T. Kim
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.
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This article extends classical one variable results about Euler products defined by integral valued polynomial or analytic functions to several variables. We show there exists a meromorphic continuation up to a presumed natural boundary, and also give a criterion, a la Estermann-Dahlquist, for the existence of a meromorphic extension to $\C^n.$ Among applications we deduce analytic properties of height zeta functions for toric varieties over $\Q$ and group zeta functions.
We study the analytic behaviour of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato-Tate conjectures we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.
Explicit Analytic Continuation of Euler Products
本文介绍了一种通过因子法对欧拉乘积进行显式解析延拓的方法，并提供了相关技术的深入阐述和研究成果。
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.
In this paper, we show that Riemann hypothesis (concerning zeros of the zeta function in the critical strip) is equivalent to the analytic continuation of Euler products obtained by restricting the Euler zeta product to suitable subsets $M_k$, $k\geq 1$ of the set of prime numbers. Each of these Euler product defines so a partial zeta function $ζ_{k}(s)$ equal to a Dirichlet series of the form $\sum ε(n)/n^s$, with coefficients $ε(n)$ equal to 0 or 1 as n belongs or not to the population of integers generated by $M_k$. We show that usual formulas of the arithmetic adapt themselves to such popu…
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.
On the analytic continuation of the Riemann zeta function
通过详细分析黎曼ζ函数的解析延拓，推导出多个可能在物理和数学中有用的新恒等式。
On Furusho's analytic continuation of Drinfeld logarithms
本文利用[Fur20]中引入的技术，建立了Drinfeld对数的解析延拓，这一结果可以看作是椭圆积分第一类的解析延拓在Drinfeld模中的类比。
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.
Analytic continuation of Kochubei multiple polylogarithms and its applications
本文提出利用Furusho发展出的技术对Kochubei多重polylogarithms进行解析延拓，并从同调角度研究其在代数元素上的线性关系和线性独立性结果。
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.
Analytic continuation of $\ell$-generalized Fibonacci zeta function
本文定义ℓ-广义斐波那契zeta函数，研究其到全复平面的解析延拓，计算奇点与留数，并证明其在负整数处取有理数值。
Analytical continuation of prime zeta function for $\Re(s)>\frac{1}{2}$ assuming (RH)
本文在RH假设下推导出素数ζ函数在Re(s)>1/2区域的解析延拓表达式，并通过数值验证和图表展示结果。
On the series expansion of k-free Dirichlet series and its analytical continuation
本文将k-自由狄利克雷级数展开为具有简单极点的洛朗级数，并推导出正则部分系数的斯蒂尔杰斯型公式，同时研究了这些级数的另一种解析延拓，并用k-自由指示函数表达ζ(1/k)。
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…
Using Langlands's {\it Beyond Endoscopy} idea and analytic number theory techniques, we study the Asai L-function associated to a real quadratic field $\mathbf{K}/\Q.$ If the Asai L-function associated to an automorphic form over $\mathbf{K}$ has a pole, then the form is a base change from $\Q$. We prove this and further prove the analytic continuation of the L-function. This is one of the first examples of using a trace formula to get such information. A hope of Langlands is that general L-functions can be studied via this method.
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.
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.
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…
In the present work we show that the Dirichlet series with the Euler product having analytical continuation to the critical strip without singularities, in some natural conditions, can be approximated by partial products of Euler type in the critical strip, if the primes over which are taken the products are distributed by a suitable way. The family of such series includes many of widely used Dirichlet series as the zeta-function, Dirichlet L-functions and etc. As a consequence the analog of the Riemann Hypothesis for such series is proven.
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 …
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