Factorization Theorems for Generalized Lambert Series and Applications
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Limiting Values and Functional and Difference Equations †
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 Peer-reviewed version available at Mathematics 2020, 8, 407; doi:10.3390/math8030407 Article Limiting values and functional and difference equations † N. -L. Wang 1, P. Agarwal 2,* and S. Kanemitsu 3 1 College of Applied Mathematics and Computer Science, ShangLuo University, Shangluo, 726000 Shaanxi, P.R. China; E-mail:[email protected] 2 Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India 3 Faculty of Engrg Kyushu Inst. Tech. 1-1Sensuicho Tobata Kitakyushu 804-8555, Japan; [email protected] * Correspondence: [email protected] † Dedicated to Professor Dr. Yumiko Hironaka with great respect and friendship Version February 17, 2020 submitted to Journal Not Specified Abstract: Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function. Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zeta-function; Laurent coefficients MSC: 11F03; 01A55; 40A30; 42A16 1. Introduction There have appeared enormous amount of papers on the Laurent coefficients of a large class of zeta-, L- and special functions. -
Overpartitions and Functions from Multiplicative Number Theory
U.P.B. Sci. Bull., Series A, Vol. 83, Iss. 3, 2021 ISSN 1223-7027 OVERPARTITIONS AND FUNCTIONS FROM MULTIPLICATIVE NUMBER THEORY Mircea Merca1 Let α and β be two nonnegative integers such that β < α. For an arbi- trary sequence fa g of complex numbers, we consider the generalized Lambert series n n>1 P αn−β αn−β n 1 anq =(1 − q ) in order to investigate linear combinations of the form P > S(αk − β; n)ak, where S(k; n) is the total number of non-overlined parts equal k>1 to k in all the overpartitions of n. The general nature of the numbers an allows us to provide connections between overpartitions and functions from multiplicative number theory. Keywords: partitions, Lambert series, theta series MSC2020: 11P81 05A19. 1. Introduction A partition of a positive integer n is a sequence of positive integers whose sum is n. The order of the summands is unimportant when writing the partitions of n, but for consistency, a partition of n will be written with the summands in a nonincreasing order [2]. An overpartition of a positive integer n is a partition of n in which the first occurrence of a part of each size may be overlined or not [6]. Let p(n) denote the number of overpartitions of an integer n. For example, p(3) = 8 because there are 8 overpartitions of 3: (3); (3)¯ ; (2; 1); (2¯; 1); (2; 1)¯ ; (2¯; 1)¯ ; (1; 1; 1); (1¯; 1; 1): Since the overlined parts form a partition into distinct parts and the non-overlined parts form an ordinary partition, the generating function for overpartitions is given by 1 X (−q; q)1 p(n)qn = : (q; q) n=0 1 Here and throughout this paper, we use the following customary q-series notation: ( 1; for n = 0, (a; q)n = (1 − a)(1 − aq) ··· (1 − aqn−1); for n > 0; (a; q)1 = lim (a; q)n: n!1 Because the infinite product (a; q)1 diverges when a 6= 0 and jqj > 1, whenever (a; q)1 appears in a formula, we shall assume jqj < 1. -
Quantum Arithmetics and the Relationship Between Real and P-Adic Physics
Quantum Arithmetics and the Relationship between Real and p-Adic Physics M. Pitk¨anen Email: [email protected]. http://tgd.wippiespace.com/public_html/. November 7, 2011 Abstract This chapter suggests answers to the basic questions of the p-adicization program, which are following. 1. Is there a duality between real and p-adic physics? What is its precice mathematic formula- tion? In particular, what is the concrete map p-adic physics in long scales (in real sense) to real physics in short scales? Can one find a rigorous mathematical formulationof canonical identification induced by the map p ! 1=p in pinary expansion of p-adic number such that it is both continuous and respects symmetries. 2. What is the origin of the p-adic length scale hypothesis suggesting that primes near power of two are physically preferred? Why Mersenne primes are especially important? The answer to these questions proposed in this chapter relies on the following ideas inspired by the model of Shnoll effect. The first piece of the puzzle is the notion of quantum arithmetics formulated in non-rigorous manner already in the model of Shnoll effect. 1. Quantum arithmetics is induced by the map of primes to quantum primes by the standard formula. Quantum integer is obtained by mapping the primes in the prime decomposition of integer to quantum primes. Quantum sum is induced by the ordinary sum by requiring that also sum commutes with the quantization. 2. The construction is especially interesting if the integer defining the quantum phase is prime. One can introduce the notion of quantum rational defined as series in powers of the preferred prime defining quantum phase. -
Quantum Arithmetics and the Relationship Between Real and P-Adic Physics
CONTENTS 1 Quantum Arithmetics and the Relationship between Real and p-Adic Physics M. Pitk¨anen, June 19, 2019 Email: [email protected]. http://tgdtheory.com/public_html/. Recent postal address: Rinnekatu 2-4 A 8, 03620, Karkkila, Finland. Contents 1 Introduction 4 1.1 Overall View About Variants Of Quantum Integers . .5 1.2 Motivations For Quantum Arithmetics . .6 1.2.1 Model for Shnoll effect . .6 1.2.2 What could be the deeper mathematics behind dualities? . .6 1.2.3 Could quantum arithmetics allow a variant of canonical identification re- specting both symmetries and continuity? . .7 1.2.4 Quantum integers and preferred extremals of K¨ahleraction . .8 1.3 Correspondence Along Common Rationals And Canonical Identification: Two Man- ners To Relate Real And P-Adic Physics . .8 1.3.1 Identification along common rationals . .8 1.3.2 Canonical identification and its variants . .9 1.3.3 Can one fuse the two views about real-p-adic correspondence . .9 1.4 Brief Summary Of The General Vision . 10 1.4.1 Two options for quantum integers . 11 1.4.2 Quantum counterparts of classical groups . 11 2 Various options for Quantum Arithmetics 12 2.1 Comparing options I and II . 12 2.2 About the choice of the quantum parameter q ..................... 14 2.3 Canonical identification for quantum rationals and symmetries . 15 2.4 More about the non-uniqueness of the correspondence between p-adic integers and their quantum counterparts . 16 2.5 The three basic options for Quantum Arithmetics . 18 CONTENTS 2 3 Could Lie groups possess quantum counterparts with commutative elements exist? 18 3.1 Quantum counterparts of special linear groups . -
Generating Special Arithmetic Functions by Lambert Series Factorizations
Volume 14, Number 1, Pages 31{45 ISSN 1715-0868 GENERATING SPECIAL ARITHMETIC FUNCTIONS BY LAMBERT SERIES FACTORIZATIONS MIRCEA MERCA AND MAXIE D. SCHMIDT Abstract. In this collaborative article, we summarize the known use- ful and interesting results and formulas we have discovered so far. Sum- marizing results from two related articles by Merca and Schmidt, we arrive at so-termed Lambert series factorization theorems. We unify the matrix representations that underlie two of our separate papers, and which commonly arise in identities involving partition functions and other functions generated by Lambert series. We provide a number of properties and open problems related to the inverse matrix entries de- fined in Schmidt's article and the Euler partition function p(n) which we prove through our new results unifying the expansions of the Lambert series factorization theorems within this article. 1. Introduction 1.1. Lambert series factorization theorems. We consider recurrence relations and matrix equations related to Lambert series expansions of the form [4, x27.7] [1, x17.10] n X anq X (1.1) = b qm; jqj < 1; 1 − qn m n≥1 m≥1 + + for prescribed arithmetic functions a : Z ! C and b : Z ! C where bm = P djm ad. There are many well-known Lambert series for special arithmetic functions of the form in (1.1). Examples include the following series where µ(n) denotes the M¨obiusfunction, φ(n) denotes Euler's phi function, σα(n) denotes the generalized sum of divisors function, λ(n) denotes Liouville's function, Λ(n) denotes von Mangoldt's function, !(n) defines the number of distinct primes dividing n, and Jt(n) is Jordan's totient function for a fixed Received by the editors October 9, 2017, and in revised form January 3, 2018. -
Rapidly Converging Formulae for $\Zeta (4K\Pm 1) $
RAPIDLY CONVERGING FORMULAE FOR ζ(4k 1) ± SHUBHO BANERJEE AND BLAKE WILKERSON Abstract. We provide rapidly converging formulae for the Riemann zeta L s n − n function at odd integers using the Lambert series q(s)= n∞=1 n q /(1 q ), s = −(4k ± 1). Our main formula for ζ(4k − 1) convergesP at rate of about √15π 4π e− per term, and the formula for ζ(4k + 1), at the rate of e− per ≈ π3√15 term. For example, the first order approximation yields ζ(3) 100 + √15π 9 4 √15π 10 e− h 4 + √15 sinh( 2 )i which has an error only of order 10− . 1. Introduction In a recent paper [3] we analyzed the q 1− asymptotic behavior of a Lambert series of the type | | → ∞ nsqnx (1.1) Lq(s, x)= , s C, 1 qn ∈ n=1 X − with q < 1 and x > 0. At x = 1, the Lambert series Lq(s, x) is the generating | | function for the divisor function σs(n), the sum of the sth powers of divisors of an integer n [1]: ∞ n (1.2) σs(n) q = Lq(s, 1). n=1 X For odd positive integer s values, the Lambert series is related to the Eisentein series 4k ∞ n arXiv:1803.03291v1 [math.NT] 9 Mar 2018 (1.3) E2k(q)=1 σ2k 1(n) q , − B k − 2 n=1 X that are modular in nature [2]. In this paper we focus on the Lambert series at negative odd integer values of s = (4k 1), k = 1, 2, 3, .. -
Minsquare Factors and Maxfix Covers of Graphs
Minsquare Factors and Maxfix Covers of Graphs Nicola Apollonio1 and Andr´as Seb˝o2 1 Department of Statistics, Probability, and Applied Statistics University of Rome “La Sapienza”, Rome, Italy 2 CNRS, Leibniz-IMAG, Grenoble, France Abstract. We provide a polynomial algorithm that determines for any given undirected graph, positive integer k and various objective functions on the edges or on the degree sequences, as input, k edges that minimize the given objective function. The tractable objective functions include linear, sum of squares, etc. The source of our motivation and at the same time our main application is a subset of k vertices in a line graph, that cover as many edges as possible (maxfix cover). Besides the general al- gorithm and connections to other problems, the extension of the usual improving paths for graph factors could be interesting in itself: the ob- jects that take the role of the improving walks for b-matchings or other general factorization problems turn out to be edge-disjoint unions of pairs of alternating walks. The algorithm we suggest also works if for any sub- set of vertices upper, lower bound constraints or parity constraints are given. In particular maximum (or minimum) weight b-matchings of given size can be determined in polynomial time, combinatorially, in more than one way. 1 Introduction Let G =(V,E) be a graph that may contain loops and parallel edges, and let k>0 be an integer. The main result of this work is to provide a polynomial algorithm for finding a subgraph of cardinality k that minimizes some pregiven objective function on the edges or the degree sequences of the graph. -
On Generalizations of Lamberts Series
ON GENERALIZATIONS OF LAMBERTS SERIES TOM OSLER AND ABDUL HASSEN Abstract. The classical Lamberts series makes it possible to generate many remarkable transformations of series. These Lamberts series are all constructed from the function z/(1 − z). In this paper we show how to generalize these series by using an arbitrary function in place of z/(1 − z). Series transformations exhibiting beautiful symmetry are obtained. In addition, a double contour integral is found which represents these series. Our method is compared to a general procedure introduced by MacMahon. Keywords. 1. INTRODUCTION Let ∞ n B(z) = bnz , n=1 X then the series ∞ b L(z) = n (1.1) 1 − zn n=1 X is called a Lamberts series. These series are mentioned briefly in the classical texts by Abramowitz and Stegun [1], Bromwich [13], Crystal [16], Hardy and Wright [28], Knopp [36], MacMahon [39], Polya and Szego [42], and Titchmarsch [48]. With all the bn = 1, the series is an example of an analytic function defined inside the unit circle, which cannot be continued to a larger domain (see Titchmarsch [48]). Starting with the classical paper by Knopp [35] in 1913 and continuing to the present day, these series have entered into the theory of numbers, the theory of Weierstrasss elliptic functions, and the theory of basic hypergeometric series. Much of this research was motivated by the ideas of Ramanujan. Lambert Series also occur in the expansion of Eisenstein series, a particular kind of modular form. Agarwal [2] gives an excellent survey of these results. The series (1.1) can be transformed into another series ∞ ∞ b L(z) = n = B (zn) (1.2) 1 − zn n=1 n=1 X X and can be expanded in a Taylors series ∞ L(z) = b zn (1.3) d n=1 X Xd|n where the inner sum is over all divisors d of n. -
Extended Higher Herglotz Functions I. Functional Equations 2
EXTENDED HIGHER HERGLOTZ FUNCTIONS I. FUNCTIONAL EQUATIONS ATUL DIXIT, RAJAT GUPTA AND RAHUL KUMAR Dedicated to Professor Don Zagier on the occasion of his 70th birthday Abstract. In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F (x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F (x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study Fk,N (x), an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series. We derive two different kinds of functional equations satisfied by Fk,N (x). Radchenko and Zagier gave a 1 log(1 + tx) beautiful relation between the integral dt and F (x) and used it to evaluate this Z0 1+ t integral at various rational as well as irrational arguments. We obtain a relation between Fk,N (x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of Fk,N (x) and some generalized Lambert series are also obtained along with other supplementary results. Contents 1. Introduction 1 2. Main results 5 2.1. Functional equations for extended higher Herglotz functions Fk,N (x) 6 2.3. Asymptotics of the extended higher Herglotz functions 8 2.4. An interesting relation between a generalized polylogarithm function and the extended higher Herglotz functions 9 2.5. -
Exact Formulas for the Generalized Sum-Of-Divisors Functions)
Exact Formulas for the Generalized Sum-of-Divisors Functions Maxie D. Schmidt School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 USA [email protected] [email protected] Abstract We prove new exact formulas for the generalized sum-of-divisors functions, σα(x) := dα. The formulas for σ (x) when α C is fixed and x 1 involves a finite sum d x α ∈ ≥ over| all of the prime factors n x and terms involving the r-order harmonic number P ≤ sequences and the Ramanujan sums cd(x). The generalized harmonic number sequences correspond to the partial sums of the Riemann zeta function when r > 1 and are related to the generalized Bernoulli numbers when r 0 is integer-valued. ≤ A key part of our new expansions of the Lambert series generating functions for the generalized divisor functions is formed by taking logarithmic derivatives of the cyclotomic polynomials, Φ (q), which completely factorize the Lambert series terms (1 qn) 1 into n − − irreducible polynomials in q. We focus on the computational aspects of these exact ex- pressions, including their interplay with experimental mathematics, and comparisons of the new formulas for σα(n) and the summatory functions n x σα(n). ≤ 2010 Mathematics Subject Classification: Primary 30B50; SecondaryP 11N64, 11B83. Keywords: Divisor function; sum-of-divisors function; Lambert series; cyclotomic polynomial. Revised: April 23, 2019 arXiv:1705.03488v5 [math.NT] 19 Apr 2019 1 Introduction 1.1 Lambert series generating functions We begin our search for interesting formulas for the generalized sum-of-divisors functions, σα(n) for α C, by expanding the partial sums of the Lambert series which generate these functions in the∈ form of [5, 17.10] [13, 27.7] § § nαqn L (q) := = σ (m)qm, q < 1. -
Lambert Series and Ramanujan
Proc. Indian Aead. Sei. (Math. Sci.), Vol. 103, No. 3, December 1993, pp. 269-293. Printed in India. Lambert series and Ramanujan R P AGARWAL Department of Mathematics and Astronomy, University of Lucknow, Lucknow 226007, India MS received 28 December 1992 Abstract. Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas. Keywords. Basic hypergeometric series; Lambert series; elliptic functions; mock theta functions. 1. Introduction The topic of this article called the Lambert series is a very well-known class of series, both as analytic function and number theories. Our present interest in these series emerges out of the works of Ramanujan who has profusely and elegantly used these series in a variety of contexts in his mystic research work. A number of mathematicians, later, in an attempt to unravel the mysticism underlying some of the unproved identities of Ramanujan, have also used these series with advantage. To appreciate the effectiveness of choice of the use of these series by Ramanujan and subsequently, by other mathematicians in proving a variety of identities, one must examine how beautifully they enter into the theory of numbers, theory of Weierstrass's elliptic functions and the basic hypergeometric theory. We, now present our work starting with the definition and then the impact of Lambert series on the development of the above areas. -
Happiness Is Integral
Hapies Is Inegral But Not Rational Austin Schmid/unsplash.com Andre Bland, Zoe Cramer, Philip de Castro, Desiree Domini, Tom Edgar, Devon Johnson, Steven Klee, Joseph Koblitz, and Ranjani Sundaresan classic problem in recreational mathemat- 1970, pages 83–84) for a proof of this fact. ics involves adding the squares of the digits A number is called happy if it eventually reaches 1 of a given number to obtain a new number. and unhappy otherwise. For example, from the number 49, we obtain In this article we explore happiness in several dif- ferent numeral systems. We include exercises along AAlthough this action may seem a bit arbitrary, some- the way that we hope will enhance your experience. thing interesting happens when we do it repeatedly. The The proofs-based exercises are meant to extend the sum of the squares of the digits in 97 is ideas we present and are completely optional. We also Continuing in this way, we obtain the sequence, include a few example-driven exercises that we strongly encourage you to complete, as they will be important At this point, the sequence produces 1s forever. later in the article. The optimistic mathematician might hope that this Everyone Can Find Happiness will always happen, but alas, it doesn’t. For example, if we begin with 2, we obtain As we have seen, only some numbers are happy. Maybe this seems unfair. However, the definition of happiness depends upon the method we use to represent posi- Once we reach 4, we enter a cycle that returns to 4 tive integers: We rely heavily on the base-10 notation every eight steps.