Dirichlet Series
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Some Questions About Spaces of Dirichlet Series
Some questions about spaces of Dirichlet series. Jos´eBonet Instituto Universitario de Matem´aticaPura y Aplicada Universitat Polit`ecnicade Val`encia Eichstaett, June 2017 Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet Series P1 an A Dirichlet Series is an expression of the form n=1 ns with s 2 C. Riemann's zeta function: If s 2 C, 1 1 1 X 1 ζ(s) := 1 + + + ::: = : 2s 3s ns n=1 Euler 1737 considered it for s real: He proved ζ(2) = π2=6, and he relation with the prime numbers: −1 Y 1 ζ(s) = 1 − ; ps p the product extended to all the prime numbers p. Jos´eBonet Some questions about spaces of Dirichlet series Riemann Riemann (1826-1866) and his Memoir \Uber¨ die Anzahl der Primzahlen unter einer gegebenen Gr¨osse", in which he formulated the \Riemann Hypothesis". Jos´eBonet Some questions about spaces of Dirichlet series Riemann The series ζ(s) converges if the real part of s is greater than 1, and ζ(s) has an analytic extension to all the complex plane except s = 1, where it has a simple pole. Obtained the functional equation. s s−1 ζ(s) = 2 π sen(πs=2)Γ(1 − s)ζ(1 − s); s 2 C: Proved that there are no zeros of ζ(s) outside 0 ≤ <(s) ≤ 1, except the trivial zeros s = −2; −4; −6; :::. Jos´eBonet Some questions about spaces of Dirichlet series Riemann's Hypothesis Riemann conjectured that all the non-trivial zeros of ζ(s) are in the line <(s) = 1=2. -
Shintani's Zeta Function Is Not a Finite Sum of Euler Products
SHINTANI’S ZETA FUNCTION IS NOT A FINITE SUM OF EULER PRODUCTS FRANK THORNE Abstract. We prove that the Shintani zeta function associated to the space of binary cubic forms cannot be written as a finite sum of Euler products. Our proof also extends to several closely related Dirichlet series. This answers a question of Wright [22] in the negative. 1. Introduction In this paper, we will prove that Shintani’s zeta function is not a finite sum of Euler products. We will also prove the same for the Dirichlet series counting cubic fields. Our work is motivated by a beautiful paper of Wright [22]. To illustrate Wright’s work, we recall a classical example, namely the Dirichlet series associated to fundamental discrim- inants. We have s 1 s s ζ(s) s L(s, χ4) (1.1) D− = 1 2− +2 4− + 1 4− , 2 − · ζ(2s) − L(2s, χ4) D>0 X as well as a similar formula for negative discriminants. These formulas may look a bit messy. However, if one combines positive and negative discriminants, one has the beautiful formulas s s s ζ(s) 1 s (1.2) D − = 1 2− +2 4− = Disc(K ) , | | − · ζ(2s) 2 | v |p p [Kv:Qp] 2 X Y X≤ and L(s, χ4) arXiv:1112.1397v3 [math.NT] 9 Nov 2012 s s (1.3) sgn(D) D − = 1 4− . | | − L(2s, χ4) X These are special cases of much more general results. Wright obtains similar formulas for quadratic extensions of any global field k of characteristic not equal to 2, which are in turn n the case n = 2 of formulas for Dirichlet series parameterizing the elements of k×/(k×) . -
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. -
SOLUTIONS of the LINEAR BOLTZMANN EQUATION and SOME DIRICHLET SERIES 1. Introduction Classical Special Functions Are Used In
SOLUTIONS OF THE LINEAR BOLTZMANN EQUATION AND SOME DIRICHLET SERIES A.V. BOBYLEV(∗) AND I.M. GAMBA(∗∗) Abstract. It is shown that a broad class of generalized Dirichlet series (includ- ing the polylogarithm, related to the Riemann zeta-function) can be presented as a class of solutions of the Fourier transformed spatially homogeneous linear Boltz- mann equation with a special Maxwell-type collision kernel. The result is based on an explicit integral representation of solutions to the Cauchy problem for the Boltz- mann equation. Possible applications to the theory of Dirichlet series are briefly discussed. 1. Introduction Classical special functions are used in different fields of mathematics and its ap- plications. Most of such functions (spherical and cylindrical functions, Hermite and Laguerre polynomials and many others) appear as solutions of some ODEs or PDEs. There is, however, another big class of functions which have no connections with differential equations. The well known examples are the Gamma and Beta functions, the Riemann Zeta-function ζ(z), obtained as an analytic continuation of the Dirichlet series [6] 1 X 1 (1.1) ζ(z) = ; Re z > 1 ; nz n=1 and some similar functions. One more example is a generalization of ζ(z), the so called polylogarithm 1 X xn (1.2) Li (x) = ; jxj < 1 ; z nz n=1 also known as de Longni´ere'sfunction [3]. Hilbert conjectured in the beginning of last century that ζ(z) and other functions of the same type do not satisfy algebraic differential relations [4]. His conjectures were proved in [7, 8]. Other references and new results on differential independence of zeta-type functions can also be found, for example, in the book [6]. -
Pell Numbers
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 38, 1877 - 1884 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.35131 On Some Identities and Generating Functions for k- Pell Numbers Paula Catarino 1 Department of Mathematics School of Science and Technology University of Trás-os-Montes e Alto Douro (Vila Real – Portugal) [email protected] Copyright © 2013 Paula Catarino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We obtain the Binet’s formula for k-Pell numbers and as a consequence we get some properties for k-Pell numbers. Also we give the generating function for k-Pell sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided. Mathematics Subject Classification: 11B37, 05A15, 11B83. Keywords : Pell numbers, k-Pell numbers, Generating functions, Binet’s formula 1. Introduction Some sequences of number have been studied over several years, with emphasis on studies of well-known Fibonacci sequence (and then the Lucas sequence) that is related to the golden ratio. Many papers and research work are dedicated to Fibonacci sequence, such as the work of Hoggatt, in [15] and Vorobiov, in [13], among others and more recently we have, for example, the works of Caldwell et al . in [4], Marques in [7], and Shattuck, in [11]. Also 1 Collaborator of CIDMA and CM-UTAD (Portuguese researches centers). 1878 Paula Catarino relating with Fibonacci sequence, Falc ın et al ., in [14], consider some properties for k-Fibonacci numbers obtained from elementary matrix algebra and its identities including generating function and divisibility properties appears in the paper of Bolat et al., in [3]. -
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. -
979 KOJIMA on DOUBLE DIRICHLET SERIES the Object of This Note Is
1933-1 KOJIMA ON DIRICHLET SERIES 979 KOJIMA ON DOUBLE DIRICHLET SERIES BY C. R. ADAMS The object of this note is to direct attention to the most ex tensive and for us one of the most interesting papers by Tet- suzô Kojima, whose death on April 7, 1921, at the early age of thirty-four deprived Japan of an able and promising mathe matician. This paper, which has only just now come to our notice* and whose existence may not be generally known, is On the double Dirichlet series, (Science Reports of the Tôhoku Imperial University, (1), (math., phys., chem.), vol. 9 (1920), pp. 351-400). It is devoted mainly to the convergence properties of double Dirichlet series, certain related double integrals, double factorial series, and double binomial coefficient series. Unaware of Kojima's work, Leja and the present writerf have recently considered some of the same questions; it may there fore be appropriate to give a brief description of Kojima's mem oir and to point out the main features of its relationship to the more recent papers cited. a r A double series ^{ =i a f° which the sequence {Smn} of ai conv partial sums, 5TOn =237/—i ^ ^rges as m, w-*oo and of which each row and column converges has been called regularly convergent by Hardy. If {Smn} is both convergent and bounded, the series may be said to be convergent-bounded. One sees at once that every series regularly convergent is convergent- bounded but that the converse is not true. The notion of regular convergence is basic in Kojima's work; that of convergence- bounded in L, AI, and AIL * An obituary in the Tôhoku Mathematical Journal, vol. -
Dirichlet-Power Series -.:: Natural Sciences Publishing
J. Ana. Num. Theor. 5, No. 1, 41-48 (2017) 41 Journal of Analysis & Number Theory An International Journal http://dx.doi.org/10.18576/jant/050107 Dirichlet-Power Series Hunduma Legesse Geleta∗ Department of Mathematics, Addis Ababa University, Ethiopia Received: 2 Sep. 2016, Revised: 18 Nov. 2016, Accepted: 23 Nov. 2016 Published online: 1 Jan. 2017 Abstract: In this paper we introduce and investigate a new type of series which we call a ”Dirichlet-power” series. This series includes both Dirichlet series and power series as a special case. It arose during the investigation of series representations of the second order hypergeometric zeta function. The Dirichlet-power series has many properties analogous to Dirichlet series. For a Dirichlet-power series we establish Domain of convergence and show the existence of zero free regions to the right half plane. Keywords: Multiplicative cone b-metric space, fixed point, contractive mapping. 1 Introduction In this paper we introduce and investigate a new type of series which we call a ”Dirichlet-power series”. This The Riemann Zeta function ζ(s) defined by series includes both Dirichlet series and power series as a ∞ special case. As in the case of Dirichlet series it has a 1 kind of abscissa of absolute convergence is attached to it ∑ s n=1 n with a right half plane convergence. On the other hand as a power series it has a kind of radius of convergence with with an integral representation D-shaped region of convergence. This kind of series actually arose during the investigation of series ∞ s−1 ζ 1 x representations of the second order hypergeometric zeta (s)= x dx Γ (s) Z0 e − 1 function. -
Fundamental Domains and Analytic Continuation of General Dirichlet Series
FUNDAMENTAL DOMAINS AND ANALYTIC CONTINUATION OF GENERAL DIRICHLET SERIES Dorin Ghisa1 with an introduction by Les Ferry2 AMS subject classification: 30C35, 11M26 Abstract: Fundamental domains are found for functions defined by general Dirichlet series and by using basic properties of conformal mappings the Great Riemann Hypothesis is studied. Introduction My interest in the topic of this paper appeared when I became aware that John Derbyshare, with whom I was in the same class as an undergraduate, had written a book on the Riemann Hypothesis and that this had won plaudits for its remarkable success in representing an abstruse topic to a lay audience. On discovering this area of mathematics for myself through different other sources, I found it utterly fascinating. More than a quarter of century before, I was taken a strong interest in the work of Peitgen, Richter, Douady and Hubbard concerning Julia, Fatou and Mandelbrot sets. From this I had learned the basic truth that, however obstruse might be the method of definition, the nature of analytic function is very tightly controlled in practice. I sensed intuitively at that time the Riemann Zeta function would have to be viewed in this light if a proof of Riemann Hypothesis was to be achieved. Many mathematicians appear to require that any fundamental property of the Riemann Zeta function and in general of any L-function must be derived from the consideration of the Euler product and implicitly from some mysterious properties of prime numbers. My own view is that it is in fact the functional equation which expresses the key property with which any method of proof of the Riemann Hypothesis and its extensions must interact. -
Contents 1 Dirichlet Series and the Riemann Zeta Function
Contents 1 Dirichlet Series and The Riemann Zeta Function 1 2 Primes in Arithmetic Progressions 4 3 Functional equations for ζ(s);L(s; χ) 9 4 Product Formula for ξ(s); ξ(s; χ) 18 5 A Zero-Free Region and the Density of Zeros for ζ(s) 23 6 A Zero-Free Region and the Density of Zeros for L(s; χ) 25 7 Explicit Formula Relating the Primes to Zeros of ζ(s) 25 8 Chebyshev Estimates and the Prime Number Theorem 28 9 Prime Number Theorem in Arithmetic Progressions 33 [Sections 1,2, and 3 are OK - the rest need work/reorganization] 1 Dirichlet Series and The Riemann Zeta Function Throughout, s = σ + it is a complex variable (following Riemann). Definition. The Riemann zeta function, ζ(s), is defined by X 1 Y ζ(s) = = (1 − p−s)−1 ns n p for σ > 1. Lemma (Summation by Parts). We have q q−1 X X anbn = An(bn − bn+1) + Aqbq − Ap−1bp n=p n=p P P where An = k≤n ak. In particular, if n≥1 anbn converges and Anbn ! 0 as n ! 1 then X X anbn = An(bn − bn+1): n≥1 n≥1 P Another formulation: if a(n) is a funciton on the integers, A(x) = n≤x a(n), and f is C1 on [y; x] then X Z x a(n)f(n) = A(x)f(x) − A(y)f(y) − A(t)f 0(t)dt y<n≤x y Proof. q q q q X X X X anbn = (An − An−1)bn = Anbn − An−1bn n=p n=p n=p n=p q q−1 q−1 X X X = Anbn − Anbn+1 = An(bn − bn+1) + Aqbq − Ap−1bp: n=p n=p−1 n=p For the second forumulation, assume y is not an integer and let N = dye;M = bxc. -
Some Periodicities in the Continued Fraction Expansions of Fibonacci and Lucas Dirichlet Series
SOME PERIODICITIES IN THE CONTINUED FRACTION EXPANSIONS OF FIBONACCI AND LUCAS DIRICHLET SERIES CHRIS K. CALDWELL AND TAKAO KOMATSU P1 ¡s Abstract. In this paper we consider the Fibonacci Zeta functions ³F (s) = n=1 Fn and P1 ¡s the Lucas Zeta functions ³L(s) = n=0 Ln . The sequences fAº gº¸0 and fBº gº¸0, which Pn ¡s are derived from º=1 Fº = An=Bn, satisfy certain recurrence formulas. We examine some properties of the periodicities of An and Bn. For example, let m and k be positive m integers. If n ¸ mk, then Bn ´¥ 0¦ (mod Fk¥) (with¦ a similar result holding for An). The n P1 n power of 2 which divides Bn is 6 + i=0 3¢2i . 1. Introduction Consider the so-called Fibonacci and Lucas Zeta functions: X1 1 X1 1 ³ (s) = ; ³ (s) = : F F s L Ls n=1 n n=0 n In [13] the analytic continuation of these series is discussed. In [4] it is shown that the numbers ³F (2); ³F (4); ³F (6) ( respectively, ³L(2); ³L(4); ³L(6)) are algebraically independent, and that each of ³F (2s) ( respectively, ³L(2s)) (s = 4; 5; 6;::: ) may be written as a rational (respectively, algebraic) function of these three numbers over Q, e.g. ³ 15 1 6 5 4 3 ³F (8) ¡ ³F (4) = 256u ¡ 3456u + 2880u + 1792u v 14 378(4u + 5)2 ´ ¡ 11100u3 + 20160u2v ¡ 10125u2 + 7560uv + 3136v2 ¡ 1050v ; where u = ³F (2) and v = ³F (6). Similar results are obtained in [4] for the alternating sums à ! X1 (¡1)n+1 X1 (¡1)n+1 ³¤ (2s) := respectively, ³¤ (2s) := (s = 1; 2; 3;::: ): F F 2s L L2s n=1 n n=1 n From the main theorem in [5] it follows that for any positive distinct integers s1; s2; s3 the numbers ³F (2s1), ³F (2s2), and ³F (2s3) are algebraically independent if and only if at least one of s1; s2; s3 is even. -
In Presenting the Dissertation As a Partial Fulfillment of The
In presenting the dissertation as a partial fulfillment of the requirements for an advanced degree from the Georgia Institute of Technology, I agree that the Library of the Institute shall make it available for inspection and circulation in accordance with its regulations governing materials of this type. I agree that permission to copy from, or to publish from, this dissertation may be granted by the professor under whose direction it was written, or, in his absence, by the Dean of the Graduate Division when such copying or publication is solely for scholarly purposes and does not involve potential financial gain. It is under stood that any copying from, or publication of, this dis sertation which involves potential financial gain will not be allowed without written permission. .1 I 7/25/6Q DIRICHLET SERIES AS A GENERALIZATION OF POWER SERIES A THESIS Presented to The Faculty of the Graduate Division by Hugh Alexander Sanders In Partial Fulfillment of the Requirements for the Degree Master of Science in Applied Mathematics Georgia Institute of Technology June, 1969 DIRICHLET SERIES AS A GENERALIZATION OF POWER SERIES Approved: Chairman rs O y . "All generalizations are dangerous, even this one. —Alexandre Dumas iii ACKNOWLEDGMENTS I would like to express my grateful appreciation to Dr. James M. Osborn, my thesis advisor, for his hard work, good humor, and expert guidance in helping me complete this thesis. I wish to thank the other members of my reading committee, Professor John P. Line and Dr. James T. Wang, for their many helpful suggestions. I also want to thank Mrs.