Journal of Number Theory最新文献

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Explicit cocycle of the Dedekind-Rademacher cohomology class and the Darmon-Dasgupta measures

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-01-21 DOI: 10.1016/j.jnt.2024.11.006

Jae Hyung Sim

The work of Darmon, Pozzi, and Vonk [3] has recently shown that the RM-values of the Dedekind-Rademacher cocycle

J_{D R}

are Gross-Stark units up to controlled torsion. The authors of [3] remarked that the measure-valued cohomology class, which we denote

μ_{D R}^{p}

, underlying

J_{D R}

is the level 1 incarnation of earlier constructions in [1]. In this paper, we make this relationship explicit by computing a concrete cocycle representative of an adelic incarnation

μ_{D R}

by tracing the construction of the cohomology class and comparing periods of weight 2 Eisenstein series. While maintaining a global perspective in our computations, we configure the appropriate method of smoothing cocycles which exactly yields the p-adic measures of [1] when applied to

μ_{D R}

. These methods will also explain the optional degree zero condition imposed in [1] which was remarked upon in [6] and [7].

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引用次数: 0

On nonzero coefficients of binary cyclotomic polynomials

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-01-21 DOI: 10.1016/j.jnt.2024.11.008

Igor E. Shparlinski , Laurence P. Wijaya

Let

ϑ (m)

be the number of nonzero coefficients in the m-th cyclotomic polynomial. For real

γ > 0

and

x \geq 2

we define

H_{γ} (x) = # {m : m = p q \leq x, p < q primes, ϑ (m) \leq m^{1 / 2 + γ}},

and show that for any fixed

η > 0

, uniformly over γ with

9 / 20 + η \leq γ \leq 1 / 2 - η,

we have an asymptotic formula

H_{γ} (x) \sim C (γ) x^{1 / 2 + γ} / \log x, x \to \infty,

where

C (γ) > 0

is an explicit constant depending only on γ. This extends the previous result of É. Fouvry (2013), which has 12/25 instead of 9/20. This improvement is based on new ingredient including work of W. Duke, J. Friedlander and H. Iwaniec (1997).

引用次数: 0

Linear x-coordinate relations of triples on elliptic curves

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-12-16 DOI: 10.1016/j.jnt.2024.11.003

Jerson Caro , Natalia Garcia-Fritz

For an elliptic curve E defined over the field

C

of complex numbers, we classify all translates of elliptic curves in

E^{3}

such that the x-coordinates satisfy a linear equation. This classification enables us to establish a relation between the rank of finite rank subgroups of E and triples in E whose x-coordinates are linearly related. The method of proof integrates complex analytic techniques on elliptic curves with results of Gao, Ge and Kühne on Uniform Mordell-Lang Conjecture for subvarieties in abelian varieties.

引用次数: 0

Higher convolutions of Ramanujan sums

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-12-10 DOI: 10.1016/j.jnt.2024.10.013

Shivani Goel , M. Ram Murty

We derive a limit formula for higher convolutions of Ramanujan sums, generalizing an old result of Carmichael. We then apply this in conjunction with the general theory of arithmetical functions of several variables to give a heuristic derivation of the Hardy-Littlewood formula for the number of prime k-tuplets less than x.

引用次数: 0

Correction to: “Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem” [J. Number Theory 130 (2010) 707–715]

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-12-03 DOI: 10.1016/j.jnt.2024.09.010

Titus W. Hilberdink

We discuss the main result of [1] which is concerned with the study of generalised prime systems for which the integer counting function

N_{P} (x)

is asymptotically very well-behaved, in the sense that

N_{P} (x) = ρ x + O (x^{β})

, where ρ is a positive constant and

β < \frac{1}{2}

. For such systems, the associated zeta function

ζ_{P} (s)

is holomorphic for

σ = ℜ s > β

. It was claimed that for

β < σ < \frac{1}{2}

\int_{0}^{T} | ζ_{P} (σ + i t) |^{2} d t = Ω (T^{2 - 2 σ - ε})

for (i) any

ε > 0

, and (ii) for

ε = 0

for all such σ except possibly one value.

The proof of these statements contains a flaw however, and in this Corrigendum we indicate where the mistake occurred but show that the proof can be rectified to still obtain (i) and get a slightly weaker result for (ii). The resulting Corollary 2 of [1] concerning the Dirichlet divisor problem for generalised integers remains essentially correct.

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引用次数: 0

Herr complex of (φ,τ)-modules

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-12-03 DOI: 10.1016/j.jnt.2024.10.012

Luming Zhao

Let p be an odd prime number and K a mixed characteristic complete discrete valuation field with perfect residue field of characteristic p. We construct a three-term complex, defined in terms of the

(φ, τ)

-module of a p-adic representation and prove its homology is isomorphic to the Galois cohomology of the representation. We further show that our complex is quasi-isomorphic to the four-term complex constructed by Tavares Ribeiro, providing an alternative proof of our result. As an application, we describe Galois cohomology of the Tate module associated to a p-divisible group in terms of corresponding Breuil-Kisin modules.

引用次数: 0

Corrigendum to “On certain maximal hyperelliptic curves related to Chebyshev polynomials” [J. Number Theory 203 (2019) 276–293] “关于切比雪夫多项式的某些极大超椭圆曲线”的勘误[J]。数论203 (2019)276-293]

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-30 DOI: 10.1016/j.jnt.2024.10.011

Saeed Tafazolian , Jaap Top

引用次数: 0

Representation functions in the set of natural numbers

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-26 DOI: 10.1016/j.jnt.2024.10.007

Shi-Qiang Chen , Csaba Sándor , Quan-Hui Yang

Let

N

be the set of all nonnegative integers. For

S \subseteq N

and

n \in N

, let

R_{S} (n)

denote the number of solutions of the equation

n = s + s^{'}

s, s^{'} \in S

s < s^{'}

. In this paper, we determine the structure of all sets A and B such that

A \cup B = N ∖ {r + m k : k \in N}

A \cap B = \emptyset

and

R_{A} (n) = R_{B} (n)

for every positive integer n, where m and r are two integers with

m \geq 2

and

r \geq 0

引用次数: 0

Sumset problem on dilated sets of integers

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-26 DOI: 10.1016/j.jnt.2024.10.006

Sandeep Singh , Ramandeep Kaur , Mamta Verma

Let A be a non-empty finite set of integers. For integers m and k, let

m \cdot A + k \cdot A = {m a_{1} + k a_{2} : a_{1}, a_{2} \in A}

. For

m = 2

and an odd prime k such that

| A | > 8 k^{k}

, Hamidoune et al. [6] proved that

| 2 \cdot A + k \cdot A | \geq (k + 2) | A | - k^{2} - k + 2

. Ljujic [7] extended this result and obtained the same bound for k to be a power of an odd prime and product of two distinct odd primes. Balog et al. [1] proved that

| p \cdot A + q \cdot A | \geq (p + q) | A | - {(p q)}^{(p + q - 3) (p + q) + 1}

, where

p < q

are relatively primes. In this article, for any odd values of k and under some certain conditions on set A, we obtain that

| 2 \cdot A + k \cdot A | \geq (k + 2) | A | - 2 k | \hat{A} |

, where

\hat{A}

is the projection of A in

Z / k Z

. This obtained bound is better than the bound given by Balog et al. We also generalize this bound for

| p \cdot A + k \cdot A |

, where p is any odd prime and k be an odd positive integer with

(p, k) = 1

引用次数: 0

Period of the Ikeda-Miyawaki lift 池田-宫崎抬升时期

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-22 DOI: 10.1016/j.jnt.2024.09.014

Tomoyoshi Ibukiyama , Hidenori Katsurada , Hisashi Kojima

In this paper, first we give a weak version of Ikeda's conjecture on the period of the Ikeda-Miyawaki lift. Next, we confirm this conjecture rigorously in some cases.

本文首先给出了池田关于池田-宫崎抬升周期猜想的一个弱版本。接下来，我们在某些情况下严格地证实了这一猜想。

引用次数: 0

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