DOCSLIB.ORG

RIEMANN's FUNCTIONAL EQUATION 1. Deriving The

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
RIEMANN's FUNCTIONAL EQUATION 1. Deriving The RIEMANN’S FUNCTIONAL EQUATION KEVIN ZHU Abstract. This paper investigates the functional equation of the Riemann zeta function ζ (s). The functional equation is useful for a few reasons. It allows us to immediately conclude that the function has zeros at the negative even integers. In the process of a proof we shall follow, the analytic continuation of the zeta function also falls out as a consequence. Furthermore, having established the functional equation, we can find formulas for the derivatives of the zeta function. The functional equation has close ties with the Riemann hypothesis, playing a role in empirically searching for zeros on the critical line. 1. Deriving the Functional Equation The aim of the first paper we shall review is to present a short proof of Riemann’s functional equation, [3] Γ(s) πs (1.1) ζ (1 − s) = 2 cos ζ (s) : (2π)s 2 In fact, the method the paper undertakes is to prove the slightly more general functional equation on the Hurwitz zeta function. Riemann’s functional equation follows directly from this relation. The proof is based upon the Lipschitz summation formula, which itself is proved using Poisson summation, a technique from Fourier analysis. The meromorphic continuation of both the periodized zeta function (an- other component of the proofs) and the Hurwitz zeta function to the whole complex plane is a corollary of the results in this paper. Riemann’s original proofs for the relation, on the other hand, use either the theta function and its Mellin transform or contour integration. Comparatively, the proof we shall follow mostly gets away with manipulating infinite series and reasoning about the analytic functions that pop out. In the words of the authors, the paper is intended to provide guidance to readers “unfamiliar with the circle of ideas” related to ζ (s). Now let us set the stage for establishing the functional equation. Recall that the P1 −s Riemann zeta function is defined as the continuation of ζ (s) = n=1 n , initially defined on < (s) > 1, where the series converges absolutely by comparison to P n−p for p > 1. The Hurwitz zeta function is an extension of the Riemann zeta function, 1 RIEMANN’S FUNCTIONAL EQUATION 2 similarly defined as 1 X 1 ζ (s; a) = : (n + a)s n=0 Here we take 0 < a ≤ 1 and < (s) > 1. We have ζ (s; 1) = ζ (s). Another relevant extension of the Riemann zeta function is what we call the periodized zeta function 1 X e2πina F (s; a) = ; ns n=1 again defined initially for < (s) > 1, a 2 R. We also define the Fourier transform ^ ^ R 1 2πixm f of f to be f (m) = −∞ f (x) e dx, so that we can use Poisson summation, which says X X f (n) = f^(m) n2Z m2Z for “sufficiently nice” f. The specifics of this are not terribly important to the result we want. Poisson summation is the main tool in the first theorem we state, the Lipschitz summation formula. Theorem 1. 1 2πiαm X s−1 Γ(s) X e (1.2) (n − α) e2πiτ(n−α) = ; (−2πi)s (τ + m)s n=1 m2Z where < (s) > 1, = (τ) > 0, and 0 ≤ α < 1. The proof is to use the Poisson summation formula on the left-hand side of (1.2), defining the summand function f (x) = (x − α)s−1 e2πiτ(x−α) to be 0 for x ≤ α. This function is nice enough to use Poisson summation on, and we end up with Z 1 2πimα Z X 2πimα s−1 2πi(τ+m)x 1 X e s−1 −y e x e dx = s s y e dy; 0 (−2πi) (τ + m) R m2Z m2Z where R is the complex ray fy : y = −2πi (m + τ) x and x ≥ 0g. Then we employ a familiar integration technique using Cauchy’s integral theorem. Consider the integral along a piece of pie formed by going along the real axis for some radius, then following a circular arc counterclockwise about the origin, then coming back to the origin along R. Since ys−1e−y is analytic, this integral is 0, and as the radius grows, the integral along the arc decays to 0. Therefore, the improper integral may be taken along the real axis, and so the theorem follows by the definition of Γ(s). The next steps involve some results that are mostly bookkeeping to justify the steps we will take in proving the relation on the Hurwitz zeta function. The results are not terribly illuminating, but we state them for use in that proof. They are mainly about being able to interchange limiting operations. RIEMANN’S FUNCTIONAL EQUATION 3 Lemma. (a) Suppose 0 ≤ a < 1 and < (s) < 0. Then 1 1 X s−1 X s−1 lim (n − a) e2πinτ = (n − a) = ζ (1 − s; 1 − a) : τ!0 n=1 n=1 (b) Let 0 ≤ a < 1 and y > 0. Write X 2πiam −s −s −s−1 Sy (s) = e (m + iy) − m + siym ; m6=0 where the sum is taken over all nonzero integers and for the complex powers we use the principle branch of the argument, with range (−π; π]. Then: (i) Sy (s) converges absolutely for < (s) > −1; (ii) Sy (s) is holomorphic in s for < (s) > −1; (iii) if < (s) > −1, then limy!0+ Sy (s) = 0. With these in hand, we can prove the Hurwitz relation, from which Riemann’s functional equation will follow. Theorem 2. e−πis=2F (s; a) + eπis=2F (s; −a) can be continued analytically into < (s) > −1. When −1 < < (s) < 0 and 0 < a ≤ 1, we have the Hurwitz relation Γ(s) (1.3) ζ (1 − s; a) = e−πis=2F (s; a) + eπis=2F (s; −a) : (2π)s Proof. In (1.2), there is a (τ + m)−s factor. The idea is to subtract the first two terms of the binomial expansion of this factor, m−s + τsm−s−1. Moving the Γ(s) = (−2πi)s factor in (1.2) to the other side and then subtracting from (τ + m)−s for nonzero indices of the sum, we get 1 X −s + e2πiαm (τ + m) − m−s + τsm−s−1 τ s m6=0 s 1 2πiαm 2πiαm (−2πi) X s−1 X e X e = (n − α) e2πiτ(n−α) − + τs : Γ(s) ms ms+1 n=1 m6=0 m6=0 for 0 ≤ α < 1 and < (s) > 1. We note that 1 1 X e2πiαm X e2πiαm X e2πiα(−m) = + ms ms (−m)s m6=0 m=1 m=1 1 −2πiαm −s X e = F (s; α) + (−1) ms m=1 = F (s; α) + e−πisF (s; −α) RIEMANN’S FUNCTIONAL EQUATION 4 and take τ = iy, y > 0 to rewrite what we have as 1 X −s (1.4) + e2πiαm (m + iy) − m−s + siym−s−1 (iy)s m6=0 s 1 (−2πi) X s−1 = (n − α) e−2πy(n−α) Γ(s) n=1 − F (s; α) + e−πisF (s; −α) + iys F (s + 1; α) − e−πisF (s + 1; −α) : P1 s−1 −2πy(n−α) The sum n=1 (n − α) e is entire in s the exponential term causes s the series to converge normally on C. If we move the (−2πi) =Γ(s) factor back to the left-hand side, the Lemma tells us that the left-hand side is meromorphic in < (s) > −1 with at most a simple pole at s = 0 from Γ(s). This tells us that the remaining terms Γ(s) iysΓ(s) − F (s; α) + e−πisF (s; −α)+ F (s + 1; α) − e−πisF (s + 1; −α) (−2πi)s (−2πi)s are meromorphic in < (s) > −1 with at most a simple pole at s = 0. Since y > 0 is arbitrary, we find that Γ(s) − F (s; α) + e−πisF (s; −α) (−2πi)s is meromorphic in < (s) > −1 with at most a simple pole at s = 0 by letting y ! 0+, since the other term vanishes uniformly on compact sets. This establishes the first part of the theorem. Now for −1 < < (s) < 0, let y ! 0+ in (1.4). The left-hand side vanishes by the Lemma and the restriction of < (s) and the sum on the right-hand side tends to ζ (1 − s; 1 − α). Hence, for 0 ≤ α < 1 we have Γ(s) ζ (1 − s; 1 − a) − F (s; α) + e−πisF (s; −α) = 0: (−2πi)s in −1 < < (s) < 0. Letting a = 1 − α gives (1.3) since F (s; 1 − a) = F (s; −a) and F (s; −1 + a) = F (s; a). Since F (s; 1) = F (s; −1) = ζ (s), 1.3 yields Γ(s) Γ(s) πs ζ (1 − s) = e−πis=2 + eπis=2 ζ (s) = 2 cos ζ (s) ; (2π)s (2π)s 2 which is the functional equation (1.1) that we want. 2. Using the Functional Equation [3] goes on to show that we can analytically continue ζ (s; a) and F (s; a) using the work done so far. The first supplementary result is that F (0; −a) + F (0; a) = −1, which we obtain by taking s ! 0− in 1.4 and using the Lemma. We also note that RIEMANN’S FUNCTIONAL EQUATION 5 lims!0− F (s; a) is finite for this to work, because a bit of manipulation shows that πis −πis πis −πis lims!0− e − e F (s; a) = 0. Since e − e has a simple zero at s = 0, the order of F (s; a) at s = 0 cannot be negative (because the order of the product is the sum of the orders); i.e.
Load more

Recommended publications
  • On the Stability of a Cauchy Type Functional Equation
    Demonstr. Math. 2018; 51:323–331 Research Article Open Access Abbas Najati, Jung Rye Lee*, Choonkil Park, and Themistocles M. Rassias On the stability of a Cauchy type functional equation https://doi.org/10.1515/dema-2018-0026 Received May 31, 2018; accepted November 11, 2018 Abstract: In this work, the Hyers-Ulam type stability and the hyperstability of the functional equation (︁ x + y )︁ (︁ x − y )︁ f + xy + f − xy = f(x) 2 2 are proved. Keywords: Hyers-Ulam stability, additive mapping, hyperstability, topological vector space MSC: 39B82, 34K20, 26D10 1 Introduction The functional equation (ξ) is called stable if any function g satisfying the equation (ξ) approximately, is near to a true solution of (ξ). Ulam, in 1940 [1], introduced the stability of homomorphisms between two groups. More precisely, he proposed the following problem: given a group (G1,.), a metric group (G2, *, d) and a positive number ϵ, does there exist a δ > 0 such that if a function f : G1 ! G2 satisfies the inequal- ity d(f (x.y), f (x) * f(y)) < δ for all x, y 2 G1, then there exists a homomorphism T : G1 ! G2 such that d(f(x), T(x)) < ϵ for all x 2 G1? If this problem has a solution, we say that the homomorphisms from G1 to G2 are stable. In 1941, Hyers [2] gave a partial solution of Ulam’s problem for the case of approximate addi- tive mappings under the assumption that G1 and G2 are Banach spaces. Aoki [3] and Rassias [4] provided a generalization of the Hyers’ theorem for additive and linear mappings, respectively, by allowing the Cauchy difference to be unbounded.
    [Show full text]
  • Lectures on Modular Forms. Fall 1997/98
    Lectures on Modular Forms. Fall 1997/98 Igor V. Dolgachev October 26, 2017 ii Contents 1 Binary Quadratic Forms1 2 Complex Tori 13 3 Theta Functions 25 4 Theta Constants 43 5 Transformations of Theta Functions 53 6 Modular Forms 63 7 The Algebra of Modular Forms 83 8 The Modular Curve 97 9 Absolute Invariant and Cross-Ratio 115 10 The Modular Equation 121 11 Hecke Operators 133 12 Dirichlet Series 147 13 The Shimura-Tanyama-Weil Conjecture 159 iii iv CONTENTS Lecture 1 Binary Quadratic Forms 1.1 The theory of modular form originates from the work of Carl Friedrich Gauss of 1831 in which he gave a geometrical interpretation of some basic no- tions of number theory. Let us start with choosing two non-proportional vectors v = (v1; v2) and w = 2 (w1; w2) in R The set of vectors 2 Λ = Zv + Zw := fm1v + m2w 2 R j m1; m2 2 Zg forms a lattice in R2, i.e., a free subgroup of rank 2 of the additive group of the vector space R2. We picture it as follows: • • • • • • •Gv • ••• •• • w • • • • • • • • Figure 1.1: Lattice in R2 1 2 LECTURE 1. BINARY QUADRATIC FORMS Let v v B(v; w) = 1 2 w1 w2 and v · v v · w G(v; w) = = B(v; w) · tB(v; w): v · w w · w be the Gram matrix of (v; w). The area A(v; w) of the parallelogram formed by the vectors v and w is given by the formula v · v v · w A(v; w)2 = det G(v; w) = (det B(v; w))2 = det : v · w w · w Let x = mv + nw 2 Λ.
    [Show full text]
  • Abelian Solutions of the Soliton Equations and Riemann–Schottky Problems
    Russian Math. Surveys 63:6 1011–1022 c 2008 RAS(DoM) and LMS Uspekhi Mat. Nauk 63:6 19–30 DOI 10.1070/RM2008v063n06ABEH004576 Abelian solutions of the soliton equations and Riemann–Schottky problems I. M. Krichever Abstract. The present article is an exposition of the author’s talk at the conference dedicated to the 70th birthday of S. P. Novikov. The talk con- tained the proof of Welters’ conjecture which proposes a solution of the clas- sical Riemann–Schottky problem of characterizing the Jacobians of smooth algebraic curves in terms of the existence of a trisecant of the associated Kummer variety, and a solution of another classical problem of algebraic geometry, that of characterizing the Prym varieties of unramified covers. Contents 1. Introduction 1011 2. Welters’ trisecant conjecture 1014 3. The problem of characterization of Prym varieties 1017 4. Abelian solutions of the soliton equations 1018 Bibliography 1020 1. Introduction The famous Novikov conjecture which asserts that the Jacobians of smooth alge- braic curves are precisely those indecomposable principally polarized Abelian vari- eties whose theta-functions provide explicit solutions of the Kadomtsev–Petviashvili (KP) equation, fundamentally changed the relations between the classical algebraic geometry of Riemann surfaces and the theory of soliton equations. It turns out that the finite-gap, or algebro-geometric, theory of integration of non-linear equa- tions developed in the mid-1970s can provide a powerful tool for approaching the fundamental problems of the geometry of Abelian varieties. The basic tool of the general construction proposed by the author [1], [2]which g+k 1 establishes a correspondence between algebro-geometric data Γ,Pα,zα,S − (Γ) and solutions of some soliton equation, is the notion of Baker–Akhiezer{ function.} Here Γis a smooth algebraic curve of genus g with marked points Pα, in whose g+k 1 neighborhoods we fix local coordinates zα, and S − (Γ) is a symmetric prod- uct of the curve.
    [Show full text]
  • Mirror Symmetry of Abelian Variety and Multi Theta Functions
    1 Mirror symmetry of Abelian variety and Multi Theta functions by Kenji FUKAYA （深谷賢治） Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto Japan Table of contents § 0 Introduction. § 1 Moduli spaces of Lagrangian submanifolds and construction of a mirror torus. § 2 Construction of a sheaf from an affine Lagrangian submanifold. § 3 Sheaf cohomology and Floer cohomology 1 (Construction of a homomorphism). § 4 Isogeny. § 5 Sheaf cohomology and Floer cohomology 2 (Proof of isomorphism). § 6 Extension and Floer cohomology 1 (0 th cohomology). § 7 Moduli space of holomorphic vector bundles on a mirror torus. § 8 Nontransversal or disconnected Lagrangian submanifolds. ∞ § 9 Multi Theta series 1 (Definition and A formulae.) § 10 Multi Theta series 2 (Calculation of the coefficients.) § 11 Extension and Floer cohomology 2 (Higher cohomology). § 12 Resolution and Lagrangian surgery. 2 § 0 Introduction In this paper, we study mirror symmetry of complex and symplectic tori as an example of homological mirror symmetry conjecture of Kontsevich [24], [25] between symplectic and complex manifolds. We discussed mirror symmetry of tori in [12] emphasizing its “noncom- mutative” generalization. In this paper, we concentrate on the case of a commutative (usual) torus. Our result is a generalization of one by Polishchuk and Zaslow [42], [41], who studied the case of elliptic curve. The main results of this paper establish a dictionary of mirror symmetry between symplectic geometry and complex geometry in the case of tori of arbitrary dimension. We wrote this dictionary in the introduction of [12]. We present the argument in a way so that it suggests a possibility of its generalization.
    [Show full text]
  • Applied Mathematics Letters Exponential Functional Equation On
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 23 (2010) 156–160 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Exponential functional equation on spheres Justyna Sikorska Institute of Mathematics, Silesian University, Bankowa 14, PL-40-007 Katowice, Poland article info a b s t r a c t Article history: We study the exponential functional equation f .x C y/ D f .x/f .y/ on spheres in real Received 28 February 2009 normed linear spaces. Regardless of the solutions of this equation, which are already Received in revised form 4 September 2009 known, we investigate its stability and consider the pexiderized version of it. Accepted 4 September 2009 ' 2009 Elsevier Ltd. All rights reserved. Keywords: Conditional exponential functional equation Stability Approximation 1. Introduction Alsina and Garcia-Roig [1] considered the conditional functional equation f .x C y/ D f .x/ C f .y/ whenever kxk D kyk (1) with a continuous function f : X ! Y mapping a real inner product space .X; h·; ·i/ with dim X ≥ 2 into a real topological vector space Y . In [2], Szabó solved (1) in case where .X; k · k/ is a real normed linear space with dim X ≥ 3 and .Y ; C/ is an Abelian group. In [3], Ger and the author proceeded with the study of (1) with the norm replaced by an abstract function fulfilling suitable conditions. We dealt also with more general structures than those considered in [1,2].
    [Show full text]
  • Theta Function Review G = 1 Case
    The genus 1 case - review Theta Function Review g = 1 case We recall the main de¯nitions of theta functions in the 1-dim'l case: De¯nition 0 Let· ¿ 2¸C such that Im¿ > 0: For "; " real numbers and z 2 C then: " £ (z;¿) = "0 n ¡ ¢ ¡ ¢ ¡ ¢ ³ ´o P 1 " " ² t ²0 l²Z2 exp2¼i 2 l + 2 ¿ l + 2 + l + 2 z + 2 The series· is uniformly¸ and absolutely convergent on compact subsets " C £ H: are called Theta characteristics "0 The genus 1 case - review Theta Function Properties Review for g = 1 case the following properties of theta functions can be obtained by manipulation of the series : · ¸ · ¸ " + 2m " 1. £ (z;¿) = exp¼i f"eg £ (z;¿) and e; m 2 Z "0 + 2e "0 · ¸ · ¸ " " 2. £ (z;¿) = £ (¡z;¿) ¡"0 "0 · ¸ " 3. £ (z + n + m¿; ¿) = "0 n o · ¸ t t 0 " exp 2¼i n "¡m " ¡ mz ¡ m2¿ £ (z;¿) 2 "0 The genus 1 case - review Remarks on the properties of Theta functions g=1 1. Property number 3 describes the transformation properties of theta functions under an element of the lattice L¿ generated by f1;¿g. 2. The same property implies that that the zeros of theta functions are well de¯ned on the torus given· ¸ by C=L¿ : In fact there is only a " unique such 0 for each £ (z;¿): "0 · ¸ · ¸ "i γj 3. Let 0 ; i = 1:::k and 0 ; j = 1:::l and "i γj 2 3 Q "i k θ4 5(z;¿) ³P ´ ³P ´ i=1 "0 k 0 l 0 2 i 3 "i + " ¿ ¡ γj + γ ¿ 2 L¿ Then i=1 i j=1 j Q l 4 γj 5 j=1 θ 0 (z;¿) γj is a meromorphic function on the elliptic curve de¯ned by C=L¿ : The genus 1 case - review Analytic vs.
    [Show full text]
  • Theta Function Identities
    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 147, 97-121 (1990) Theta Function Identities RONALD J. EVANS Deparlment of Mathematics, University of California, San Diego, La Jolla, California 92093 Submitted by Bruce C. Berndt Received June 3. 1988 1. INTR~D~JcTI~N By 1986, all but one of the identities in the 21 chapters of Ramanujan’s Second Notebook [lo] had been proved; see Berndt’s books [Z-4]. The remaining identity, which we will prove in Theorem 5.1 below, is [ 10, Chap. 20, Entry 8(i)] 1 1 1 V2(Z/P) (1.1) G,(z) G&l + G&J G&J + G&l G,(z) = 4 + dz) ’ where q(z) is the classical eta function given by (2.5) and 2 f( _ q2miP, - q1 - WP) G,(z) = G,,,(z) = (- 1)” qm(3m-p)‘(2p) f(-qm,p, -q,-m,p) 9 (1.2) with q = exp(2niz), p = 13, and Cl(k2+k)/2 (k2pkV2 B . (1.3) k=--13 The author is grateful to Bruce Berndt for bringing (1.1) to his attention. The quotients G,(z) in (1.2) for odd p have been the subject of interest- ing investigations by Ramanujan and others. Ramanujan [ 11, p. 2071 explicitly wrote down a version of the famous quintuple product identity, f(-s’, +)J-(-~*q3, -w?+qfF~, -A2q9) (1.4) f(-43 -Q2) f(-Aq3, -Pq6) ’ which yields as a special case a formula for q(z) G,(z) as a linear combina- tion of two theta functions; see (1.7). In Chapter 16 of his Second 97 0022-247X/90 $3.00 Copyright % 1993 by Academc Press, Inc.
    [Show full text]
  • An Addition Formula for the Jacobian Theta Function and Its Applications
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Advances in Mathematics 212 (2007) 389–406 www.elsevier.com/locate/aim An addition formula for the Jacobian theta function and its applications Zhi-Guo Liu 1 East China Normal University, Department of Mathematics, Shanghai 200062, People’s Republic of China Received 13 July 2005; accepted 17 October 2006 Available online 28 November 2006 Communicated by Alain Lascoux Abstract In this paper, we prove an addition formula for the Jacobian theta function using the theory of elliptic functions. It turns out to be a fundamental identity in the theory of theta functions and elliptic function, and unifies many important results about theta functions and elliptic functions. From this identity we can derive the Ramanujan cubic theta function identity, Winquist’s identity, a theta function identities with five parameters, and many other interesting theta function identities; and all of which are as striking as Winquist’s identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma function. A new identity about the Ramanujan cubic elliptic function is given. The proofs are self contained and elementary. © 2006 Elsevier Inc. All rights reserved. MSC: 11F11; 11F12; 11F27; 33E05 Keywords: Weierstrass sigma function; Addition formula; Elliptic functions; Jacobi’s theta function; Winquist’s identity; Ramanujan’s cubic theta function theory E-mail addresses: [email protected], [email protected]. 1 The author was supported in part by the National Science Foundation of China. 0001-8708/$ – see front matter © 2006 Elsevier Inc.
    [Show full text]
  • Square Root Singularities of Infinite Systems of Functional Equations Johannes F
    Square root singularities of infinite systems of functional equations Johannes F. Morgenbesser To cite this version: Johannes F. Morgenbesser. Square root singularities of infinite systems of functional equations. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’10), 2010, Vienna, Austria. pp.513-526. hal-01185580 HAL Id: hal-01185580 https://hal.inria.fr/hal-01185580 Submitted on 20 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. AofA’10 DMTCS proc. AM, 2010, 513–526 Square root singularities of infinite systems of functional equations Johannes F. Morgenbesser† Institut f¨ur Diskrete Mathematik und Geometrie, Technische Universit¨at Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria Infinite systems of equations appear naturally in combinatorial counting problems. Formally, we consider functional equations of the form y(x) = F (x, y(x)), where F (x, y) : C × ℓp → ℓp is a positive and nonlinear function, and analyze the behavior of the solution y(x) at the boundary of the domain of convergence. In contrast to the finite dimensional case different types of singularities are possible. We show that if the Jacobian operator of the function F is compact, then the occurring singularities are of square root type, as it is in the finite dimensional setting.
    [Show full text]
  • Transformation Groups for Soliton Equations V —
    Publ. RIMS, Kyoto Univ. 18 (1982), 1111-1119 Quasi- Periodic Solutions of the Orthogonal KP Equation — Transformation Groups for Soliton Equations V — By Etsuro DATE*, Michio JIMBO?, Masaki KASHIWARAI and Tetsuji MiWAf § 0. Introduction In this note we study quasi-periodic solutions of the BKP hierarchy in- troduced in [1]. Our main result is the Theorem in Section 2, which states that quasi-periodic T-functions for the BKP hierarchy are the theta functions on the Prym varieties of algebraic curves admitting involutions with two fixed points. The rational and soliton solutions of the BKP hierarchy were studied in part IV [2] together with its operator formalism. We also showed that the BKP hierarchy is the compatibility condition for the following system of linear equations for w(x), x = (xl9 x3, .T5,...)^ (1) ^7=B'W' '='.3,5,... where 5, is a linear ordinary differential operator with respect to XL without constant term. dl l~2 dm One of the specific properties of the BKP hierarchy is the fact that squares of T-functions for the BKP hierarchy are T-functions for the KP hierarchy with x2j- = 0. Now we explain why the Prym varieties and the theta functions on them ap- pear in our present study. Received November 20, 1981. Faculty of General Education, Kyoto University, Kyoto 606, Japan. Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan. 1112 ETSURO DATE, MICHIO JIMBO, MASAKI KASHIWARA AND TETSUJI MIWA One derivation is given through the examination of the geometrical prop- erties of the wave functions associated with soliton solutions.
    [Show full text]
  • 2D Theta Functions and Crystallization Among Bravais Lattices Laurent Bétermin
    2D Theta Functions and Crystallization among Bravais Lattices Laurent Bétermin To cite this version: Laurent Bétermin. 2D Theta Functions and Crystallization among Bravais Lattices. 2015. hal- 01116262v2 HAL Id: hal-01116262 https://hal.archives-ouvertes.fr/hal-01116262v2 Preprint submitted on 9 Apr 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial| 4.0 International License 2D Theta Functions and Crystallization among Bravais Lattices Laurent B´etermin∗ Universit´eParis-Est Cr´eteil LAMA - CNRS UMR 8050 61, Avenue du G´en´eral de Gaulle, 94010 Cr´eteil, France April 9, 2015 Abstract In this paper, we study minimization problems among Bravais lattices for finite energy per point. We prove – as claimed by Cohn and Kumar – that if a function is completely monotonic, then the triangular lattice minimizes energy per particle among Bravais lattices with density fixed for any density. Furthermore we give an example of convex decreasing positive potential for which triangular lattice is not a minimizer for some densities. We use the Montgomery method presented in our previous work to prove minimality of triangular lattice among Bravais lattices at high density for some general potentials.
    [Show full text]
  • The Riemann Zeta Function and Its Functional Equation (And a Review of the Gamma Function and Poisson Summation)
    Math 259: Introduction to Analytic Number Theory The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler's identity: 1 1 1 s 0 cps1 [ζ(s) :=] n− = p− = s : (1) X Y X Y 1 p− n=1 p prime @cp=1 A p prime − We showed that this holds as an identity between absolutely convergent sums and products for real s > 1. Riemann's insight was to consider (1) as an identity between functions of a complex variable s. We follow the curious but nearly universal convention of writing the real and imaginary parts of s as σ and t, so s = σ + it: s σ We already observed that for all real n > 0 we have n− = n− , because j j s σ it log n n− = exp( s log n) = n− e − and eit log n has absolute value 1; and that both sides of (1) converge absolutely in the half-plane σ > 1, and are equal there either by analytic continuation from the real ray t = 0 or by the same proof we used for the real case. Riemann showed that the function ζ(s) extends from that half-plane to a meromorphic function on all of C (the \Riemann zeta function"), analytic except for a simple pole at s = 1. The continuation to σ > 0 is readily obtained from our formula n+1 n+1 1 1 s s 1 s s ζ(s) = n− Z x− dx = Z (n− x− ) dx; − s 1 X − X − − n=1 n n=1 n since for x [n; n + 1] (n 1) and σ > 0 we have 2 ≥ x s s 1 s 1 σ n− x− = s Z y− − dy s n− − j − j ≤ j j n so the formula for ζ(s) (1=(s 1)) is a sum of analytic functions converging absolutely in compact subsets− of− σ + it : σ > 0 and thus gives an analytic function there.
    [Show full text]