Tchebotarev theorems for function fields

Universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function

In this paper, we prove the universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function on some line parallel to the real axis.

Rudnick and Soundarajan's Theorem over Prime Polynomials for the Rational Function Field

In this paper, we use the methods of Andrade, Rudnick and Soundarajan to prove a Theorem about Lower bounds of moments of quadratic Dirichlet L-functions associated to monic irreducible polynomials over function fields.

An inverse theorem for Gowers norms of trace functions over prime fields

We prove estimates for the Gowers uniformity norms of functions over $\Zz/p\Zz$ which are trace functions of certain $\ell$-adic sheaves, and establish in particular a strong inverse theorem for these functions.

An extension of Rohrlich's Theorem to the $j$-function

In this paper, we extend Rohrlich's Theorem on the integral of logarithms of meromorphic functions to compute the inner product between such functions and polynomials in the $j$-function. We then show that the generating function for these inner products with $j_n$ is part of a weight two modular object.

A decomposition theorem on differential polynomials of theta functions of high level

In this article, we prove a decomposition theorem on differential polynomials of theta functions of high level.

Mordell and Tuenter formulas, Kubert functions and reciprocity theorems for Apostol-Dedekind sums

We connect Dedekind sums and some formulas for numerical semigroups.

An effective Schmidt's subspace theorem for hypersurfaces in subgeneral position in projective varieties over function fields

We deduce an effective version of Schmidt's subspace theorem on a smooth projective variety X over function fields of characteristic zero for hypersurfaces located in N-subgeneral position with respect to X.

Some new density theorems for Dirichlet L-functions

We prove some new log-free density theorems for zeros of Dirichlet L-functions (which accordingly are more sharp than earlier ones near to the boundary line of the critical strip). The results can be applied in several problems of prime number theory.

When does the Bombieri-Vinogradov Theorem hold for a given multiplicative function?

Let $f$ and $g$ be $1$-bounded multiplicative functions for which $f*g=1_{.=1}$. The Bombieri-Vinogradov Theorem holds for both $f$ and $g$ if and only if the Siegel-Walfisz criterion holds for both $f$ and $g$, and the Bombieri-Vinogradov Theorem holds for $f$ restricted to the primes.

Omega-theorems for the Riemann zeta function and its derivatives near the line $\mathrm{Re}\,s=1$

We introduce a generalization of the method of S. P. Zaitsev. This generalization allows us to prove omega-theorems for the Riemann zeta function and its derivatives in some regions near the line $\mathrm{Re}\,s=1$.

函数域上关于移动超平面目标的 Schmidt 子空间定理

A Schmidt's subspace theorem for moving hyeprplane targets over function fields

本文在函数域上的射影空间中建立了关于移动超平面目标的 Schmidt 子空间定理。

Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH, II

We have proved recently several explicit versions of the prime ideal theorem under GRH. Here we prove a version with optimal asymptotic behaviour.

Explicit estimates for Artin $L$-functions: Duke's short-sum theorem and Dedekind Zeta Residues

Under GRH, we establish a version of Duke's short-sum theorem for entire Artin $L$-functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerical constants in this work are explicit.

整函数的差分“abc”定理与Nevanlinna第二主定理截断版本的差分类似

Difference "abc" theorem for entire functions and Difference analogue of truncated version of Nevanlinna second main theorem

本文研究了阶小于1的整函数的差分$abc$定理及Nevanlinna第二主定理截断版本的差分类似，揭示了次正规亚纯函数在复平面上具有长长度点的数量限制，并应用于差分Fermat泛函方程的整解。

Ax-Schanuel Type Theorems on Functional Transcendence via Nevanlinna Theory

We will apply Nevanlinna Theory to prove several Ax-Schanuel type Theorems for functional transcendence when the exponential map is replaced by other meromorphic functions. We also show that analytic dependence will imply algebraic dependence for certain classes of entire functions. Finally, some links to transcendental number theory and geometric Ax-Schanuel Theorem will be discussed.

Kronecker theta function and a decomposition theorem for theta functions I

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's $_1ψ_1$ summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanuja…

Generalized ABC theorems for non-Archimedean entire functions of several variables in arbitrary characteristic

We prove generalized ABC theorems for vanishing sums of non-Archimedean entire functions of several variables in arbitrary characteristic.

A Shafarevich-Faltings Theorem For Rational Functions

Using an alternative notion of good reduction, an analog of the Shafarevich theorem for elliptic curves is proved for morphisms of the projective line over number fields.

The Distribution Relation and Inverse Function Theorem in Arithmetic Geometry

We study arithmetic distribution relations and the inverse function theorem in algebraic and arithmetic geometry, with an emphasis on versions that can be applied uniformly across families of varieties and maps. In particular, we prove two explicit versions of the inverse function theorem, the first via general distribution and separation inequalities that may be of independent interest, the second via a careful implementation of classical Newton iteration.

Arithmetic duality theorems for 1-motives over function fields

In this paper we obtain a Poitou-Tate exact sequence for finite and flat group schemes over a global function field. We also extend the duality theorems for 1-motives over number fields obtained by D.Harari and T.Szamuely to the function field case.

Dirichlet's Theorem in function fields

We study metric Diophantine approximation in local fields of positive characteristic. Specifically, we study the problem of improving Dirichlet's theorem in Diophantine approximation and prove very general results in this context.