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Corrigendum to “On p-adic Siegel Eisenstein series” [J. Number Theory 251 (2023) 3–30] J. Number Theory (IF 0.7) Pub Date : 2024-02-26 H. Katsurada, S. Nagaoka
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Algebraicity modulo p of generalized hypergeometric series [formula omitted] J. Number Theory (IF 0.7) Pub Date : 2024-02-21 D, a, n, i, e, l, , V, a, r, g, a, s, -, M, o, n, t, o, y, a
Let be the hypergeometric series with parameters and in , let be the least common multiple of the denominators of , written in lowest form and let be a prime number such that does not divide and . Recently in , it was shown that if for all , then the reduction of modulo is algebraic over . A standard way to measure the complexity of an algebraic power series is to estimate its degree and its height
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The finitude of tamely ramified pro-p extensions of number fields with cyclic p-class groups J. Number Theory (IF 0.7) Pub Date : 2024-02-20 Yoonjin Lee, Donghyeok Lim
Let be an odd prime and be a number field whose -class group is cyclic. Let be the maximal pro- extension of which is unramified outside a single non--adic prime ideal of . In this work, we study the finitude of the Galois group of over . We prove that is finite for the majority of 's such that the generator rank of is two, provided that for , is not a complex quartic field containing the primitive
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Rational self-maps with a regular iterate on a semiabelian variety J. Number Theory (IF 0.7) Pub Date : 2024-02-20 Jason Bell, Dragos Ghioca, Zinovy Reichstein
Let be a semiabelian variety defined over an algebraically closed field of characteristic 0. Let be a dominant rational self-map. Assume that an iterate is regular for some and that there exists no non-constant homomorphism of semiabelian varieties such that for some . We show that under these assumptions Φ itself must be a regular. We also prove a variant of this assertion in prime characteristic
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Higher Order Turán Inequalities for the Distinct Partition Function J. Number Theory (IF 0.7) Pub Date : 2024-02-20 Janet J.W. Dong, Kathy Q. Ji
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Divisibility properties of polynomial expressions of random integers J. Number Theory (IF 0.7) Pub Date : 2024-02-20 Zakhar Kabluchko, Alexander Marynych
We study divisibility properties of a set , where are polynomials in variables over and is a point picked uniformly at random from the set . We show that, as , the GCD and the suitably normalized LCM of this set converge in distribution to a.s. finite random variables under mild assumptions on . Our approach is based on the known fact that the uniform distribution on converges to the Haar measure on
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Bounds for the quartic Weyl sum J. Number Theory (IF 0.7) Pub Date : 2024-02-20 D, ., R, ., , H, e, a, t, h, -, B, r, o, w, n
We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that for any and any quadratic irrational . Classically one would have had the exponent for such . In contrast to the author's earlier work on cubic Weyl sums (which was conditional on the -conjecture), we show that the van der Corput -steps are sufficient for the
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Shifted convolution sums motivated by string theory J. Number Theory (IF 0.7) Pub Date : 2024-02-20 Ksenia Fedosova, Kim Klinger-Logan
In , it was conjectured that a particular shifted sum of even divisor sums vanishes, and in , a formal argument was given for this vanishing. Shifted convolution sums of this form appear when computing the Fourier expansion of coefficients for the low energy scattering amplitudes in type IIB string theory and have applications to subconvexity bounds of -functions. In this article, we generalize the
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Mod-p Galois representations not arising from abelian varieties J. Number Theory (IF 0.7) Pub Date : 2024-02-20 S, h, i, v, a, , C, h, i, d, a, m, b, a, r, a, m
It is known that any Galois representation with determinant equal to the mod- cyclotomic character, arises from the -torsion of an elliptic curve over , if and only if . In dimension , when , it is again known that any Galois representation valued in with cyclotomic similitude character arises from an abelian surface. In this paper, we study this question for all primes and dimensions . When and ,
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Eventual log-concavity of k-rank statistics for integer partitions J. Number Theory (IF 0.7) Pub Date : 2024-02-20 N, i, a, n, , H, o, n, g, , Z, h, o, u
Let denote the number of partitions of with Garvan -rank . It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the -rank for and , respectively. In this paper, we prove that the sequences are log-concave for all sufficiently large integers and each integer . In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was
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On a Ramanujan-type series associated with the Heegner number 163 J. Number Theory (IF 0.7) Pub Date : 2024-02-20 J, o, h, n, , M, ., , C, a, m, p, b, e, l, l
Using the Wolfram package and the command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series related to the class number for a quadratic form with discriminant . This conjectured series is of level one and has positive terms, and recalls the Chudnovsky brothers' alternating series
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Numerically explicit estimates for the distribution of rough numbers J. Number Theory (IF 0.7) Pub Date : 2024-02-20 Kai Fan
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The pair correlation function of multi-dimensional low-discrepancy sequences with small stochastic error terms J. Number Theory (IF 0.7) Pub Date : 2024-02-20 Anja Schmiedt, Christian Weiß
In any dimension , there is no known example of a low-discrepancy sequence which possesses Poisssonian pair correlations. This is in some sense rather surprising, because low-discrepancy sequences always have -Poissonian pair correlations for all and are therefore arbitrarily close to having Poissonian pair correlations (which corresponds to the case ). In this paper, we further elaborate on the closeness
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Residue of special functions of Anderson A-modules at the characteristic graph J. Number Theory (IF 0.7) Pub Date : 2024-02-20 Quentin Gazda, Andreas Maurischat
Let be an Anderson -module over . The period lattice of is related to its module of special functions by means of a non-canonical isomorphism introduced by the authors in . In this paper, we explain how a modification of the inverse map is canonical by interpreting it as a residue morphism along the characteristic graph. This phenomenon has already been observed in various situations. The main innovation
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A note on Galois groups of linearized polynomials J. Number Theory (IF 0.7) Pub Date : 2024-02-19 P, e, t, e, r, , M, ü, l, l, e, r
Let be a monic -linearized polynomial over of degree , where is an odd prime. In , Gow and McGuire showed that the Galois group of over the field of rational functions is unless . The case of even remained open, but it was conjectured that the result holds too and partial results were given. In this note we settle this conjecture. In fact we use Hensel's Lemma to give a unified proof for all prime
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The resolution of three exponential Diophantine equations in several variables J. Number Theory (IF 0.7) Pub Date : 2024-02-19 Csanád Bertók, Lajos Hajdu
We find all solutions of three exponential Diophantine equations, arising from certain quadratic, cubic and quartic identities. The first identity comes from a painting of the famous Russian painter Nikolay Bogdanov-Belsky, highlighted by Ja. I. Perelman. The equations have five, four and six terms, respectively, so they cannot be handled by classical tools based upon Baker's method. To solve the equations
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Lie invariant Frobenius lifts J. Number Theory (IF 0.7) Pub Date : 2024-02-19 A, l, e, x, a, n, d, r, u, , B, u, i, u, m
We begin with the observation that the -adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts that are “Lie invariant mod ” in the sense that the “normalized” action of on 1-forms preserves mod the space of invariant 1-forms. Our main result is that, after removing the 2-torsion sections, the above situation can be “infinitesimally deformed” in the sense that
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Sato–Tate type distributions for matrix points on elliptic curves and some K3 surfaces J. Number Theory (IF 0.7) Pub Date : 2024-02-19 Avalon Blaser, Molly Bradley, Daniel A.N. Vargas, Kathy Xing
Generalizing the problem of counting rational points on curves and surfaces over finite fields, we consider the setting of matrix points with finite field entries. We obtain exact formulas for matrix point counts on elliptic curves and certain 3 surfaces for “supersingular” primes. These exact formulas, which involve partitions of integers up to , essentially coincide with the expected value for the
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Quotients of the Bruhat-Tits tree by function field analogs of the Hecke congruence subgroups J. Number Theory (IF 0.7) Pub Date : 2024-02-08 C, l, a, u, d, i, o, , B, r, a, v, o
Let be a smooth, projective and geometrically integral curve defined over a finite field . For each closed point of , let be the ring of functions that are regular outside , and let be the completion at of the function field of . In order to study groups of the form , Serre describes the quotient graph , where is the Bruhat-Tits tree defined from .
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Computations on overconvergence rates related to the Eisenstein family J. Number Theory (IF 0.7) Pub Date : 2024-01-24 B, r, y, a, n, , A, d, v, o, c, a, a, t
We provide for primes a method to compute valuations appearing in the “formal” Katz expansion of the family derived from the family of Eisenstein series . We will describe two algorithms: the first one to compute the Katz expansion of an overconvergent modular form and the second one, which uses the first algorithm, to compute valuations appearing in the “formal” Katz expansion. Based on data obtained
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The number of preimages of iterates of ϕ and σ J. Number Theory (IF 0.7) Pub Date : 2024-01-24 A, g, b, o, l, a, d, e, , A, k, a, n, d, e
Paul Erdos and Carl Pomerance have proofs on an asymptotic upper bound on the number of preimages of Euler's totient function and the sum-of-divisors functions . In this paper, we will extend the upper bound to the number of preimages of iterates of and . Using these new asymptotic upper bounds, a conjecture in de Koninck and Kátai's paper, “On the uniform distribution of certain sequences involving
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Quartic integral polynomial Pell equations J. Number Theory (IF 0.7) Pub Date : 2024-01-24 Zachary Scherr, Katherine Thompson
In this paper we classify all monic, quartic, polynomials for which the Pell equation has a non-trivial solution with .
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On certain kernel functions and shifted convolution sums of the Fourier coefficients J. Number Theory (IF 0.7) Pub Date : 2024-01-22 Kampamolla Venkatasubbareddy, Ayyadurai Sankaranarayanan
We study the behavior of the shifted convolution sum involving even power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the k-full kernel function for any fixed integer k≥2.
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Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL2 J. Number Theory (IF 0.7) Pub Date : 2024-01-23 Kazuki Oshita, Masao Tsuzuki
We introduce a local zeta-function for an irreducible admissible supercuspidal representation of the metaplectic double cover of over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of . We obtain an expression of the gamma factor, which shows its
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On quotients of derivatives of L-functions inside the critical strip J. Number Theory (IF 0.7) Pub Date : 2024-01-22 R, a, s, h, i, , L, u, n, i, a
In , Gun, Murty and Rath studied non-vanishing and transcendental nature of special values of a varying class of -functions and their derivatives. This led to a number of works by several authors in different set-ups including studying higher derivatives. However, all these works were focused around the central point of the critical strip. In this article, we extend the study to arbitrary points in
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Sparse sets that satisfy the prime number theorem J. Number Theory (IF 0.7) Pub Date : 2024-01-22 Olivier Bordellès, Randell Heyman, Dion Nikolic
For arbitrary real we examine the set . Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the
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Universal sums of triangular numbers and squares J. Number Theory (IF 0.7) Pub Date : 2024-01-22 Z, i, c, h, e, n, , Y, a, n, g
In this paper, we study universal sums of triangular numbers and squares. Specifically, we prove that a sum of triangular numbers and squares is universal if and only if it represents , and 48.
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The factorial function and generalizations, extended J. Number Theory (IF 0.7) Pub Date : 2024-01-22 Jeffrey C. Lagarias, Wijit Yangjit
This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset of . Bhargava's factorials are invariants, constructed using the notion of -orderings of where is a prime. This paper defines -orderings of any nonempty subset of for all integers , as well as “extreme” cases and . It defines generalized factorials and generalized binomial coefficients as nonnegative
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One-level density of zeros of Dirichlet L-functions over function fields J. Number Theory (IF 0.7) Pub Date : 2024-01-08 Hua Lin
Text We compute the one-level density of zeros of order ℓ Dirichlet L-functions over function fields Fq[t] for ℓ=3,4 in the Kummer setting (q≡1(modℓ)) and for ℓ=3,4,6 in the non-Kummer setting (q≢1(modℓ)). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and lower order terms not predicted by RMT. We also confirm the symmetry type of the families is unitary, supporting Katz
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Collision of orbits for a one-parameter family of Drinfeld modules J. Number Theory (IF 0.7) Pub Date : 2023-12-28 Dragos Ghioca
We prove a result (see Theorem 1.1) regarding unlikely intersections of orbits for a given 1-parameter family of Drinfeld modules. We also advance a couple of general conjectures regarding unlikely intersections for algebraic families of Drinfeld modules (see Conjecture 1.3, Conjecture 2.3).
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On the variance of the Fibonacci partition function J. Number Theory (IF 0.7) Pub Date : 2023-12-28 Sam Chow, Owen Jones
We determine the order of magnitude of the variance of the Fibonacci partition function. The answer is different to the most naive guess. The proof involves a diophantine system and an inhomogeneous linear recurrence.
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On integers of the form p+2k1r1+⋯+2ktrt J. Number Theory (IF 0.7) Pub Date : 2023-12-29 Yong-Gao Chen, Ji-Zhen Xu
Let r1,…,rt be positive integers and let R2(r1,…,rt) be the set of positive odd integers that can be represented as p+2k1r1+⋯+2ktrt, where p is a prime and k1,…,kt are positive integers. It is easy to see that if r1−1+⋯+rt−1<1, then the set R2(r1,…,rt) has asymptotic density zero. In this paper, we prove that if r1−1+⋯+rt−1≥1, then the set R2(r1,…,rt) has a positive lower asymptotic density. Several
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Mean of the product of derivatives of Hardy's Z-function with Dirichlet polynomial J. Number Theory (IF 0.7) Pub Date : 2023-12-29 Mithun Kumar Das, Sudhir Pujahari
Inspired by the work of Balasubramanian, Conrey and Heath-Brown [1], we obtain an asymptotic expression for the mean of the product of any two finite order derivatives of Hardy's Z-function times Dirichlet polynomials in short intervals.
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On real zeros of the Hurwitz zeta function J. Number Theory (IF 0.7) Pub Date : 2023-12-29 Karin Ikeda
In this paper, we present results on the uniqueness of the real zeros of the Hurwitz zeta function in given intervals. The uniqueness in question, if the zeros exist, has already been proved for the intervals (0,1) and (−N,−N+1) for N≥5 by Endo-Suzuki and Matsusaka, respectively. We prove the uniqueness of the real zeros in the remaining intervals by examining the behavior of certain associated polynomials
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Moments of ideal class counting functions J. Number Theory (IF 0.7) Pub Date : 2023-12-29 Kam Cheong Au
We consider the counting function of ideals in a given ideal class of a number field of degree d. This describes, at least conjecturally, the Fourier coefficients of an automorphic form on GL(d), typically not a Hecke eigenform and not cuspidal. We compute its moments, and also investigate the moments of the corresponding cuspidal projection.
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Class numbers of multinorm-one tori J. Number Theory (IF 0.7) Pub Date : 2023-12-29 Fan-Yun Hung, Chia-Fu Yu
We present a formula for the class number of a multinorm one torus TL/k associated to any étale algebra L over a global field k. This is deduced from a formula for analogues of invariants introduced by T. Ono, which are interpreted as a generalization of Gauss genus theory. This paper includes the variants of Ono's invariant for arbitrary S-ideal class numbers and the narrow version, generalizing results
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A basis for the space of weakly holomorphic Drinfeld modular forms of level T J. Number Theory (IF 0.7) Pub Date : 2023-12-28 Tarun Dalal
In this article, we explicitly construct a canonical basis for the space of certain weakly holomorphic Drinfeld modular forms for Γ0(T) (resp., for Γ0+(T)) and compute the generating function satisfied by the basis elements. We also give an explicit expression for the action of the Θ-operator, which depends on the divisor of meromorphic Drinfeld modular forms.
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Massey products in Galois cohomology and the elementary type conjecture J. Number Theory (IF 0.7) Pub Date : 2023-12-28 Claudio Quadrelli
Let p be a prime. We prove that a positive solution to Efrat's Elementary Type Conjecture implies a positive solution to a strengthened version of Minač–Tân's Massey Vanishing Conjecture in the case of finitely generated maximal pro-p Galois groups whose pro-p cyclotomic character has torsion-free image. Consequently, the maximal pro-p Galois group of a field K containing a root of 1 of order p (and
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Attractor-repeller construction of Shintani domains for totally complex quartic fields J. Number Theory (IF 0.7) Pub Date : 2023-12-28 Alex Capuñay, Milton Espinoza, Eduardo Friedman
The units of a number field k act naturally on the real vector space k⊗QR, and so on open subsets of (k⊗QR)⁎ that are stable under the units. A Shintani domain for this action consists of a finite number of polyhedral cones, all having generators in k, whose union is a fundamental domain. Aside from the trivial case of imaginary quadratic fields, no practical method for computing Shintani domains for
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Construction of modular differential equations by using Rankin-Cohen brackets J. Number Theory (IF 0.7) Pub Date : 2023-12-28 Shotaro Kimura
In this paper, we construct higher-order modular differential equations for elliptic modular forms, holomorphic Jacobi forms, and skew-holomorphic Jacobi forms by using the Rankin-Cohen brackets extended to the Eisenstein series of weight two. Rankin-Cohen brackets are typical tools in the construction of modular forms by differential operators. We introduce the extended Rankin-Cohen brackets for holomorphic
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Ramanujan type congruences for quotients of Klein forms J. Number Theory (IF 0.7) Pub Date : 2023-12-28 Timothy Huber, Nathaniel Mayes, Jeffery Opoku, Dongxi Ye
In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are
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Growth of torsion groups of elliptic curves over number fields without rationally defined CM J. Number Theory (IF 0.7) Pub Date : 2023-12-28 Bo-Hae Im, Hansol Kim
For a quadratic field K without rationally defined complex multiplication, we prove that there exists of a prime pK depending only on K such that if d is a positive integer whose minimal prime divisor is greater than pK, then for any extension L/K of degree d and any elliptic curve E/K, we have E(L)tors=E(K)tors. By not assuming the GRH, this is a generalization of the results by Genao, and Gonález-Jiménez
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Extreme values of Dirichlet polynomials with multiplicative coefficients J. Number Theory (IF 0.7) Pub Date : 2023-12-27 Max Wenqiang Xu, Daodao Yang
We study extreme values of Dirichlet polynomials with multiplicative coefficients, namelyDN(t):=Df,N(t)=1N∑n⩽Nf(n)nit, where f is a completely multiplicative function with |f(n)|=1 for all n∈N. We use Soundararajan's resonance method to produce large values of |DN(t)| uniformly for all such f. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds
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Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity J. Number Theory (IF 0.7) Pub Date : 2023-12-27 Peter Vang Uttenthal
This paper concerns the distribution of Selmer ranks in a family of even Galois representations in residual characteristic p=2 obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as
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Rarefied Thue-Morse sums via automata theory and logic J. Number Theory (IF 0.7) Pub Date : 2023-11-24 Jeffrey Shallit
Let t(n) denote the number of 1-bits in the base-2 representation of n, taken modulo 2. We show how to prove the classic conjecture of Leo Moser, on the rarefied sum ∑0≤i
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Plectic structures in p-adic de Rham cohomology J. Number Theory (IF 0.7) Pub Date : 2023-11-28 David Loeffler, Sarah Livia Zerbes
Given a Hilbert modular form for a totally real field F, and a prime p split completely in F, the f-eigenspace in p-adic de Rham cohomology has a family of partial filtrations and partial Frobenius maps, indexed by the primes of F above p. The general plectic conjectures of Nekovář and Scholl suggest a “plectic comparison isomorphism” comparing these structures to étale cohomology. We prove this conjecture
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Roth-type theorem for quadratic system in Piatetski-Shapiro primes J. Number Theory (IF 0.7) Pub Date : 2023-11-24 Xiumin Ren, Qingqing Zhang, Rui Zhang
Let c1,…,cs be nonzero integers satisfying c1+…+cs=0. We consider the rational quadratic system c1x12+…+csxs2=0 where xi are restricted in subset A of Piatetski-Shapiro primes not exceeding x and corresponding to c. We show that for c∈(1,min{ss−1,2928}), if the system has only K-trivial solutions in A, then |A|≪x1/c(logx)−1(loglogloglogx)(2−s)/(2c)+ε holds for s⩾7.
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Solving Skolem's problem for the k–generalized Fibonacci sequence with negative indices J. Number Theory (IF 0.7) Pub Date : 2023-11-29 Jonathan García, Carlos A. Gómez, Florian Luca
The k–generalized Fibonacci sequence F(k) is the linearly recurrence sequence of order k whose first k terms are 0,…,0,1 and each term afterward is the sum of the preceding k terms. In this paper, we extend F(k) to negative indices obtaining a sequence denoted by H(k). Then, we solve Skolem's problem by finding all the zeros of H(k). To this end, we find very useful identities concerning H(k) similar
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Asymptotic equidistribution for partition statistics and topological invariants J. Number Theory (IF 0.7) Pub Date : 2023-11-24 Giulia Cesana, William Craig, Joshua Males
We provide a general framework for proving asymptotic equidistribution, convexity, and log-concavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proven by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution
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On a variant of the prime number theorem J. Number Theory (IF 0.7) Pub Date : 2023-11-25 Wei Zhang
In this paper, we can show thatSΛ(x)=∑1≤n≤xΛ([xn])=∑n=1∞Λ(n)n(n+1)x+O(x7/15+1/195+ε), where Λ(n) is the von Mangdolt function. Moreover, we can also give similar results related to the divisor function, which improve previous results.
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Mass formulas and Eisenstein congruences in higher rank J. Number Theory (IF 0.7) Pub Date : 2023-11-24 Kimball Martin, Satoshi Wakatsuki
We use mass formulas to construct minimal parabolic Eisenstein congruences for algebraic modular forms on reductive groups compact at infinity. For unitary groups of prime degree, this construction yields Eisenstein congruences for non-endoscopic cuspidal automorphic forms on quasi-split unitary groups. In supplementary sections, we also generalize previous weight 2 Eisenstein congruences for Hilbert
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The 3-rd unramified cohomology for norm one torus J. Number Theory (IF 0.7) Pub Date : 2023-11-24 Hanqing Long, Dasheng Wei
For an algebraic torus S, Blinstein and Merkurjev have given an estimate of 3-rd unramified cohomology H¯nr3(F(S),Q/Z(2)) obtained from a flasque resolution of S. Based on their work, for the norm one torus W=RK/F(1)Gm with K/F abelian, we compute the 3-rd unramified cohomology H¯nr3(F(W),Q/Z(2)).
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Optimizing the coefficients of the Ramanujan expansion J. Number Theory (IF 0.7) Pub Date : 2023-11-24 Dawei Lu, Ruoyi Wang
In this paper, we provide the coefficients of the Ramanujan expansion proposed by Wang [23] with a different method. Then, we present the rates of convergence of the expansion for the harmonic numbers, the values of the parameter h of the coefficients and some relevant inequalities. To demonstrate the superiority of our expansion with new coefficients over the classic Ramanujan formula, some numerical
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On the integrality of locally algebraic representations of GL2(D) J. Number Theory (IF 0.7) Pub Date : 2023-11-24 Santosh Nadimpalli, Mihir Sheth
Emerton's theory of Jacquet modules for locally analytic representations provides necessary conditions for the existence of integral structures in locally analytic representations. These conditions are also expected to be sufficient for the integrality of generic irreducible locally algebraic representations. In this article, we prove the sufficiency of Emerton's conditions for some tamely ramified
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Möbius Disjointness Conjecture on the product of a circle and the Heisenberg nilmanifold J. Number Theory (IF 0.7) Pub Date : 2023-11-24 Jing Ma, Ronghui Wu
Let T be the unit circle and Γ﹨G the 3-dimensional Heisenberg nilmanifold. We prove that the Möbius function is linearly disjoint from a class of distal skew products on T×Γ﹨G. These results generalize a recent work of Huang-Liu-Wang.
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On the sum of a prime and a square-free number with divisibility conditions J. Number Theory (IF 0.7) Pub Date : 2023-11-22 Shehzad Hathi, Daniel R. Johnston
Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For example, we show for odd k≤105 and even k≤2⋅105 that any even integer n≥40 can be expressed as the sum of a prime and a square-free number coprime to k. We also discuss
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A bound for Stieltjes constants J. Number Theory (IF 0.7) Pub Date : 2023-11-23 S. Pauli, F. Saidak
The main goal of this note is to improve the best known bounds for the Stieltjes constants, using the method of steepest descent that was applied in 2011 by Coffey and Knessl in order to approximate these constants.
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Ankeny-Artin-Chowla and Mordell conjectures in terms of p-rationality J. Number Theory (IF 0.7) Pub Date : 2023-11-23 Y. Benmerieme, A. Movahhedi
We interpret the two old and still unsettled conjectures of Ankeny-Artin-Chowla and of Mordell, concerning the unit of the real quadratic field Q(p), in terms of its p-rationality.