p-adic q-expansion of alternating sums of powers

A Stickleberger theorem for p-adic Gauss sums

We define "splitting functions of level l" for any integer l>0. These functions generalize Dwork's splitting functions : they allow us to represent additive characters of order $p^l$. Then we use these functions to obtain a Stickleberger theorem on the $p$-adic valuation of new Gauss sums.

Dedekind sums in the $p$-adic number field

In a recent note W. Kohnen asks whether the values of Dedekind sums are dense in the field of $p$-adic numbers. The present paper answers this question. Dedekind sums do not approximate units of $\mathbb Z_2$ or $\mathbb Z_3$, so they are not dense in $\mathbb Q_2$ or $\mathbb Q_3$. But they are dense in $\mathbb Q_p$ if $p\ge 5$.

Non-vanishing of certain cyclotomic multiple harmonic sums and application to the non-vanishing of certain $p$-adic cyclotomic multiple zeta values

We define and apply a method to study the non-vanishing of $p$-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses a cyclotomic multiple harmonic sums as an infinite sum of products of $p$-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain $p$-adic cyclotomic multiple zeta values.

L-functions of symmetric powers of Kloosterman sums (unit root L-functions and p-adic estimates)

The L-function of symmetric powers of classical Kloosterman sums is a polynomial whose degree is now known, as well as the complex absolute values of the roots. In this paper, we provide estimates for the p-adic absolute values of these roots. Our method is indirect. We first develop a Dwork-type p-adic cohomology theory for the two-variable infinite symmetric power L-function associated to the Kloosterman family, and then study p-adic estimates of the eigenvalues of Frobenius. A continuity argument then provides the desired p-adic estimates.

On harmonic sums and alternating Euler sums

The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.

On the $\ell$-adic valuation of certain Jacobi sums

Fix distinct primes $\ell$ and $f$, a finite field $\mathbf{F}_{q}$ such that $q \equiv 1 \pmod{\ell f}$, and a character $χ: \mathbf{F}_{q}^{\times} \to \mathbf{C}^{\times}$ of exact order $\ell f$. We present a new $\ell$-adic congruence for the Jacobi sum $J(χ^{\ell}, χ^{f})$. These Jacobi sums are Frobenius eigenvalues of the curve $y^{\ell} = x^{f} + 1$.

A precise description of the p-adic valuation of the number of alternating sign matrices

Following Sun and Moll, we study v_p(T(N)), the p-adic valuation of the counting function of the alternating sign matrices. We find an exact analytic expression for it that exhibits the fluctuating behaviour, by means of Fourier coefficients. The method is the Mellin-Perron technique, which is familiar in the analysis of the sum-of-digits function and related quantities.

Certain character sums and hypergeometric series

We prove two transformations for the $p$-adic hypergeometric series which can be described as $p$-adic analogues of a Kummer's linear transformation and a transformation of Clausen. We first evaluate two character sums, and then relate them to the $p$-adic hypergeometric series to deduce the transformations. We also find another transformation for the $p$-adic hypergeometric series from which many special values of the $p$-adic hypergeometric series as well as finite field hypergeometric functions are obtained.

Extended q-Dedekind-type Daehee-Changhee sums associated with Extended q-Euler polynomials

In the present paper, our goal is to introduce a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order dedekind-type sums with weight alpha related to Extended q-Euler polynomials by using p-adic q-integral in the p-adic integer ring.

p-adic Dedekind and Hardy-Berndt type sum related to Volkenborn Integral on Z_p

The purpose of this paper is to construct p-adic Dedekind sums and Hardy-Berndt type sums. We also construct generating function of the twisted Bernoulli polynomials and functions. Furthermore, we give some discussions on elliptic analogue of the Apostol-Dedekind sums.

L-functions of symmetric powers of the generalized Airy family of exponential sums: ell-adic and p-adic methods

For ψa nontrivial additive character on the finite field F_q, the map t \mapsto \sum_{x \in F_q} ψ(f(x)+tx) is the Fourier transform of the map t \mapsto ψ(f(t))$. As is well-known, this has a cohomological interpretation, producing a continuous ell-adic Galois representation. This paper studies the L-function attached to the k-th symmetric power of this representation using both ell-adic and p-adic methods. Using ell-adic techniques, we give an explicit formula for the degree of this L-function and determine the complex absolute values of its roots. Using p-adic techniques, we study the p-adi…

Alternating Euler $T$-sums and Euler $\tilde S$-sums

In this paper, we study the alternating Euler $T$-sums and related sums by using the method of contour integration. We establish the explicit formulas for all linear and quadratic Euler $T$-sums and related sums. Some interesting new consequences and illustrative examples are considered.

A note on the alternating sums of powers of consecutive integers

Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.

The $p$-adic Riemann Hypothesis For Expnonential Sums

The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is defined. A $p$-adic Riemann hypothesis is formulated. It asserts that the Newton polygon of the $L$-function coincides with the Frobenius polygon when $p$ is large enough. This $p$-adic Riemann hypothesis is proved when $n=2$ and $p\equiv-1({\rm mod }\ d)$. In general, it is proved that the Newton p…

Euler sums of generalized alternating hyperharmonic numbers II

In this paper, we introduce a new type of generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$, and show that Euler sums of the generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$ can be expressed in terms of linear combinations of classical (alternating) Euler sums.

Cubic alternating harmonic number sums

A recent paper of A. Sofo proves some results about sums of products of quadratic alternating harmonic numbers and reciprocal binomial coefficients. In this paper, we extend his result to cubic alternating harmonic number sums and develop new closed form representations of sums of cubic alternating harmonic numbers and reciprocal binomial coefficients. Some inter- esting (known or new) illustrative special cases as well as immediate consequences of the main results are also considered.

惯性素数处虚二次域的局部预解式、广义高斯和与反分圆Hecke $L$-值的$p$-adic赋值

The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes

本文证明了虚二次域在惯性素数处沿反分圆塔的Hecke $L$-值 $p$-adic 赋值的渐近公式，通过确定广义高斯和与局部预解式的 $p$-adic 赋值，回答了 Rubin 的问题。

Some 3-adic congruences for binomial sums

We prove some 3-adic congruences for binomial sums, which were conjectured by Sun.

The $p$-adic Order of Power Sums, the Erdös-Moser Equation, and Bernoulli Numbers

The Erdös-Moser equation is a Diophantine equation proposed more than 60 years ago which remains unresolved to this day. In this paper, we consider the problem in terms of divisibility of power sums and in terms of certain Egyptian fraction equations. As a consequence, we show that solutions must satisfy strong divisibility properties and a restrictive Egyptian fraction equation. Our studies lead us to results on the Bernoulli numbers and allow us to motivate Moser's original approach to the problem.

On the alternating sums of powers of consecutive q-integers

In this paper we construct $q$-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the $q$-analogue of alternating sums of powers of consecutive integers due to Euler.

T-adic exponential sums under diagonal base change

Twistec T-adic exponential sums are studied. As an application, the behavior of the L-function under diagonal base chang is explicitely given.

Positive alternating sums of integer partitions

We combine an extended version of Bailey's transform with an identity of Bressoud and with some identities of Berkovich and Warnaar to prove a variety of positivity results for alternating sums involving partition functions.

$L$-functions for families of generalized Kloosterman sums and $p$-adic differential equations

In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperber's construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the theory of GKZ system, we derive the Dwork's deformation equation for our family. Furthermore, with the help of Dwork's dual theory and deformation theory, the strong Frobenius structure of this equ…

On alternating sums of binomial coefficients and $q$-binomial coefficients

In this paper we shall evaluate two alternating sums of binomial coefficients by a combinatorial argument. Moreover, by combining the same combinatorial idea with partition theoretic techniques, we provide $q$-analogues involving the $q$-binomial coefficients.

A note on the alternating sums of powers of consecutive q-integers

In this paper we construct a new q-Euler numbers and polynomials. By using these numbers and polynomials, we give the interesting formulae related to alternating sums of powers of consecutive q-integers following an idea due to Euler.

A p-adic analogue of Siegel's Theorem on sums of squares

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally stud…

指标为1和3交替、以1开头以3结尾的$t$-进对称多重zeta值

$t$-adic symmetric multiple zeta values for indices with alternating $1$ and $3$, starting with $1$ and ending with $3$

本文针对Hirose等人提出的部分$t$-进对称多重zeta值相关猜想，证明了以1开头、以3结尾且1和3交替的指标对应的这类值可表示为Riemann zeta值的多项式，推进了相关数论问题的研究。

T-adic exponential sums over finite fields

$T$-adic exponential sums associated to a Laurent polynomial $f$ are introduced. They interpolate all classical $p^m$-power order exponential sums associated to $f$. The Hodge bound for the Newton polygon of $L$-functions of $T$-adic exponential sums is established. This bound enables us to determine, for all $m$, the Newton polygons of $L$-functions of $p^m$-power order exponential sums associated to an $f$ which is ordinary for $m=1$. Deeper properties of $L$-functions of $T$-adic exponential sums are also studied. Along the way, new open problems about the $T$-adic exponential sum itself ar…

Note on q-Dedekind type sums related to q-Euler polynomials

it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.

$p$-adic理论中的部分托里克指数和

$p$-adic Theory for Partial Toric Exponential Sums

本文通过Dwork方法提出$p$-adic证明，展示部分$L$-函数可表示为交替乘积的扭曲Fredholm行列式，并构造$p$-adic上同调理论及公式，证明其具有唯一的$p$-adic单位根。

A note on the q-Dedekind-type Daehee-Changhee sums with weight alpha arising from modified q-Genocchi polynomials with weight alpha

In the present paper, our objective is to treat a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight in connection with modified q-Genocchi polynomials with weight alpha by using p-adic invariant q-integral on Zp.

Generic twisted $T$-adic exponential sums of binomials

The twisted $T$-adic exponential sum associated to $x^{d}+λx$ is studied. If $λ\neq0,$ then an explicit arithmetic polygon is proved to be the Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives the Newton polygons of the $L$-functions of twisted $p$-power order exponential sums.

The $p$-Adic Valuations of Weil Sums of Binomials

We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=\sum_{x \in F} ψ(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $ψ$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^\times|$, and $a$ is an element of $F$. Such sums often arise in arithmetical calculations and also have applications in information theory. For each $F$ and $d$ one would like to know $V_{F,d}$, the minimum $p$-adic valuation of $W_{F,d}(a)$ as $a$ runs through the elements of $F$. We exclude exponents $d$ that are congruent to a power of $p$ modulo …

Congruences of alternating multiple harmonic sums

In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials play the key roles, in the alternating setting the Euler numbers and the Euler polynomials are also essential.

Integer ratios of consecutive alternating power sums

We give a characterization of all pairs $(k,n)$ of positive integers for which the ratio $$ \frac{1^k-2^k+3^k-\dots+(-1)^{n+1} n^k}{1^k-2^k+3^k-\dots+(-1)^{n}(n-1)^k} $$ of two consecutive alternating power sums is an integer.

$p$-adic estimates of exponential sums on curves

The purpose of this article is to prove a ``Newton over Hodge'' result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. For a regular function $\overline{f}$ on $V$, we may form the $L$-function $L(\overline{f},V,s)$ associated to the exponential sums of $\overline{f}$. In this article, we prove a lower estimate on the Newton polygon of $L(\overline{f},V,s)$. The estimate depends on the local monodromy of $f$ around each point $x \in X-V$. This confirms a hope of Deligne tha…

包含下降阶乘、上升阶乘、组合和与特殊数的p进q积分公式

Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special numbers

本文提出推导含Volkenborn积分、p进费米子积分的p进q积分公式的新方法，结合阶乘、二项式系数得到多类组合和与特殊数的恒等式，给出新组合和及新特殊数与多项式族的生成函数。

Generic twisted $T$-adic exponential sums of polynomials

The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of the twisted $L$-functions of $p^m$-power order exponential sums, for all $m$.

Partial zeta functions, partial exponential sums, and p-adic estimates

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, …

Generic T-adic exponential sums in one variable

The $T$-adic exponential sum associated to a Laurent polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of $L$-functions of $p$-power order exponential sums.

$T$-adic exponential sums of polynomials in one variable

The $T$-adic exponential sum of a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the T-adic exponential sum. This bound gives lower bounds for the Newton polygon of the $L$-function of exponential sums of $p$-power order.

Kloosterman sums and Hecke polynomials in characteristics 2 and 3

In this paper we give a modular interpretation of the $k$-th symmetric power $L$-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of $p=2$ and $k$ odd give the precise 2-adic Newton polygon. We also give a $p$-adic modular interpretation of Dwork's unit root $L$-function of the Kloosterman family, and give the precise 2-adic Newton polygon when $k$ is odd. In a previous paper, we gave an estimate for the $q$-adic Newton polygon of the symmetric power $L$-function of the Kloosterman family when $p \geq 5$. We discuss how this restriction on prim…

调和和的$p$进赋值及其与Wolstenholme素数的联系

$p$-adic valuation of harmonic sums and their connections with Wolstenholme primes

本研究围绕Eswarathasan-Levine关于调和数$p$进赋值的猜想展开，用模算术方法分别证明了非Wolstenholme素数与Wolstenholme素数情形下，$H(p^mn)$的$p$进赋值在$H(n)$的$p$进赋值为3时的反常渐近行为。

The p-adic valuations of sequences counting alternating sign matrices

The p-adic valuations of a sequence of integers T(n) counting alternating sign matrices is examined for p=2 and p=3. Symmetry properties of their graphs produce a new proof of the result that characterizes the indices for which T(n) is odd.

On inverted Kloosterman sums over finite fields

The classical $n$-variable Kloosterman sums over finite fields are well understood by Deligne's theorem from complex point of view and by Sperber's theorem from $p$-adic point of view. In this paper, we study the complex and $p$-adic estimates of inverted $n$-variable Kloosterman sums, addressing a question of N. Katz (1995). We shall give two complex estimates. The first one is elementary based on Gauss sums. The second estimate is deeper, depending on the cohomological results of Adolphson-Sperber, Denef-Loeser and Fu for twisted toric exponential sums. This deeper result assumes that the ch…

素数幂模下二元二次型的特征和

Character sums to prime power moduli evaluated at binary quadratic forms

研究素数幂模下二元二次型的特征和估计，采用p进分析方法，补充Heath-Brown对平方自由模的估计。

Igusa and Denef-Sperber Conjectures on nondegenerate p-adic exponential sums

We prove the intersection of Igusa's Conjecture of [Igusa, J., "Lectures on forms of higher degree", Lect. math. phys., Springer-Verlag, 59 (1978)] and the Denef - Sperber Conjecture of [Denef, J. and Sperber, S., "Exponential sums mod p^n and Newton polyhedra", Bull. Belg. Math. Soc., suppl. (2001) 55-63] on nondegenerate exponential sums modulo p^m.

Uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums

This thesis contains some new results on the uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums

On p-adic approximation of sums of binomial coefficients

We propose higher-order generalizations of Jacobsthal's $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\binom{ip}{p}$ ($i=1,2,\dots$) that are divisible by arbitrarily large powers of prime $p$.

$p$-adic estimates for multiplicative character sums

This article is an expanded version of the talk given by the first author at the conference "Exponential sums over finite fields and applications" (ETH, Zürich, November, 2010). We state some conjectures on archimedian and $p$-adic estimates for multiplicative character sums over smooth projective varieties. We also review some of the results of J. Dollarhide, which formed the basis for these conjectures. Applying his results, we prove one of the conjectures when the smooth projective variety is ${\mathbb P}^n$ itself.

Alternating Euler sums at the negative integers

We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.

Alternating Multiple $T$-Values: Weighted Sums, Duality, and Dimension Conjecture

In this paper, we define some weighted sums of the alternating multiple $T$-values (AMTVs), and study several duality formulas for them by using the tools developed in our previous papers. Then we introduce the alternating version of the convoluted $T$-values and Kaneko-Tsumura $ψ$-function, which are proved to be closely related to the AMTVs. At the end of the paper, we study the $\Q$-vector space generated by the AMTVs of any fixed weight $w$ and provide some evidence for the conjecture that their dimensions $\{d_w\}_{w\ge 1}$ form the tribonacci sequence 1, 2, 4, 7, 13, ....

Parametric Euler $T$-sums of odd harmonic numbers

In this paper, we define a parametric variant of generalized Euler sums and call them the (alternating) parametric Euler $T$-sums. By using the contour integration method and residue theorem, we establish several explicit formulae for the linear parametric Euler $T$-sums. Furthermore, by applying the results, we obtain explicit formulae for the Hoffman's (alternating) double $t$-values and Kaneko-Tsumura's (alternating) double $T$-values.

Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums

Let $p$ be a prime. In this short note we study some families of super congruences involving the following alternating sums \begin{equation*} \sum_{\substack{j_1+j_2+\cdots+j_n=2 p^r p\nmid j_1 j_2 \cdots j_n}} \frac{(-1)^{j_1+\cdots+j_b}}{j_1\cdots j_n} \pmod{p^r}, \end{equation*} which extend similar statements proved by Shen and Cai who treated the cases when $n=4,5$.