Journal of Number Theory最新文献

Let $k\in N$ and suppose we are given k integers $1\le a_1,\dots ,a_k\le n$. If $\sqrt{a_1}+\dots +\sqrt{a_k}$ is not an integer, how close can it be to one? When $k=1$, the distance to the nearest integer is $\gtrsim n^{-1/2}$. Angluin-Eisenstat observed the bound $\gtrsim n^{-3/2}$ when $k=2$. We prove there is a universal $c>0$ such that, for all $k\ge 2$, there exists a $c_k>0$ and k integers in $\{1,2,\dots ,n\}$ with$0<\Vert \sqrt{a_1}+\dots +\sqrt{a_k}\Vert \le c_k\cdot n^{-c\cdot k^{1/3}},$ where $\Vert \cdot \Vert$ denotes the distance to the nearest integer. This is a case of the square-root sum problem in numerical analysis where the usual cancellation constructions do not apply: already for $k=3$, the problem appears hard.

Let $T_m(N,2k)$ be the mth Hecke operator on the space $S(N,2k)$ of cuspforms of weight 2k and level N. This paper shows that in all but finitely many cases, which we list, the second coefficient of the characteristic polynomial of $T_2(N,2k)$ does not vanish when 2 and N are coprime.

We count the number of rational elliptic curves of bounded naive height that have a rational N-isogeny, for $N\in \{2,3,4,5,6,8,9,12,16,18\}$. For some N, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack.

Assuming the Generalized Riemann Hypothesis, we calculate ${\sum }_{0<{\gamma }_{\chi }\le T}{L}^{\left(\mu \right)}({\rho }_{\chi }+i\delta ,\chi )L^{\left(\nu \right)}(1-{\rho }_{\chi }-i\delta ,\bar{\chi })$, which extends the result proposed by Gonek in 1984.

We prove a log-free zero density estimate for automorphic L-functions defined over a number field k. This work generalizes and sharpens the method of pseudo-characters and the large sieve used earlier by Kowalski and Michel. As applications, we demonstrate for a particular family of number fields of degree n over k (for any n) that an effective Chebotarev density theorem and a bound on ℓ-torsion in class groups hold for almost all fields in the family.

Deep work by Shintani in the 1970's describes Hecke L-functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for $[F:Q]=n\ge 3$, Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on $R_{+}^n$, so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L-functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F, our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant.

Let $d_{\left(1\right)}\left(n\right)$ be the n-th coefficient of the Dirichlet series ${\left({\zeta }^{\prime }\right(s\left)\right)}^2={\sum }_{n=1}^{\infty }{d}_{\left(1\right)}\left(n\right)n^{-s}$ in $\Re s>1$, and ${\Delta }_{\left(1\right)}\left(x\right)$ be the error term of ${\sum }_{n\le x}{d}_{\left(1\right)}\left(n\right)$. In this paper, we will study the fifth-power moment of ${\Delta }_{\left(1\right)}\left(x\right)$.

The question of whether or not a given integral polynomial takes infinitely many square-free values has only been addressed unconditionally for polynomials of degree at most 3. We address this question, on average, for polynomials of arbitrary degree.

We investigate the relations between the rings E, G and D of values taken at algebraic points by arithmetic Gevrey series of order either −1 (E-functions), 0 (analytic continuations of G-functions) or 1 (renormalization of divergent series solutions at ∞ of E-operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D, and is the limit at infinity of some E-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E-functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for E-functions) and the conjecture $D\cap E=\bar{Q}$ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of E-functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E.