Search ResultsShowing 1-20 of 46
-
arXiv:2505.03224 (Published 2025-05-06)
Analytic continuation of Kochubei multiple polylogarithms and its applications
Comments: 25 pagesCategories: math.NTIn the present paper, we propose an analytic continuation of Kochubei multiple polylogarithms using the techniques developed by Furusho. Moreover, we produce a family of linear relations and a linear independence result for values of our analytically continued Kochubei polylogarithms at algebraic elements from a cohomological aspect.
-
arXiv:2411.02517 (Published 2024-11-04)
The $i\varepsilon$-Prescription for String Amplitudes and Regularized Modular Integrals
Comments: 33 pages + appendicesWe study integrals appearing in one-loop amplitudes in string theory, and in particular their analytic continuation based on a string theoretic analog of the $i\varepsilon$-prescription of quantum field theory. For various zero- and two-point one-loop amplitudes of both open and closed strings, we prove that this analytic continuation is equivalent to a regularization using generalized exponential integrals. Our approach provides exact expressions in terms of the degeneracies at each mass level. For one-loop amplitudes with boundaries, our result takes the form of a linear combination of three partition functions at different temperatures depending on a variable $T_0$, yet their sum is independent of this variable. The imaginary part of the amplitudes can be read off in closed form, while the real part is amenable to numerical evaluation. While the expressions are rather different, we demonstrate agreement of our approach with the contour put forward by Eberhardt-Mizera (2023) following the Hardy-Ramanujan-Rademacher circle method.
-
arXiv:2407.20509 (Published 2024-07-30)
Interpolant of truncated multiple zeta functions
Categories: math.NTWe introduce an analytic function $\Psi(s_1,\ldots,s_r;w)$ that interpolates truncated multiple zeta functions $\zeta_N(s_1,\ldots,s_r)$. We represent this interpolant as a Mellin transform of a function $G(q_1,\ldots,q_r;w)$ and, using this expression, give the analytic continuation. Further, the harmonic product relations for $\Psi$ and $G$ are established via relevant Hopf algebra structures, and some properties of the function $G$ are provided.
-
arXiv:2305.16184 (Published 2023-05-25)
Analytic continuation of $\ell$-generalized Fibonacci zeta function
Categories: math.NTIn this paper, for any positive integer $\ell\geq2,$ we define $\ell$-generalized Fibonacci zeta function. We then study its analytic continuation to the whole complex plane $\mathbb{C}.$ Further, we compute a possible list of singularities and residues of the function at these simple poles. Moreover, we deduce that the special values of $\ell$-generalized Fibonacci zeta function at negative integer arguments are rational.
-
arXiv:2302.05857 (Published 2023-02-12)
Sums, series, and products in Diophantine approximation
Comments: 81 pagesThere is not much that can be said for all $x$ and for all $n$ about the sum \[ \sum_{k=1}^n \frac{1}{|\sin k\pi x|}. \] However, for this and similar sums, series, and products, we can establish results for almost all $x$ using the tools of continued fractions. We present in detail the appearance of these sums in the singular series for the circle method. One particular interest of the paper is the detailed proof of a striking result of Hardy and Littlewood, whose compact proof, which delicately uses analytic continuation, has not been written freshly anywhere since its original publication. This story includes various parts of late 19th century and early 20th century mathematics.
-
arXiv:2207.03822 (Published 2022-07-08)
A Conjecture of Coleman on the Eisenstein Family
Comments: 15 pagesCategories: math.NTWe prove for primes $p\ge 5$ a conjecture of Coleman on the analytic continuation of the family of modular functions $\frac{E^\ast_\kappa}{V(E^\ast_\kappa)}$ derived from the family of Eisenstein series $E^\ast_\kappa$. The precise, quantitative formulation of the conjecture involved a certain on $p$ depending constant. We show by an example that the conjecture with the constant that Coleman conjectured cannot hold in general for all primes. On the other hand, the constant that we give is also shown not to be optimal in all cases. The conjecture is motivated by its connection to certain central statements in works by Buzzard and Kilford, and by Roe, concerning the ``halo'' conjecture for the primes $2$ and $3$, respectively. We show how our results generalize those statements and comment on possible future developments.
-
arXiv:2206.09234 (Published 2022-06-18)
Analytic continuation of the Lerch zeta function
We give two results on the Lerch zeta function $\Phi(z,\,s,\,w)$. The first is to give an explicitly brief proof providing both the analytic continuation of $\Phi$ in $n$-variables $(n \in \{1,\,2,\,3\})$ to maximal domains of holomorphy in $\mathbb{C}^n$ and an extended formula for the special values of $\Phi$ at non-positive integers in the variable $s$. The second is to show that Lerch's functional equation is essentially the same as Apostol's functional equation using the first result.
-
On the Epstein zeta function and the zeros of a class of Dirichlet series
Comments: typos corrected; added corollary 4.5 and example 5.6By generalizing the classical Selberg-Chowla formula, we establish the analytic continuation and functional equation for a large class of Epstein zeta functions. This continuation is studied in order to provide new classes of theorems regarding the distribution of zeros of Dirichlet series in their critical lines and to produce a new method for the study of these problems. Due to the symmetries provided by the representation via the Selberg-Chowla formula, some generalizations of well-known formulas in analytic number theory are also deduced as examples.
-
arXiv:2102.11820 (Published 2021-02-23)
Global Parabolic Induction and Abstract Automorphicity
In arXiv:2011.03313, the author has constructed a category of abstractly automorphic representations for $\mathrm{GL}(2)$ over a function field $F$. This is a symmetric monoidal Abelian category, constructed with the goal of having the irreducible automorphic representations as its simple objects. The goal of this paper is to systematically study this category. We will prove several structural theorems about this category. We will show that it admits an adjoint pair $(r^\mathrm{aut},i^\mathrm{aut})$ of automorphic parabolic restriction and induction functors, respectively. This will allow us to show that the category of abstractly automorphic representations decomposes into cuspidal and Eisenstein components, in analogy with the Bernstein decomposition of the category of $p$-adic representations. Moreover, along the way, we will give a new perspective on the intertwining operator of $\mathrm{GL}(2)$ (and on the functional equation for Eisenstein series), as a form of self-duality of the functor of parabolic induction. We will also illustrate how the role of analytic continuation in this theory can be thought of as trivializing a twist by a certain line bundle, which corresponds to an L-function via the results of arXiv:2012.03068. If one chooses to keep the twist as a part of the theory, then one avoids the need for analytic continuation.
-
arXiv:2101.06835 (Published 2021-01-18)
The Lerch's $Φ$ Analytic Continuation
Comments: arXiv admin note: text overlap with arXiv:2012.08558Categories: math.NTWe provide alternative expressions for the Lerch $\Phi$ and the polylogarithm functions, $\Phi(e^m,k,b)$ and $\mathrm{Li}_{k}(e^{m})$, respectively, and for their partial sums, valid when $\Re{(k)}>0$. These expressions are obtained through analytic continuation of the formulae valid at the positive integers.
-
arXiv:2011.00142 (Published 2020-10-30)
Analytic continuation of multiple polylogarithms in positive characteristic
Comments: 20 pagesCategories: math.NTWe introduce a method of analytic continuation of Carlitz multiple (star) polylogarithms to the whole space and present a treatment of their branches. 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.
-
arXiv:2010.10974 (Published 2020-10-21)
Eisenstein series of weight $k \geq 2$ and integral binary quadratic forms
Comments: Ongoing work, 13 pages, no figuresCategories: math.NTIn a recent paper arXiv:2003.12354v2, Matsusaka investigated parabolic, elliptic, and hyperbolic Eisenstein series in weight $2$. He provided the analytic continuation to $s=0$ in the elliptic case, and conjectured an expression describing the same continuation in the hyperbolic case. We extend Matsusakas setting to general weight $k \geq 2$, and embed his Eisenstein series into a framework based on discriminants of integral binary quadratic forms. Lastly, we compute the Fourier expansion of our Eisenstein series in the hyperbolic case by adapting Zagiers method, and using results of Andersen, Duke.
-
arXiv:2008.13079 (Published 2020-08-30)
Probabilistic renormalization and analytic continuation
Comments: 20 pgWe 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 is given by the Riemann zeta function.
-
arXiv:2004.00332 (Published 2020-04-01)
Multiple Lucas Dirichlet series associated to additive and Dirichlet characters
Comments: 20 pagesCategories: math.NTIn this article, we obtain the analytic continuation of the multiple shifted Lucas zeta function, multiple Lucas $L$-function associated to Dirichlet characters and additive characters. We then compute a complete list of exact singularities and residues of these functions at these poles. Further, we show the rationality of the multiple Lucas $L$-functions associated with quadratic characters at negative integer arguments.
-
arXiv:1905.04193 (Published 2019-05-10)
A uniqueness property of general Dirichlet series
Let $F(s)=\sum_n a_n/\lambda_n^s$ be a general Dirichlet series which is absolutely convergent on $\Re(s)>1$. Assume that $F(s)$ has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely the degree $d_F$ and conductor $\alpha_F$. In this paper, we show that there are at most $2d_F$ general Dirichlet series with a given degree $d_F$, conductor $\alpha_F$ and residue $\rho_F$ at $s=1$. As a corollary, we get that elements in the extended Selberg class with positive Dirichlet coefficients are determined by their degree, conductor and the residue at $s=1$.
-
arXiv:1902.10911 (Published 2019-02-28)
Analytic continuation of differential operators and applications to Galois representations
Comments: 25 pagesCategories: math.NTThe main goal of this paper is to describe the effect of certain differential operators on mod $p$ Galois representations associated to automorphic forms on unitary and symplectic groups. As intermediate steps, we analytically continue the mod $p$ reduction of certain $p$-adic differential operators, defined a priori only over the ordinary locus in some of the authors' earlier work, to the whole Shimura variety associated to a unitary or symplectic group; and we explicitly describe the commutation relations between Hecke operators and our differential operators. The motivation for this investigation comes from the special case of $\mathrm{GL}_2$, where similar operators were used to study the weight part of Serre's conjecture. In that case, one has a convenient description in terms of $q$-expansions, but such a formula-driven approach is not readily accessible in our settings. So, building on ideas of Gross and Katz, we pursue an intrinsic approach that avoids a need for $q$-expansions or analogues.
-
arXiv:1809.07605 (Published 2018-09-20)
Renormalization of integrals of Eisenstein series and analytic continuation of representations
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.
-
arXiv:1806.03642 (Published 2018-06-10)
Petersson scalar products and L-functions arising from modular forms
Comments: 41 pages, 2 figuresCategories: math.NTAnalytic continuation and functional equation of a Dirichlet series constructed from two (not necessarily cuspidal) holomorphic modular forms is discussed, where either weights of the modular forms or characters are not necessarily equal to each other. The applications to quadratic forms are also given.
-
arXiv:1712.00737 (Published 2017-12-03)
Explicit formulae for averages of Goldbach representations
Comments: 16 pagesCategories: math.NTWe prove an explicit formula, analogous to the classical explicit formula for $\psi(x)$, for the Ces\`aro-Riesz mean of any order $k>0$ of the number of representations of $n$ as a sum of two primes. Our approach is based on a double Mellin transform and the analytic continuation of certain functions arising therein.
-
arXiv:1701.06608 (Published 2017-01-23)
Linear correlations of the divisor function
Comments: 49 pagesCategories: math.NTMotivated by arithmetic applications on the number of points in a bihomogeneous variety and on moments of Dirichlet $L$-functions, we provide analytic continuation for the series $\mathcal A_{\boldsymbol{a}}(s):=\sum_{n_1,\dots,n_k\geq1}\frac{d(n_1)\cdots d(n_k)}{(n_1\cdots n_k)^{s}}$ with the sum restricted to solutions of a non-trivial linear equation $a_1n_1+\cdots+a_kn_k=0$. The series $\mathcal A_{\boldsymbol{a}}(s)$ converges absolutely for $\Re(s)>1-\frac1k$ and we show it can be meromorphically continued to $\Re(s)>1-\frac 2{k+1}$ with poles at $s=1-\frac1{k-j}$ only, for $1\leq j< (k-1)/2$. As an application, we obtain an asymptotic formula with power saving error term for the number of points in the variety $a_1x_1y_1+\cdots+a_kx_ky_k=0$ in $\mathbb P^{k-1}(\mathbb Q)\times \mathbb P^{k-1}(\mathbb Q)$.