Our aim of this paper is to propose a method of analytic continuation of Carlitz multiple (star) polylogarithms to the whole space by using Artin-Schreier equation and present a treatment of their branches by introducing the notion of monodromy modules. As applications of this method, we obtain (1) a method of continuation of the logarithms of higher tensor powers of Carlitz module, (2) the orthogonal property (Chang-Mishiba functional relations), (3) a branch independency of the Eulerian property.
In this paper we shall define the analytic continuation of the multiple polylogarithms by using Chen's theory of iterated path integrals and compute the monodromy of all multiple logarithms explicitly.
The fractional polylogarithms, depending on a complex parameter $\a$, are defined by a series which is analytic inside the unit disk. After an elementary conversion of the series into an integral presentation, we show that the fractional polylogarithms are multivalued analytic functions in the complex plane minus 0 and 1. For non-integer values of $\a$, we prove the analytic continuation, compute the monodromy around 0 and 1, give a Mittag-Leffler decomposition and compute the asymptotic behavior for large values of the complex variable. The fractional polylogarithms are building blocks of res…
This is an expository paper on the meromorphic continuation of zeta functions with Euler products (for example zeta functions of groups and height zeta functions) or without (for example the Goldbach zeta function). As an application we show how a natural boundary of analytic continuation can give asymptotic results.
We prove two theorems. Theorem 1 gives the meromorphic continuation of the multiple zeta function to the whole space. In Theorem 2, we prove asymptotic behavior near the non-positive integers.
We combine Zagier's theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of $\mathrm{SL}(2,\mathbb{R})$ due to Bernstein and Reznikov to study triple products of Eisenstein series of arbitrary (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.
In the present paper, we analyse analytic continuation of weighted q-Genocchi numbers and polynomials. A novel formula for weighted q-Genocchi- Zeta function ζG,q (s | α) in terms of nested series of ζG,q (n | α) is derived. Moreover, we introduce a novel concept of dynamics of the zeros of analytically continued weighted q-Genocchi polynomials.
In this paper we study that the $q$-Euler numbers and polynomials are analytically continued to $E_q(s)$. A new formula for the Euler's $q$-Zeta function $ζ_{E,q}(s)$ in terms of nested series of $ζ_{E,q}(n)$ is derived. Finally we introduce the new concept of the dynamics of analytically continued $q$-Euler numbers and polynomials.
We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf algebra.
We study the geometry of unitary Shimura varieties without assuming the existence of an ordinary locus. We prove, by a simple argument, the existence of canonical subgroups on a strict neighborhood of the $μ$-ordinary locus (with an explicit bound). We then define the overconvergent modular forms (of classical weight), as well as the relevant Hecke operators. Finally, we show how an analytic continuation argument can be adapted to this case to prove a classicality theorem, namely that an overconvergent modular form which is an eigenform for the Hecke operators is classical under certain assump…
The paper deals with the analytic entire function Chi(s) closely related to Riemann Zeta Function Zeta(s). A formula is obtained for Chi(s) essentially within the so-called critical strip. This is achieved by applying Cauchy integral formula to an infinite loop encircling the critical strip. In the obtained formula a remarkable role is played by a special type of complex function known as Incomplete Gamma Function. Numerical examples verifying the obtained formula are included.
We introduce a symbolic representation of $r$-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber's formula, and as the result of a natural renorma…
It's well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall {\em explicitly} determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications: First we study of the limit mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple {\em poly}logarithms. Then following Deligne and Beilinson we …
In this paper we consider certain 1-parametric family of Dirichlet series. For a particular value of the parameter the series turns into the Dirichlet series for the Riemann zeta function. We prove that almost every series of the family has analytic continuation to the half plane Re s > 1/2 where it doesn't vanish. The result was obtained before by different authors. We give its simple proof in terms of estimates of some trigonometric sums.
We investigate the elliptic analogs of multi-indexed polylogarithms that appear in the theory of the fundamental group of the projective line minus three points as sections of a universal nilpotent bundle with regular singular connection. We use an analytic uniformisation to derive the fundamental nilpotent De Rham torsor of a single elliptic curve in terms of a double Jacobi form introduced by Kronecker. We then extend this result to any smooth family, relatively to the base, i.e., to the moduli stack $M_{1,2}$ over $M_{1,1}$. Everything relies on explicit formulas that turn out to be algebra…
The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in a neighborhood of zero, and we postulate their arithmetic resurgence. By the latter term, we mean analytic continuation as a multivalued analytic function in the complex numbers minus a discrete set of points, with restricted singularities, local and global monodromy. We point ou…
The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for $ζ(2)$ and $ζ(3),$ and those of the second author for Euler's constant $γ$ and its alternating analog $\ln(4/π),$ and on the other hand the infinite products of the first author for $e$, and of the second author for $π$ and $e^γ.$ We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch's transcendent of Hadjicostas's double integral formula for the Riemann zeta function, and logarithmic series for the digamma a…
We introduce a theory of probabilistic renormalization for series, the renormalized values being encoded in the expectation of a certain random variable on the set of natural numbers. We identify a large class of weakly renormalizable series of Dirichlet type, whose analysis depends on the properties of a (infinite order) difference operator that we call Bernoulli operator. For the series in this class, we show that the probabilistic renormalization is compatible with analytic continuation. The general zeta series for $s\neq 1$ is found to be strongly renormalizable and its renormalized value …
