TOWARDS the PRIME NUMBER THEOREM Contents 1. The
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Applications of the Mellin Transform in Mathematical Finance
University of Wollongong Research Online University of Wollongong Thesis Collection 2017+ University of Wollongong Thesis Collections 2018 Applications of the Mellin transform in mathematical finance Tianyu Raymond Li Follow this and additional works at: https://ro.uow.edu.au/theses1 University of Wollongong Copyright Warning You may print or download ONE copy of this document for the purpose of your own research or study. The University does not authorise you to copy, communicate or otherwise make available electronically to any other person any copyright material contained on this site. You are reminded of the following: This work is copyright. Apart from any use permitted under the Copyright Act 1968, no part of this work may be reproduced by any process, nor may any other exclusive right be exercised, without the permission of the author. Copyright owners are entitled to take legal action against persons who infringe their copyright. A reproduction of material that is protected by copyright may be a copyright infringement. A court may impose penalties and award damages in relation to offences and infringements relating to copyright material. Higher penalties may apply, and higher damages may be awarded, for offences and infringements involving the conversion of material into digital or electronic form. Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong. Recommended Citation Li, Tianyu Raymond, Applications of the Mellin transform in mathematical finance, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2018. https://ro.uow.edu.au/theses1/189 Research Online is the open access institutional repository for the University of Wollongong. -
APPENDIX. the MELLIN TRANSFORM and RELATED ANALYTIC TECHNIQUES D. Zagier 1. the Generalized Mellin Transformation the Mellin
APPENDIX. THE MELLIN TRANSFORM AND RELATED ANALYTIC TECHNIQUES D. Zagier 1. The generalized Mellin transformation The Mellin transformation is a basic tool for analyzing the behavior of many important functions in mathematics and mathematical physics, such as the zeta functions occurring in number theory and in connection with various spectral problems. We describe it first in its simplest form and then explain how this basic definition can be extended to a much wider class of functions, important for many applications. Let ϕ(t) be a function on the positive real axis t > 0 which is reasonably smooth (actually, continuous or even piecewise continuous would be enough) and decays rapidly at both 0 and , i.e., the function tAϕ(t) is bounded on R for any A R. Then the integral ∞ + ∈ ∞ ϕ(s) = ϕ(t) ts−1 dt (1) 0 converges for any complex value of s and defines a holomorphic function of s called the Mellin transform of ϕ(s). The following small table, in which α denotes a complex number and λ a positive real number, shows how ϕ(s) changes when ϕ(t) is modified in various simple ways: ϕ(λt) tαϕ(t) ϕ(tλ) ϕ(t−1) ϕ′(t) . (2) λ−sϕ(s) ϕ(s + α) λ−sϕ(λ−1s) ϕ( s) (1 s)ϕ(s 1) − − − We also mention, although we will not use it in the sequel, that the function ϕ(t) can be recovered from its Mellin transform by the inverse Mellin transformation formula 1 C+i∞ ϕ(t) = ϕ(s) t−s ds, 2πi C−i∞ where C is any real number. -
Generalizsd Finite Mellin Integral Transforms
Available online a t www.scholarsresearchlibrary.com Scholars Research Library Archives of Applied Science Research, 2012, 4 (2):1117-1134 (http://scholarsresearchlibrary.com/archive.html) ISSN 0975-508X CODEN (USA) AASRC9 Generalizsd finite mellin integral transforms S. M. Khairnar, R. M. Pise*and J. N. Salunke** Deparment of Mathematics, Maharashtra Academy of Engineering, Alandi, Pune, India * R.G.I.T. Versova, Andheri (W), Mumbai, India **Department of Mathematics, North Maharastra University, Jalgaon-India ______________________________________________________________________________ ABSTRACT Aim of this paper is to see the generalized Finite Mellin Integral Transforms in [0,a] , which is mentioned in the paper , by Derek Naylor (1963) [1] and Lokenath Debnath [2]..For f(x) in 0<x< ∞ ,the Generalized ∞ ∞ ∞ ∞ Mellin Integral Transforms are denoted by M − [ f( x ), r] = F− ()r and M + [ f( x ), r] = F+ ()r .This is new idea is given by Lokenath Debonath in[2].and Derek Naylor [1].. The Generalized Finite Mellin Integral Transforms are extension of the Mellin Integral Transform in infinite region. We see for f(x) in a a a a 0<x<a, M − [ f( x ), r] = F− ()r and M + [ f( x ), r] = F+ ()r are the generalized Finite Mellin Integral Transforms . We see the properties like linearity property , scaling property ,power property and 1 1 propositions for the functions f ( ) and (logx)f(x), the theorems like inversion theorem ,convolution x x theorem , orthogonality and shifting theorem. for these integral transforms. The new results is obtained for Weyl Fractional Transform and we see the results for derivatives differential operators ,integrals ,integral expressions and integral equations for these integral transforms. -
Extended Riemann-Liouville Type Fractional Derivative Operator with Applications
Open Math. 2017; 15: 1667–1681 Open Mathematics Research Article P. Agarwal, Juan J. Nieto*, and M.-J. Luo Extended Riemann-Liouville type fractional derivative operator with applications https://doi.org/10.1515/math-2017-0137 Received November 16, 2017; accepted November 30, 2017. Abstract: The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modied Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by dening the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented. Keywords: Gamma function, Extended beta function, Riemann-Liouville fractional derivative, Hypergeomet- ric functions, Fox H-function, Generating functions, Mellin transform, Integral representations MSC: 26A33, 33B15, 33B20, 33C05, 33C20, 33C65 1 Introduction Extensions and generalizations of some known special functions are important both from the theoretical and applied point of view. Also many extensions of fractional derivative operators have been developed and applied by many authors (see [2–6, 11, 12, 19–21] and [17, 18]). These new extensions have proved to be very useful in various elds such as physics, engineering, statistics, actuarial sciences, economics, nance, survival analysis, life testing and telecommunications. The above-mentioned applications have largely motivated our present study. The extended incomplete gamma functions constructed by using the exponential function are dened by z p α, z; p tα−1 exp t dt R p 0; p 0, R α 0 (1) t 0 ( ) = − − ( ( ) > = ( ) > ) and S ∞ p Γ α, z; p tα−1 exp t dt R p 0 (2) t z ( ) = − − ( ( ) ≥ ) with arg z π, which have been studiedS in detail by Chaudhry and Zubair (see, for example, [2] and [4]). -
Prime Number Theorem
Prime Number Theorem Bent E. Petersen Contents 1 Introduction 1 2Asymptotics 6 3 The Logarithmic Integral 9 4TheCebyˇˇ sev Functions θ(x) and ψ(x) 11 5M¨obius Inversion 14 6 The Tail of the Zeta Series 16 7 The Logarithm log ζ(s) 17 < s 8 The Zeta Function on e =1 21 9 Mellin Transforms 26 10 Sketches of the Proof of the PNT 29 10.1 Cebyˇˇ sev function method .................. 30 10.2 Modified Cebyˇˇ sev function method .............. 30 10.3 Still another Cebyˇˇ sev function method ............ 30 10.4 Yet another Cebyˇˇ sev function method ............ 31 10.5 Riemann’s method ...................... 31 10.6 Modified Riemann method .................. 32 10.7 Littlewood’s Method ..................... 32 10.8 Ikehara Tauberian Theorem ................. 32 11ProofofthePrimeNumberTheorem 33 References 39 1 Introduction Let π(x) be the number of primes p ≤ x. It was discovered empirically by Gauss about 1793 (letter to Enke in 1849, see Gauss [9], volume 2, page 444 and Goldstein [10]) and by Legendre (in 1798 according to [14]) that x π(x) ∼ . log x This statement is the prime number theorem. Actually Gauss used the equiva- lent formulation (see page 10) Z x dt π(x) ∼ . 2 log t 1 B. E. Petersen Prime Number Theorem For some discussion of Gauss’ work see Goldstein [10] and Zagier [45]. In 1850 Cebyˇˇ sev [3] proved a result far weaker than the prime number theorem — that for certain constants 0 <A1 < 1 <A2 π(x) A < <A . 1 x/log x 2 An elementary proof of Cebyˇˇ sev’s theorem is given in Andrews [1]. -
Fourier-Type Kernels and Transforms on the Real Line
FOURIER-TYPE KERNELS AND TRANSFORMS ON THE REAL LINE GOONG CHEN AND DAOWEI MA Abstract. Kernels of integral transforms of the form k(xy) are called Fourier kernels. Hardy and Titchmarsh [6] and Watson [15] studied self-reciprocal transforms with Fourier kernels on the positive half-real line R+ and obtained nice characterizing conditions and other properties. Nevertheless, the Fourier transform has an inherent complex, unitary structure and by treating it on the whole real line more interesting properties can be explored. In this paper, we characterize all unitary integral transforms on L2(R) with Fourier kernels. Characterizing conditions are obtained through a natural splitting of the kernel on R+ and R−, and the conversion to convolution integrals. A collection of concrete examples are obtained, which also include "discrete" transforms (that are not integral transforms). Furthermore, we explore algebraic structures for operator-composing integral kernels and identify a certain Abelian group associated with one of such operations. Fractional powers can also be assigned a new interpretation and obtained. 1. Introduction The Fourier transform is one of the most powerful methods and tools in mathematics (see, e.g., [3]). Its applications are especially prominent in signal processing and differential equations, but many other applications also make the Fourier transform and its variants universal elsewhere in almost all branches of science and engineering. Due to its utter significance, it is of interest to investigate the Fourier transform -
Multiplication Modulo N Along the Primorials with Its Differences And
Multiplication Modulo n Along The Primorials With Its Differences And Variations Applied To The Study Of The Distributions Of Prime Number Gaps A.K.A. Introduction To The S Model Russell Letkeman r. letkeman@ gmail. com Dedicated to my son Panha May 19, 2013 Abstract The sequence of sets of Zn on multiplication where n is a primorial gives us a surprisingly simple and elegant tool to investigate many properties of the prime numbers and their distributions through analysis of their gaps. A natural reason to study multiplication on these boundaries is a construction exists which evolves these sets from one primorial boundary to the next, via the sieve of Eratosthenes, giving us Just In Time prime sieving. To this we add a parallel study of gap sets of various lengths and their evolution all of which together informs what we call the S model. We show by construction there exists for each prime number P a local finite probability distribution and it is surprisingly well behaved. That is we show the vacuum; ie the gaps, has deep structure. We use this framework to prove conjectured distributional properties of the prime numbers by Legendre, Hardy and Littlewood and others. We also demonstrate a novel proof of the Green-Tao theorem. Furthermore we prove the Riemann hypoth- esis and show the results are perhaps surprising. We go on to use the S model to predict novel structure within the prime gaps which leads to a new Chebyshev type bias we honorifically name the Chebyshev gap bias. We also probe deeper behavior of the distribution of prime numbers via ultra long scale oscillations about the scale of numbers known as Skewes numbers∗. -
An Elementary Proof of the Prime Number Theorem
AN ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM ABHIMANYU CHOUDHARY Abstract. This paper presents an "elementary" proof of the prime number theorem, elementary in the sense that no complex analytic techniques are used. First proven by Hadamard and Valle-Poussin, the prime number the- orem states that the number of primes less than or equal to an integer x x asymptotically approaches the value ln x . Until 1949, the theorem was con- sidered too "deep" to be proven using elementary means, however Erdos and Selberg successfully proved the theorem without the use of complex analysis. My paper closely follows a modified version of their proof given by Norman Levinson in 1969. Contents 1. Arithmetic Functions 1 2. Elementary Results 2 3. Chebyshev's Functions and Asymptotic Formulae 4 4. Shapiro's Theorem 10 5. Selberg's Asymptotic Formula 12 6. Deriving the Prime Number Theory using Selberg's Identity 15 Acknowledgments 25 References 25 1. Arithmetic Functions Definition 1.1. The prime counting function denotes the number of primes not greater than x and is given by π(x), which can also be written as: X π(x) = 1 p≤x where the symbol p runs over the set of primes in increasing order. Using this notation, we state the prime number theorem, first conjectured by Legendre, as: Theorem 1.2. π(x) log x lim = 1 x!1 x Note that unless specified otherwise, log denotes the natural logarithm. 1 2 ABHIMANYU CHOUDHARY 2. Elementary Results Before proving the main result, we first introduce a number of foundational definitions and results. -
Arxiv:1707.01315V3 [Math.NT]
CORRELATIONS OF THE VON MANGOLDT AND HIGHER DIVISOR FUNCTIONS I. LONG SHIFT RANGES KAISA MATOMAKI,¨ MAKSYM RADZIWIL L, AND TERENCE TAO Abstract. We study asymptotics of sums of the form X<n≤2X Λ(n)Λ(n+h), d (n)d (n + h), Λ(n)d (n + h), and Λ(n)Λ(N n), X<n≤2X k l X<n≤2X k P n − where Λ is the von Mangoldt function, d is the kth divisor function, and N,X P P k P are large. Our main result is that the expected asymptotic for the first three sums holds for almost all h [ H,H], provided that Xσ+ε H X1−ε for 8 ∈ − ≤ ≤ some ε > 0, where σ := 33 =0.2424 ..., with an error term saving on average an arbitrary power of the logarithm over the trivial bound. This improves upon results of Mikawa, Perelli-Pintz, and Baier-Browning-Marasingha-Zhao, who ob- 1 tained statements of this form with σ replaced by 3 . We obtain an analogous result for the fourth sum for most N in an interval of the form [X,X + H] with Xσ+ε H X1−ε. Our≤ method≤ starts with a variant of an argument from a paper of Zhan, using the circle method and some oscillatory integral estimates to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials asso- ciated to “Type d3” and “Type d4” sums (as well as some other sums that are easier to treat). After applying H¨older’s inequality to the Type d3 sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. -
EQUIDISTRIBUTION ESTIMATES for the K-FOLD DIVISOR FUNCTION to LARGE MODULI
EQUIDISTRIBUTION ESTIMATES FOR THE k-FOLD DIVISOR FUNCTION TO LARGE MODULI DAVID T. NGUYEN Abstract. For each positive integer k, let τk(n) denote the k-fold divisor function, the coefficient −s k of n in the Dirichlet series for ζ(s) . For instance τ2(n) is the usual divisor function τ(n). The distribution of τk(n) in arithmetic progression is closely related to distribution of the primes. Aside from the trivial case k = 1 their distributions are only fairly well understood for k = 2 (Selberg, Linnik, Hooley) and k = 3 (Friedlander and Iwaniec; Heath-Brown; Fouvry, Kowalski, and Michel). In this note I'll survey and present some new distribution estimates for τk(n) in arithmetic progression to special moduli applicable for any k ≥ 4. This work is a refinement of the recent result of F. Wei, B. Xue, and Y. Zhang in 2016, in particular, leads to some effective estimates with power saving error terms. 1. Introduction and survey of known results Let n ≥ 1 and k ≥ 1 be integers. Let τk(n) denote the k-fold divisor function X τk(n) = 1; n1n2···nk=n where the sum runs over ordered k-tuples (n1; n2; : : : ; nk) of positive integers for which n1n2 ··· nk = −s n. Thus τk(n) is the coefficient of n in the Dirichlet series 1 k X −s ζ(s) = τk(n)n : n=1 This note is concerned with the distribution of τk(n) in arithmetic progressions to moduli d that exceed the square-root of length of the sum, in particular, provides a sharpening of the result in [53]. -
PHD THESIS a Class of Equivalent Problems Related to the Riemann
PHDTHESIS Sadegh Nazardonyavi A Class of Equivalent Problems Related to the Riemann Hypothesis Tese submetida `aFaculdade de Ci^enciasda Universidade do Porto para obten¸c~ao do grau de Doutor em Matem´atica Departamento de Matem´atica Faculdade de Ci^enciasda Universidade do Porto 2013 To My Parents Acknowledgments I would like to thank my supervisor Prof. Semyon Yakubovich for all the guidance and support he provided for me during my studies at the University of Porto. I would also like to thank Professors: Bagher Nashvadian-Bakhsh, Abdolhamid Ri- azi, Abdolrasoul Pourabbas, Jos´eFerreira Alves for their teaching and supporting me in academic stuffs. Also I would like to thank Professors: Ana Paula Dias, Jos´eMiguel Urbano, Marc Baboulin, Jos´ePeter Gothen and Augusto Ferreira for the nice courses I had with them. I thank Professors J. C. Lagarias, C. Calderon, J. Stopple, and M. Wolf for useful discussions and sending us some relevant references. My sincere thanks to Profes- sor Jean-Louis Nicolas for careful reading some parts of the manuscript, helpful comments and suggestions which rather improved the presentation of the last chap- ter. Also I sincerely would like to thank Professor Paulo Eduardo Oliveira for his kindly assistances and advices as a coordinator of this PhD program. My thanks goes to all my friends which made a friendly environment, in particular Mohammad Soufi Neyestani for what he did before I came to Portugal until now. My gratitude goes to Funda¸c~aoCalouste Gulbenkian for the financial support dur- ing my PhD. I would like to thank all people who taught me something which are useful in my life but I have not mentioned their names. -
Laplace Transform - Wikipedia, the Free Encyclopedia 01/29/2007 07:29 PM
Laplace transform - Wikipedia, the free encyclopedia 01/29/2007 07:29 PM Laplace transform From Wikipedia, the free encyclopedia In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. The Laplace transform is an important concept from the branch of mathematics called functional analysis. In actual physical systems the Laplace transform is often interpreted as a transformation from the time- domain point of view, in which inputs and outputs are understood as functions of time, to the frequency- domain point of view, where the same inputs and outputs are seen as functions of complex angular frequency, or radians per unit time. This transformation not only provides a fundamentally different way to understand the behavior of the system, but it also drastically reduces the complexity of the mathematical calculations required to analyze the system. The Laplace transform has many important applications in physics, optics, electrical engineering, control engineering, signal processing, and probability theory. The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory. The transform was discovered originally by Leonhard Euler, the prolific eighteenth-century Swiss mathematician. See also moment-generating function.