Highly Composite Numbers | concept | 1915

Highly Composite Numbers

Highly Composite Numbers is catalogued as a concept in Highly Composite Numbers.

m < n \Rightarrow d (m) < d (n)

Formula permalink

What Ramanujan Discovered

Highly Composite Numbers belongs to the Phase 2 contribution catalogue of major Ramanujan discoveries, named results and formula families. The entry avoids claiming that all 3,900 notebook results are individually transcribed.

Catalogue Position

This entry sits in Highly Composite Numbers, one of the 37 Phase 2 contribution areas.

Reading the Entry

Formula panels show canonical notation where available. Broad families are grouped editorially rather than expanded into unsupported individual notebook entries.

How to Study This Entry

Start with the statement above, then compare its category with nearby entries in the archive. Use the formula permalink when a canonical expression is available, and follow related discoveries to see how the idea connects with partitions, q-series, modular forms, continued fractions, or number theory.

Difficulty is marked as intermediate, and the editorial status is source-supplied. Records marked for review preserve the contribution but await tighter primary-source citation.

Original Source

Published Papers. Entries marked for review await the unavailable primary markdown catalogue for exact citation detail.

Archive Companion

Glossary

Namagiri DeviLanguage

The family deity Ramanujan revered. In the site, the term marks the devotional context he associated with mathematical insight.

Faith, Namakkal, goddess tradition

KumbakonamPlaces

The Tamil Nadu temple town where Ramanujan was raised, educated, married, and later died in 1920.

Town High School, notebooks, final years

ErodePlaces

Ramanujan's birthplace in present-day Tamil Nadu. He was born there on 22 December 1887.

Birthplace

MadrasPlaces

The colonial-era name for Chennai. Ramanujan worked around Madras before his Cambridge journey and had patrons there.

University of Madras, Port Trust

CambridgePlaces

The English university city where Ramanujan worked with G. H. Hardy at Trinity College from 1914 to 1919.

Hardy, Littlewood, Royal Society

PartitionMathematics

A way of writing a positive integer as a sum of positive integers where order is not counted separately.

p(n), congruences, asymptotic formula

CongruenceMathematics

A modular arithmetic relation saying two quantities leave the same remainder after division by a chosen modulus.

Partition congruences

Mock Theta FunctionMathematics

A mysterious q-series family from Ramanujan's final work, later understood through mock modular forms.

Lost Notebook, final letter

q-SeriesMathematics

A power series in the variable q. Ramanujan used q-series throughout his work on theta functions, products, and partitions.

Rogers-Ramanujan identities

Modular FormMathematics

A highly symmetric complex analytic function. Many later proofs of Ramanujan's results use modular forms.

Tau function, eta products

Srinivasa RamanujanPeople

Indian mathematician whose notebooks, formulas, and conjectures reshaped number theory, analysis, partitions, and q-series.

1887-1920

G. H. HardyPeople

Cambridge mathematician who recognized Ramanujan's genius, invited him to England, and collaborated with him.

Hardy-Ramanujan asymptotic formula

Janaki AmmalPeople

Ramanujan's wife, whose life intersected with his poverty, illness, separation during Cambridge years, and posthumous legacy.

Biography

NotebookConcepts

Ramanujan's handwritten collections of results, often stated without proofs and later studied by many mathematicians.

Three Notebooks, Lost Notebook

Lost NotebookConcepts

A manuscript collection rediscovered after Ramanujan's death, containing important late material including mock theta functions.

George Andrews, 1976 rediscovery

YantraLanguage

A sacred geometric diagram. Here it informs a restrained design vocabulary for mathematical symmetry and order.

Sacred geometry, mandala