choudhry, Bibekananda Maji
This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers n such that n,n+h and n+k are all sums of two squares where h and k are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers that are sums of two squares. Similarly we obtain n in parametric
-
Rational points on x3 + x2y2 + y3 = k Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-05 Xiaoan Lang, Jeremy Rouse
We study the problem of determining, given an integer k, the rational solutions to Ck:x3z+x2y2+y3z=kz4. For k≠0, the curve Ck has genus 3 and its Jacobian is isogenous to the product of three elliptic curves E1,k, E2,k, E3,k. We explicitly determine the rational points on Ck under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result
-
The genus of a quotient of several types of numerical semigroups Int. J. Number Theory (IF 0.7) Pub Date : 2024-04-05 Kyeongjun Lee, Hayan Nam
Finding the Frobenius number and the genus of any numerical semigroup S is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of
-
On the Diophantine equation σ2(X¯n) = σn(X¯n) Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-26 Piotr Miska, Maciej Ulas
In this paper, we investigate the set S(n) of positive integer solutions of the title Diophantine equation. In particular, for a given n we prove boundedness of the number of solutions, give precise upper bound on the common value of σ2(X¯n) and σn(X¯n) together with the biggest value of the variable xn appearing in the solution. Moreover, we enumerate all solutions for n≤16 and discuss the set of
-
Some separable integer partition classes Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-26 Y. H. Chen, Thomas Y. He, F. Tang, J. J. Wei
Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus 2. We also extend separable integer partition classes with modulus 1 to overpartitions, called separable overpartition classes. We study overpartitions
-
Multiplier systems for Siegel modular groups Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-26 Eberhard Freitag, Adrian Hauffe-Waschbüsch
Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel
-
The minimal odd excludant and Euler’s partition theorem Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-26 Andrew Y. Z. Wang, Zheng Xu
In this work, we establish two interesting partition identities involving the minimal odd excludant, which has attracted great attention in recent years. In particular, we find a strong refinement of Euler’s celebrated theorem that the number of partitions of an integer into odd parts equals the number of partitions of that integer into distinct parts.
-
Reciprocity formulae for generalized Dedekind–Rademacher sums attached to three Dirichlet characters and related polynomial reciprocity formulae Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-26 Brad Isaacson
We define a three-character analogue of the generalized Dedekind–Rademacher sum introduced by Hall, Wilson, and Zagier and prove its reciprocity formula which contains all of the reciprocity formulas in the literature for generalized Dedekind–Rademacher sums attached (and not attached) to Dirichlet characters as special cases. Additionally, we prove related polynomial reciprocity formulas which contain
-
On almost-prime k-tuples Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-26 Bin Chen
Let τ denote the divisor function and ℋ={h1,…,hk} be an admissible set. We prove that there are infinitely many n for which the product ∏i=1k(n+hi) is square-free and ∑i=1kτ(n+hi)≤⌊ρk⌋, where ρk is asymptotic to 21262853k2. It improves a previous result of Ram Murty and Vatwani, replacing 3/4 by 2126/2853. The main ingredients in our proof are the higher rank Selberg sieve and Irving–Wu–Xi estimate
-
Variance of primes in short residue classes for function fields Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-26 Stephan Baier, Arkaprava Bhandari
Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not.2014(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions
-
Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-26 Jiseong Kim
By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for SL(n,ℤ). As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for SL(n,ℤ). Furthermore, we present
-
Fourier coefficients of cusp forms on special sequences Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-21 Weili Yao
In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms f and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals.
-
Density questions in rings of the form 𝒪K[γ] ∩ K Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-21 Deepesh Singhal, Yuxin Lin
We fix a number field K and study statistical properties of the ring 𝒪K[γ]∩K as γ varies over algebraic numbers of a fixed degree n≥2. Given k≥1, we explicitly compute the density of γ for which 𝒪K[γ]∩K=𝒪K[1/k] and show that this does not depend on the number field K. In particular, we show that the density of γ for which 𝒪K[γ]∩K=𝒪K is ζ(n+1)ζ(n). In a recent paper [Singhal and Lin, Primes in
-
Variations on a theorem of Capelli Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-20 Pradipto Banerjee
Elementary irreducibility criteria are established for f(xp) where f(x)∈ℤ[x] is irreducible over ℚ and p is a prime. For instance, our main criterion implies that if f(xp) is reducible over ℚ, then f(x) divides f(xp) modulo p2. Among several applications, it is shown that if f(x) has coefficients in {−1,1}, then f(x2) is irreducible over ℚ excluding a couple of obvious exceptions. As another application
-
Corrigendum to “The discriminant of compositum of algebraic number fields” Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-20 Sudesh Kaur Khanduja
We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not.
-
The Barnes–Hurwitz zeta cocycle at s = 0 and Ehrhart quasi-polynomials of triangles Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-20 Milton Espinoza
Following a theorem of Hayes, we give a geometric interpretation of the special value at s=0 of certain 1-cocycle on PGL2(ℚ) previously introduced by the author. This work yields three main results: an explicit formula for our cocycle at s=0, a generalization and a new proof of Hayes’ theorem, and an elegant summation formula for the zeroth coefficient of the Ehrhart quasi-polynomial of certain triangles
-
Fast computation of generalized dedekind sums Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-20 Preston Tranbarger, Jessica Wang
We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory.
-
The Manin–Peyre conjecture for certain multiprojective hypersurfaces Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-20 Xiaodong Zhao
By the circle method, an asymptotic formula is established for the number of integer points on certain hypersurfaces within multiprojective space. Using Möbius inversion and the modified hyperbola method, we prove the Manin–Peyre conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for certain smooth hypersurfaces in the multiprojective space of sufficiently
-
Multi-partition analogue of q-binomial coefficients Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-20 Byungchan Kim, Hayan Nam, Myungjun Yu
We introduce the multi-Gaussian polynomial Gk(M,N), a multi-partition analogue of the Gaussian polynomial (also known as q-binomial coefficient), as the generating function for certain restricted multi-color partitions. We study basic properties of multi-Gaussian polynomials and non-symmetric properties of Gk(M,N). We also derive a Sylvester-type identity and its application.
-
Linear algebra and congruences for MacMahon’s k-rowed plane partitions Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-20 Shi-Chao Chen
In this paper, we provide an algorithm to detect linear congruences of plk(n), the number of MacMahon’s k-rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the k-rowed plane partition functions. We also show p(n,m) that the number of partitions at most m parts always admits linear congruences.
-
Higher Mertens constants for almost primes II Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-20 Jonathan Bayless, Paul Kinlaw, Jared Duker Lichtman
For k≥1, let ℛk(x) denote the reciprocal sum up to x of numbers with k prime factors, counted with multiplicity. In prior work, the authors obtained estimates for ℛk(x), extending Mertens’ second theorem, as well as a finer-scale estimate for ℛ2(x) up to (logx)−N error for any N>0. In this paper, we establish the limiting behavior of the higher Mertens constants from the ℛ2(x) estimate. We also extend
-
A conjecture of Hegyvári Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-16 Xing-Wang Jiang, Wu-Xia Ma
For a given sequence A of nonnegative integers, let P(A) be the set of all finite subsequence sums of A. A is called complete if P(A) contains all sufficiently large integers. A real number α>0 is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of α. Hegyvári conjectured that Aα,β is complete if α or β is i.d.f. and
-
Values of certain Dirichlet series and higher derivative formulas of trigonometric functions Int. J. Number Theory (IF 0.7) Pub Date : 2024-03-13 Dominic Lanphier, Allen Lin
We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series.
-
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
-
On some sums involving the integral part function Int. J. Number Theory (IF 0.7) Pub Date : 2023-12-16 Kui Liu, Jie Wu, Zhishan yang
Denote by τk(n), ω(n) and μ2(n) the number of representations of n as a product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f=ω,2ω,μ2,τk, we prove that ∑n≤xfxn=x∑d≥1f(d)d(d+1)+O𝜀(x𝜃f+𝜀) for x→∞, where 𝜃ω=53110, 𝜃2ω=919, 𝜃μ2=25, 𝜃τk=5k−110k−1 and 𝜀>0
-
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
-
Determination of normalized extremal quasimodular forms of depth 1 with integral Fourier coefficients Int. J. Number Theory (IF 0.7) Pub Date : 2023-12-08 Tomoaki Nakaya
The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from q=e2πiτ to the reciprocal of the elliptic modular j-function, we prove that all normalized extremal quasimodular forms of depth 1 have a hypergeometric series expression and that integrality is not affected by
-
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
-
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.
-
On some discrete mean values of higher derivatives of Hardy’s Z-function Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-23 Hirotaka Kobayashi
Yıldırım obtained an asymptotic formula of the discrete moment of |ζ(12+it)| over the zero of the higher derivatives of Hardy’s Z-function. We give a generalization of his result on Hardy’s Z-function.
-
The Ostrowski quotient of an elliptic curve Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-23 Abbas Maarefparvar
For K/F a finite Galois extension of number fields, the relative Pólya group Po(K/F) is the subgroup of the ideal class group of K generated by all the strongly ambiguous ideal classes in K/F. The notion of Ostrowski quotient Ost(K/F), as the cokernel of the capitulation map into Po(K/F), has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions
-
Iwasawa theory of plus/minus Selmer groups with non-co-free plus/minus local conditions Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-23 Byoung Du Kim
We often use the plus/minus Selmer groups to study Iwasawa theory for elliptic curves with good supersingular reduction at p. But, if the plus/minus local conditions are not co-free, it can be difficult to use them effectively. In this paper, we introduce some technical improvements so that we can use the plus/minus Selmer groups even when the plus/minus local conditions are not co-free. In particular
-
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.
-
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
-
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
-
Locally cyclic extensions with Galois group GL2(p) Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-21 Sara Arias-de-Reyna, Joachim König
Using Galois representations attached to elliptic curves, we construct Galois extensions of ℚ with group GL2(p) in which all decomposition groups are cyclic. This is the first such realization for all primes p.
-
An explicit Voronoĭ formula for SL3(ℝ) newforms underlying the symmetric lifts in the level aspect Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-21 Fei Hou, GuangShi Lü
Let F be a newform for SL3(ℝ) underlying the symmetric square lift of a (either holomorphic or Hecke–Maaß) newform f of square-free level and trivial nebentypus. In this paper, we present the classical level aspect version of the Voronoĭ formula for the symmetric square lift is established in an alternative way by tracing back to its geometric nature, compared with Zhou’s work [F. Zhou, The Voronoi
-
The twisted second moment of L-functions associated to Hecke–Maass forms Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-21 Sheng-Chi Liu, Jakob Streipel
In this paper, we establish an asymptotic formula for the twisted second moment of L-functions associated to Hecke–Maass forms, which can be used to deduce a zero-density estimate for these L-functions in the spectral aspect.
-
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
-
A Dirichlet series related to the error term in the Prime Number Theorem Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-18 Ertan Elma
For a natural number n, let Z1(n):=∑ρnρρ where the sum runs over the nontrivial zeros of the Riemann zeta function. For a primitive Dirichlet character χ modulo q≥3, we define Z1(s,χ):=∑n=1∞χ(n)Z1(n)ns for ℜ(s)>2 and obtain the meromorphic continuation of the function Z1(s,χ) to the region ℜ(s)>12. Our main result indicates that the poles of Z1(s,χ) in the region 12<ℜ(s)<1, if they exist, are related
-
Determination of all imaginary cyclic quartic fields of prime class number p ≡ 3(mod4), and non-divisibility of class numbers Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-18 Mahesh Kumar Ram
Let p be a prime such that p≡3(mod4). Then, we show that there is no imaginary cyclic quartic extension K of ℚ whose class number is p. Suppose L/ℚ is a cyclic extension of number fields with an odd degree. Then, we show that 2 does not divide the class number of L if the class group of L is cyclic. We also construct some families of number fields whose class number is not divisible by a fixed prime
-
A two-parametric family of high rank Mordell curves Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-18 Mikhail A. Reynya
We present an elementary construction of an explicit two-parametric family of elliptic curves over ℚ of the form y2=x3+k such that the rank of each member of the family is at least seven. The main new ingredient is Ramanujan’s identity for sums of rational cubes.
-
Short interval results for powerfree polynomials over finite fields Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-18 Angel V. Kumchev, Nathan G. McNew, Ariana Park
Let k≥2 be an integer and 𝔽q be a finite field with q elements. We prove several results on the distribution in short intervals of polynomials in 𝔽q[x] that are not divisible by the kth power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all k≥2. We also develop polynomial versions of the classical
-
A note on the size of iterated sumsets in ℤd Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-16 Ilija Vrećica
In a recent paper by Curran and Goldmakher, the cardinality of h-fold sumsets hA was given when A⊂ℤd has d+2 elements. In this paper we provide a different method for doing this and obtain a more general result. We also obtain an upper bound for the value of |hA| when A⊂ℤd is a set of d+3 elements with simplicial hull.
-
Intermediate modular curves with infinitely many cubic points over ℚ Int. J. Number Theory (IF 0.7) Pub Date : 2023-11-16 Tarun Dalal
In this paper, we determine all intermediate modular curves XΔ(N) that admit infinitely many cubic points over the rational field ℚ.