Fundamental Domains and Analytic Continuation of General Dirichlet Series
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Calculus and Differential Equations II
Calculus and Differential Equations II MATH 250 B Sequences and series Sequences and series Calculus and Differential Equations II Sequences A sequence is an infinite list of numbers, s1; s2;:::; sn;::: , indexed by integers. 1n Example 1: Find the first five terms of s = (−1)n , n 3 n ≥ 1. Example 2: Find a formula for sn, n ≥ 1, given that its first five terms are 0; 2; 6; 14; 30. Some sequences are defined recursively. For instance, sn = 2 sn−1 + 3, n > 1, with s1 = 1. If lim sn = L, where L is a number, we say that the sequence n!1 (sn) converges to L. If such a limit does not exist or if L = ±∞, one says that the sequence diverges. Sequences and series Calculus and Differential Equations II Sequences (continued) 2n Example 3: Does the sequence converge? 5n 1 Yes 2 No n 5 Example 4: Does the sequence + converge? 2 n 1 Yes 2 No sin(2n) Example 5: Does the sequence converge? n Remarks: 1 A convergent sequence is bounded, i.e. one can find two numbers M and N such that M < sn < N, for all n's. 2 If a sequence is bounded and monotone, then it converges. Sequences and series Calculus and Differential Equations II Series A series is a pair of sequences, (Sn) and (un) such that n X Sn = uk : k=1 A geometric series is of the form 2 3 n−1 k−1 Sn = a + ax + ax + ax + ··· + ax ; uk = ax 1 − xn One can show that if x 6= 1, S = a . -
Topic 7 Notes 7 Taylor and Laurent Series
Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
Zeros and Coefficients
Zeros and coefficients Alexandre Eremenko∗ 10.28.2015 Abstract Asymptotic distribution of zeros of sequences of analytic functions is studied. Let fn be a sequence of analytic functions in a region D in the complex plane. The case of entire functions, D = C will be the most important. Sup- pose that there exist positive numbers Vn → ∞, such that the subharmonic functions 1 log |fn| areuniformlyboundedfromabove (1) Vn on every compact subset of D, and for some z0 ∈ D the sequence 1 log |fn(z0)| isboundedfrombelow. (2) Vn Under these conditions, one can select a subsequence such that for this sub- sequence 1 log |fn| → u, (3) Vn where u is a subharmonic function in D, u =6 −∞. This convergence can be understood in various senses, for example, quasi-everywhere, which means at every point except a set of points of zero logarithmic capacity, and also in ′ p D (Schwartz distributions), or in Lloc, 1 ≤ p< ∞. The corresponding Riesz measures converge weakly. The limit measure, −1 which is (2π) ∆u, describes the limit distribution of zeros of fn. ∗Supported by NSF grant DMS-1361836. 1 For all these facts we refer to [9] or [10]. This setting is useful in many cases when one is interested in asymptotic distribution of zeros of sequences of analytic functions. Let us mention several situations to which our results apply. 1. Let fn be monic polynomials of degree n, and Vn = n. We denote by µn the so-called empirical measures, which are the Riesz measures of (1/n)log |fn|. The measure µn is discrete, with finitely many atoms, with an atom of mass m at every zero of fn of multiplicity m. -
Dirichlet Series
数理解析研究所講究録 1088 巻 1999 年 83-93 83 Overconvergence Phenomena For Generalized Dirichlet Series Daniele C. Struppa Department of Mathematical Sciences George Mason University Fairfax, VA 22030 $\mathrm{U}.\mathrm{S}$ . A 84 1 Introduction This paper contains the text of the talk I delivered at the symposium “Resurgent Functions and Convolution Equations”, and is an expanded version of a joint paper with T. Kawai, which will appear shortly and which will contain the full proofs of the results. The goal of our work has been to provide a new approach to the classical topic of overconvergence for Dirichlet series, by employing results in the theory of infinite order differential operators with constant coefficients. The possibility of linking infinite order differential operators with gap theorems and related subjects such as overconvergence phenomena was first suggested by Ehrenpreis in [6], but in a form which could not be brought to fruition. In this paper we show how a wide class of overconvergence phenomena can be described in terms of infinite order differential operators, and that we can provide a multi-dimensional analog for such phenomena. Let us begin by stating the problem, as it was first observed by Jentzsch, and subsequently made famous by Ostrowski (see [5]). Consider a power series $f(z)= \sum_{=n0}^{+\infty}anZ^{n}$ (1) whose circle of convergence is $\triangle(0, \rho)=\{z\in C : |z|<\rho\}$ , with $\rho$ given, therefore, by $\rho=\varliminf|a_{n}|^{-1/}n$ . Even though we know that the series given in (1) cannot converge outside of $\Delta=\Delta(0, \rho)$ , it is nevertheless possible that some subsequence of its sequence of partial sums may converge in a region overlapping with $\triangle$ . -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Power Series and Convergence Issues
POWER SERIES AND CONVERGENCE ISSUES MATH 153, SECTION 55 (VIPUL NAIK) Corresponding material in the book: Section 12.8, 12.9. What students should definitely get: The meaning of power series. The relation between Taylor series and power series, the notion of convergence, interval of convergence, and radius of convergence. The theorems for differentiation and integration of power series. Convergence at the boundary and Abel’s theorem. What students should hopefully get: How notions of power series provide an alternative interpre- tation for many of the things we have studied earlier in calculus. How results about power series combine with ideas like the root and ratio tests and facts about p-series. Executive summary Words ... P∞ k (1) The objects of interest here are power series, which are series of the form k=0 akx . Note that for power series, we start by default with k = 0. If the set of values of the index of summation is not specified, assume that it starts from 0 and goes on to ∞. The exception is when the index of summation occurs in the denominator, or some other such thing that forces us to exclude k = 0. 0 (2) Also note that x is shorthand for 1. When evaluating a power series at 0, we simply get a0. We do not actually do 00. (3) If a power series converges for c, it converges absolutely for all |x| < |c|. If a power series diverges for c, it diverges for all |x| > |c|. P k (4) Given a power series akx , the set of values where it converges is either 0, or R, or an interval that (apart from the issue of inclusion of endpoints) is symmetric about 0. -
Wei Zhang – Positivity of L-Functions and “Completion of the Square”
Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Positivity of L-functions and “Completion of square" Wei Zhang Massachusetts Institute of Technology Bristol, June 4th, 2018 1 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Outline 1 Riemann hypothesis 2 Positivity of L-functions 3 Completion of square 4 Positivity on surfaces 2 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Riemann Hypothesis (RH) Riemann zeta function 1 X 1 1 1 ζ(s) = = 1 + + + ··· ns 2s 3s n=1 Y 1 = ; s 2 ; Re(s) > 1 1 − p−s C p; primes Analytic continuation and Functional equation −s=2 Λ(s) := π Γ(s=2)ζ(s) = Λ(1 − s); s 2 C Conjecture The non-trivial zeros of the Riemann zeta function ζ(s) lie on the line 1 Re(s) = : 3 2 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Riemann Hypothesis (RH) Riemann zeta function 1 X 1 1 1 ζ(s) = = 1 + + + ··· ns 2s 3s n=1 Y 1 = ; s 2 ; Re(s) > 1 1 − p−s C p; primes Analytic continuation and Functional equation −s=2 Λ(s) := π Γ(s=2)ζ(s) = Λ(1 − s); s 2 C Conjecture The non-trivial zeros of the Riemann zeta function ζ(s) lie on the line 1 Re(s) = : 3 2 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Equivalent statements of RH Let π(X) = #fprimes numbers p ≤ Xg: Then Z X dt 1=2+ RH () π(X) − = O(X ): 2 log t Let X θ(X) = log p: p<X Then RH () jθ(X) − Xj = O(X 1=2+): 4 Riemann hypothesis Positivity of L-functions Completion of square -
Lagrange Inversion Theorem for Dirichlet Series Arxiv:1911.11133V1
Lagrange Inversion Theorem for Dirichlet series Alexey Kuznetsov ∗ November 26, 2019 Abstract We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4]. Keywords: Dirichlet series, Lagrange Inversion Theorem, convolution polynomials 2010 Mathematics Subject Classification : Primary 30B50, Secondary 11M41 1 Introduction 0 The Lagrange Inversion Theorem states that if φ(z) is analytic near z = z0 with φ (z0) 6= 0 then the function η(w), defined as the solution to φ(η(w)) = w, is analytic near w0 = φ(z0) and is represented there by the power series X bn η(w) = z + (w − w )n; (1) 0 n! 0 n≥0 where the coefficients are given by n−1 n d z − z0 bn = lim n−1 : (2) z!z0 dz φ(z) − φ(z0) 0 This result can be stated in the following simpler form. We assume that φ(z0) = z0 = 0 and φ (0) = 1. We define A(z) = z=φ(z) and check that A is analytic near z = 0 and is represented there by the power series arXiv:1911.11133v1 [math.NT] 25 Nov 2019 X n A(z) = 1 + anz : (3) n≥1 Let us define B(z) implicitly via A(zB(z)) = B(z): (4) Then the function η(z) = zB(z) solves equation η(z)=A(η(z)) = z, which is equivalent to φ(η(z)) = z. Applying Lagrange Inversion Theorem in the form (2) we obtain X an(n + 1) B(z) = zn; (5) n + 1 n≥0 ∗Dept. -
Notes on the Riemann Hypothesis Ricardo Pérez-Marco
Notes on the Riemann Hypothesis Ricardo Pérez-Marco To cite this version: Ricardo Pérez-Marco. Notes on the Riemann Hypothesis. 2018. hal-01713875 HAL Id: hal-01713875 https://hal.archives-ouvertes.fr/hal-01713875 Preprint submitted on 21 Feb 2018 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. NOTES ON THE RIEMANN HYPOTHESIS RICARDO PEREZ-MARCO´ Abstract. Our aim is to give an introduction to the Riemann Hypothesis and a panoramic view of the world of zeta and L-functions. We first review Riemann's foundational article and discuss the mathematical background of the time and his possible motivations for making his famous conjecture. We discuss some of the most relevant developments after Riemann that have contributed to a better understanding of the conjecture. Contents 1. Euler transalgebraic world. 2 2. Riemann's article. 8 2.1. Meromorphic extension. 8 2.2. Value at negative integers. 10 2.3. First proof of the functional equation. 11 2.4. Second proof of the functional equation. 12 2.5. The Riemann Hypothesis. 13 2.6. The Law of Prime Numbers. 17 3. On Riemann zeta-function after Riemann. -
Of Triangles, Gas, Price, and Men
OF TRIANGLES, GAS, PRICE, AND MEN Cédric Villani Univ. de Lyon & Institut Henri Poincaré « Mathematics in a complex world » Milano, March 1, 2013 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemannian (= non-Euclidean) geometry At each location, the units of length and angles may change Shortest path (= geodesics) are curved!! Geodesics can tend to get closer (positive curvature, fat triangles) or to get further apart (negative curvature, skinny triangles) Hyperbolic surfaces Bernhard Riemann 1826-1866 List of topics named after Bernhard Riemann From Wikipedia, the free encyclopedia Riemann singularity theorem Cauchy–Riemann equations Riemann solver Compact Riemann surface Riemann sphere Free Riemann gas Riemann–Stieltjes integral Generalized Riemann hypothesis Riemann sum Generalized Riemann integral Riemann surface Grand Riemann hypothesis Riemann theta function Riemann bilinear relations Riemann–von Mangoldt formula Riemann–Cartan geometry Riemann Xi function Riemann conditions Riemann zeta function Riemann curvature tensor Zariski–Riemann space Riemann form Riemannian bundle metric Riemann function Riemannian circle Riemann–Hilbert correspondence Riemannian cobordism Riemann–Hilbert problem Riemannian connection Riemann–Hurwitz formula Riemannian cubic polynomials Riemann hypothesis Riemannian foliation Riemann hypothesis for finite fields Riemannian geometry Riemann integral Riemannian graph Bernhard -
Interplay Between Number Theory and Analysis for Dirichlet Series
Mathematisches Forschungsinstitut Oberwolfach Report No. 51/2017 DOI: 10.4171/OWR/2017/51 Mini-Workshop: Interplay between Number Theory and Analysis for Dirichlet Series Organised by Fr´ed´eric Bayart, Aubi`ere Kaisa Matom¨aki, Turku Eero Saksman, Helsinki Kristian Seip, Trondheim 29 October – 4 November 2017 Abstract. In recent years a number of challenging research problems have crystallized in the analytic theory of Dirichlet series and its interaction with function theory in polydiscs. Their solutions appear to require unconventional combinations of expertise from harmonic, functional, and complex analysis, and especially from analytic number theory. This MFO workshop provided an ideal arena for the exchange of ideas needed to nurture further progress and to solve important problems. Mathematics Subject Classification (2010): 11xx, 30xx. Introduction by the Organisers The workshop ’Interplay between number theory and analysis for Dirichlet series’, organised by Fr´ed´eric Bayart (Clermont Universit´e), Kaisa Matom¨aki (Univer- sity of Turku), Eero Saksman (University of Helsinki) and Kristian Seip (NTNU, Trondheim) was held October 29th – November 4th, 2017. This meeting was well attended with around 25 participants coming from a number of different countries, including participants form North America. The group formed a nice blend of re- searchers with somewhat different mathematical backgrounds which resulted in a fruitful interaction. About 17 talks, of varying lengths, were delivered during the five days. The talks were given by both leading experts in the field as well as by rising young re- searchers. Given lectures dealt with e.g. connections between number theory and 3036 Oberwolfach Report 51/2017 random matrix theory, operators acting on Dirichlet series, distribution of Beurl- ing primes, growth of Lp-norms of Dirichlet polynomials, growth and density of values of the Riemann zeta on boundary of the critical strip, Rado’s criterion for kth powers, Sarnak’s and Elliot’s conjectures, and Hardy type spaces of general Dirichlet series.