Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field

Fermat-Wilson Supercongruences, arithmetic derivatives and strange factorizations

In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld additive') analogs each, for the well-known basic congruences of Fermat and Wilson, in the case of polynomials over finite fields. When we look at them modulo higher powers of primes, i.e. at `supercongruences', we find interesting relations linking them together, as well as linking them with arithmetic derivatives and zeta values. In the current work, we expand on the first analog and connections with arithmetic derivatives more systematically, giving many more equivalent conditions linking the two, now using `mixed derivati…

Evolution Equations and Functions of Hypergeometric Type over Fields of Positive Characteristic

We consider a class of partial differential equations with Carlitz derivatives over a local field of positive characteristic, for which an analog of the Cauchy problem is well-posed. Equations of such type correspond to quasi-holonomic modules over the ring of differential operators with Carlitz derivatives. The above class of equations includes some equations of hypergeometric type. Building on the work of Thakur, we develop his notion of the hypergeometric function of the first kind (whose parameters belonged initially to $\mathbb Z$) in such a way that it becomes fully an object of the func…

Multiplicative relations between coefficients of logarithmic derivatives of $\mathbb{F}_q$-linear functions and applications

We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from $\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives connect power sums to elementary symmetric functions via the Newton identities, we establish, as applications, new identities between important quantities of function field arithmetic, such as the Bernoulli-Carlitz fractions and power sums as well as their multi-variable generalizations. Resulting understanding of their factorizations has arithmetic significance, as w…

Higher order generalized geometric polynomials

According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and Euler polynomials. Furthermore, we find new formulas for the Carlitz's (Carlitz) and Howard's (Howard2) finite sums. Finally, we evaluate several series in closed forms, one of which has the coefficients include values of the Riemann zeta function. Moreover, we calculate some integrals in terms of generalized geometric polynomials.

Differential Equations for $F_q$-Linear Functions, II: Regular Singularity

We study some classes of equations with Carlitz derivatives for $F_q$-linear functions, which are the natural function field counterparts of linear ordinary differential equations with a regular singularity. In particular, an analog of the equation for the power function, the Fuchs and Euler type equations, and Thakur's hypergeometric equation are considered. Some properties of the above equations are similar to the classical case while others are different. For example, a simple model equation shows a possibility of existence of a non-trivial continuous locally analytic $F_q$-linear solution …

Universal Gauss-Thakur sums and L-series

In this paper we study the behavior of the function omega of Anderson-Thakur evaluated at the elements of the algebraic closure of the finite field with q elements F_q. Indeed, this function has quite a remarkable relation to explicit class field theory for the field K=F_q(T). We will see that these values, together with the values of its divided derivatives, generate the maximal abelian extension of K which is tamely ramified at infinity. We will also see that omega is, in a way that we will explain in detail, an universal Gauss-Thakur sum. We will then use these results to show the existence…

Lehmer pairs and derivatives of Hardy's $Z$-function

Occurrences of very close zeros of the Riemann zeta function are strongly connected with Lehmer pairs and with the Riemann Hypothesis. The aim of the present note is to derive a condition for a pair of consecutive simple zeros of the $ζ$-function to be a Lehmer pair in terms of derivatives of Hardy's $Z$-function. Furthermore, we connect Newman's conjecture with stationary points of the $Z$-function, and present some numerical results.

更高周期积分与L-函数的高阶导数

Higher Period Integrals and Derivatives of L-functions

本文提出几何框架，将高阶周期积分与L-函数的高阶中心导数联系起来，扩展了相对Langlands对偶框架，通过构造Hecke特征sheaf的几何周期积分作用，恢复了L-函数的高阶中心导数，并应用于Rankin-Selberg情况。