BBC TRIBUNE
  • Home
  • News
    • Politics
    • National
    • Culture
  • World
    • Law
    • Opinion
  • Business
    • Real Estate
  • Technology
  • Entertainment
    • Sports
    • Movie
    • Music
  • Lifestyle
    • Fashion
    • Travel
    • Social Media
  • Health & Fitness
    • Coronavirus
    • Parenting
  • Education
No Result
View All Result
  • Home
  • News
    • Politics
    • National
    • Culture
  • World
    • Law
    • Opinion
  • Business
    • Real Estate
  • Technology
  • Entertainment
    • Sports
    • Movie
    • Music
  • Lifestyle
    • Fashion
    • Travel
    • Social Media
  • Health & Fitness
    • Coronavirus
    • Parenting
  • Education
No Result
View All Result
BBC TRIBUNE
No Result
View All Result
Home Education

Ramanujan’s Dilemma: Finding a Closed Form for Partitions 100 Years On

admin by admin
December 15, 2022
in Education
0
Ramanujan’s Dilemma: Finding a Closed Form for Partitions 100 Years On
1.4k
SHARES
3.2k
VIEWS
Share on FacebookShare on Twitter

Dr Jonathan Kenigson, FRSA Don of Research, Athanasian Hall, Cambridge Limited

Mathematicians have frequently debated the existence of a closed-form solution to the number of partitions of an integer n into k distinct parts. If n is a positive integer, a partition of n is a way of writing n using only positive summands without regard to the order of the summands. For instance, the partitions of 3 are 1+1+1, 2+1, and 3; the partitions of 4 are 1+1+1+1, 2+1+1, 2+2, 3+1, and 4. Finding a closed form for the number of partitions of an integer n was elusive until 1937, when Rademacher provided a complicated but convergent series. Rademacher’s solution was based heavily on Hardy and Ramanujan’s work from 1918. His solution – though stunningly elegant and deeply inspired by Apollonian Geometry – is very difficult to generalize. Here is how I might propose to generate new closed forms for all k-weighted partitions of n (partitions of n into k parts) while avoiding explicitly geometric notions present in the original proof and making use of modern Algebraic Number Theory, Analytic Combinatorics, and Complex Analysis. My training lies partially in the Combinatorics of the Riemann Zeta Function, so I shall restrict my attention to advances in this paradigm made in Russia since the time of Prudnikov in the 1980s and culminating in Hurwitz generalizations of Zeta that have admitted globally convergent canonical representations since 2018.

Scholium I. Haec dicta sunt summatim affirmant. Symmetries of the upper half of the coordinate plane are essential to Rademacher’s proof. Several key elements of the proof are the behavior of modular forms on the upper-half-plane, which is in turn effectively equivalent to the study of representations of the Dedekind Eta Function in terms of modular discriminants (Rademacher 1937). This approach was featured in the Proceedings of the London Mathematical Society and was a strict generalization of the Ramanujan-Hardy paradigm.

Scholium II. Haec dicta sunt summatim affirmant. The functions in the proof can be defined in terms of differential equations on the Complex Plane. By “Differential Equation,” I connote an equation whose rate of change depends upon a given function whose behavior must be determined. Modular discriminants are, in turn, intimately related to Weierstrass (W) Elliptic functions representable by first-order nonhomogenous differential equations whose discriminant is a difference of cubes. It is instructive to consider the Elliptic Functions over the entire Complex domain modulo the lattice of Gaussian Integers (Apostol 1976). In this case, uniform convergence is guaranteed.

Scholium III. Haec dicta sunt summatim affirmant. I can, with sufficient care, write every admissible W- Function as an infinite sum of other infinite sums, the behavior of both of which is well-known. The W-Function then has a Laurent Representation on the restricted domain whose coefficients are the Complex Eisenstein Series. These series are themselves intimately related to integral representations of modular forms (Rodriguez & Kra 2012).

Scholium IV. Haec dicta sunt summatim affirmant. I can write every admissible W-function as a waveform whose behavior is defined by quantities from basic Trigonometry – the study of triangles. In fact, every W-function has two distinct invariants that are expressible as Fourier Series of weight functions of the Eisenstein Series. Because the convergence of Fourier Series is well-known, explicit values can be obtained for the coefficients. These are the k-weighted Divisor Functions multiplied by powers of the nome, which is to say, exponentials of a Complex argument (Apostol 1976), which in Complex Geometry is representable in-turn as a trigonometric expansion by Euler’s Theorem (see Moskowitz 2002 for an exploration of Euler’s Theorem).

Scholium V. Haec dicta sunt summatim affirmant. I can use Ramanujan Summation to represent the Divisor Functions as infinite series. Each of these Divisor Functions is, in turn, representable in a manner that Ramanujan would have understood in 1918 – the so-called Ramanujan Sums c_m(n). Each of these sums runs over integers comprime to m and bounded by above by m (Nicol 1962). By Fermat’s Little Theorem, these sums form a multiplicative finite field whose cardinality is the Euler Totient of m multiplied by powers of the nome. One may then consider the representation of the Divisor Functions in terms of the Ramanujan Sums (Krätzel 1981).

Scholium VI. Haec dicta sunt summatim affirmant. I can relate Ramanujan Summation to Zeta Analysis. A theorem – won after some detailed arguments from Complex Analysis – is that every k-weighted Divisor Function of a positive integer n is expressible in terms of a product of the Riemann Zeta Function evaluated at an integer value (in particular 1+m) and a Dirichlet-Like Series running over all possible values of the Ramanujan Sums multiplied pointwise by the terms of Zeta(1+m) (Krätzel 1981). Because Zeta(1+m) is always compactly representable in terms of special sequences (Harmonic, Bernoulli, Cauchy, etc.), a host of new closed forms may be afforded by the representations of the Zeta Function (Hasse 1930).

Works Consulted.

N.B. The list furnished here is by no means exhaustive. The primary objective in the selection of these sources is the degree of clarity and economy in exposition present in each.   

Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 11

Hasse, Helmut (1930). “Ein Summierungsverfahren für die Riemannsche ζ-Reihe” [A summation method for the Riemann ζ series]. Mathematische Zeitschrift (in German). 32 (1): 458–464.

Krätzel, E (1981). Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften. p. 130. (German)

Moskowitz, Martin A. (2002). A Course in Complex Analysis in One Variable. World Scientific Publishing Co. p. 7.

Nicol, C. A. (1962). “Some formulas involving Ramanujan sums”. Can. J. Math. 14: 284–286.

Rademacher, Hans (1937), “On the partition function p(n)“, Proceedings of the London Mathematical Society, Second Series, 43 (4): 241–254

Rodriguez, Rubi; Kra, Irwin; Gilman, Jane P. (2012), Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics, vol. 245, Springer, p. 12

Tags: Jonathan KenigsonRamanujanRamanujan’s Dilemma
admin

admin

Related Posts

TECH becomes the official online university of the NBA
Education

TECH becomes the official online university of the NBA

November 30, 2022
TECH leads the employability ranking with a rate of 99% in the first year for its students
Education

TECH leads the employability ranking with a rate of 99% in the first year for its students

November 29, 2022
The success of the Harvard ‘Case Method’ and how to improve knowledge using it
Education

The success of the Harvard ‘Case Method’ and how to improve knowledge using it

November 28, 2022
Next Post
Ken Ken, Owner of Great Shape Training, Is Making Moves

Ken Ken, Owner of Great Shape Training, Is Making Moves

Baka Prase tricked people on his Instagram Account for crypto fraud

Baka Prase tricked people on his Instagram Account for crypto fraud

Artist Jake Beck

Artist Jake Beck

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Follow Us

Recommended

ZEV Wealth – Good, but Controversial

ZEV Wealth – Good, but Controversial

2 years ago
Chibuike Promise Obi is fast becoming a musical sensation in Africa

Chibuike Promise Obi is fast becoming a musical sensation in Africa

2 years ago
Boris Johnson says that BBC’s Diana talk with failings should never happen again

Boris Johnson says that BBC’s Diana talk with failings should never happen again

4 years ago
cat toilet training seat

5 Reasons Why a Cat Toilet Training Seat is a Game Changer for Pet Owners

6 months ago

Instagram

    Please install/update and activate JNews Instagram plugin.

Categories

  • Business
  • Coronavirus
  • Culture
  • Education
  • Entertainment
  • Fashion
  • Health & Fitness
  • Lifestyle
  • Marketing
  • Movie
  • Music
  • Music
  • National
  • News
  • Opinion
  • Parenting
  • Politics
  • Real Estate
  • Social Media
  • Sports
  • Technology
  • Travel
  • Uncategorized
  • World

Topics

@abigailspenserhu Abigail Spenser Hu American Elite athos salome baba vanga best SEO companies in Indiana cat toilet training seat Celebrity Vacations concrete moisture meter director Franklin Livingston Engine MRO Entertainment filing a dba in arizona Franklin Livingston’s health benefits Hollywood Hollywood movie in home alcohol detox Izack Izack Artist Izack Songwriter Izack Under Izack Under New Song Jonathan Kenigson Lifestyle luxury office space Magliner Pallet Dolly Maz De Roxas nostradamus property in Nottingham UK psychic Purchasing in China Ramanujan Ramanujan’s Dilemma refrigerated trucking companies Roomates Roommates American Drama rooting compound science uk news UN Usa news vegan anniversary Vegan Journey world news
No Result
View All Result

Highlights

How Do Dermal Infusion Serums Improve Skin Hydration and Texture?

What Are the Key Features of a Zipline Harness for Safe Adventures?

Can In-Home Alcohol Detox Help You Achieve Lasting Sobriety?

How Can a Magliner Pallet Dolly Improve Your Warehouse Operations?

What Does Engine MRO (Maintenance, Repair, and Overhaul) Entail?

How Can an Intellectual Property Attorney Help Protect Your Creations?

Trending

Remedial Massage Therapy
Health & Fitness

Revitalize and Recover: Exploring Remedial Massage Therapy and Sports Injury Massage in Adelaide

by sanayakushwaha
December 16, 2024
0

When life’s daily grind or an intense workout leaves you sore, stiff, or stressed, massage therapy can...

tylenol autism lawsuit

How Does the Tylenol Autism Lawsuit Impact Families and Legal Precedents?

December 13, 2024
luxury office space

What Makes Luxury Office Space a Smart Investment for Businesses?

December 13, 2024
dermal infusion serums

How Do Dermal Infusion Serums Improve Skin Hydration and Texture?

December 12, 2024
zipline harness

What Are the Key Features of a Zipline Harness for Safe Adventures?

December 12, 2024

BBC TRIBUNE

BBC(Bulletin Box Corporation) TRIBUNE is a Professional News Platform. Here we will provide you only interesting content, which you will like very much. We’re dedicated to providing you the best of News, with a focus on dependability and News.

Follow us on social media:

Recent News

  • Revitalize and Recover: Exploring Remedial Massage Therapy and Sports Injury Massage in Adelaide
  • How Does the Tylenol Autism Lawsuit Impact Families and Legal Precedents?
  • What Makes Luxury Office Space a Smart Investment for Businesses?

Category

  • Business
  • Coronavirus
  • Culture
  • Education
  • Entertainment
  • Fashion
  • Health & Fitness
  • Lifestyle
  • Marketing
  • Movie
  • Music
  • Music
  • National
  • News
  • Opinion
  • Parenting
  • Politics
  • Real Estate
  • Social Media
  • Sports
  • Technology
  • Travel
  • Uncategorized
  • World

© 2022 BBC TRIBUNE

  • About
  • Advertise
  • Careers
  • Contact
No Result
View All Result
  • Home
  • Politics
  • News
  • Business
  • Culture
  • National
  • Sports
  • Lifestyle
  • Travel
  • Opinion

© 2022 BBC TRIBUNE