Journal of Number Theory最新文献

We give asymptotics for shifted convolutions of the form$\sum _{n<X}\frac{{\sigma }_{2u}(n,\chi ){\sigma }_{2v}(n+k,\psi )}{n^{u+v}}$ for nonzero complex numbers $u,v$ and nontrivial Dirichlet characters $\chi ,\psi$. We use the technique of automorphic regularization to find the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable. The error term we obtain is in some cases smaller than what the method we use typically yields.

In this paper, we study the asymptotic distribution of coefficients of general L-functions over arithmetic progressions without the Ramanujan conjecture. As an application, we consider the high mean of Fourier coefficients of holomorphic forms or Maass forms for $\Gamma =\mathrm{SL}(2,Z)$ over arithmetic progressions, and improve the results of Jiang and Lü [10]. Our new results remove the restriction to prime module and improve the interval length of module q.

Let L be a lattice. We exhibit algorithms for calculating Tits buildings and orbits of vectors in L for certain subgroups of the orthogonal group $O\left(L\right)$. We discuss how these algorithms can be applied to determine the configuration of boundary components in the Baily-Borel compactification of orthogonal modular varieties and to improve the performance of computer arithmetic of orthogonal modular forms.

For all ${\alpha }_1,{\alpha }_2\in (1,2)$ with $1/{\alpha }_1+1/{\alpha }_2>5/3$, we show that the number of pairs $(n_1,n_2)$ of positive integers with $N=\lfloor n_1^{{\alpha }_1}\rfloor +\lfloor n_2^{{\alpha }_2}\rfloor$ is equal to $\Gamma (1+1/{\alpha }_1)\Gamma (1+1/{\alpha }_2)\Gamma {(1/{\alpha }_1+1/{\alpha }_2)}^{-1}{N}^{1/{\alpha }_1+1/{\alpha }_2-1}+o\left(N^{1/{\alpha }_1+1/{\alpha }_2-1}\right)$ as $N\to \infty$, where Γ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when ${\alpha }_1,{\alpha }_2\in (1,2)$ lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the p-th Fermat curve where 2 is a quadratic residue modulo p, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.

We prove a conjecture of B. Gross and D. Prasad about determination of generic L-packets in terms of the analytic properties of the adjoint L-function for p-adic general even spin groups of semi-simple ranks 2 and 3. We also explicitly write the adjoint L-function for each L-packet in terms of the local Langlands L-functions for the general linear groups.

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete “stick breaking”) subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior.

In this article we investigate the action of (ramified and unramified) Hecke operators on automorphic forms for the function field of the projective line defined over $F_q$ and for the group ${\mathrm{GL}}_2$. We first compute the dimension of the Hecke eigenspaces for every generator of the unramified Hecke algebra. Thus, we consider the ramification in a point of degree one and explicitly describe the action of certain ramified Hecke operators on automorphic forms. Moreover, we also compute the dimensions of its eigenspaces for those ramified Hecke operators. We finish the article considering more general ramifications, namely those one attached to a closed point of higher degree.

A conjecture recently stated by Flach and Morin relates the action of the monodromy on the Galois invariant part of the p-adic Beilinson–Hyodo–Kato cohomology of the generic fiber of a scheme defined over a DVR of mixed characteristic to (the cohomology of) its special fiber. We prove the conjecture in the case that the special fiber of the given arithmetic scheme is also a fiber of a geometric family over a curve in positive characteristic.