Useful Links
Mathematics
Number Theory
Special Functions and Number Theory
Bernoulli Numbers
Definition and properties
Generating function of Bernoulli numbers
Recurrence relations
Connection to Faulhaber's formula
Applications in number theory
Relation to sums of powers
Role in the calculation of the Riemann zeta function at negative integers
Connection to the Euler-Maclaurin formula
Euler Numbers
Definition and distinctive properties
Alternating sum property
Generating function for Euler numbers
Bernoulli Polynomials
Definition and relation to Bernoulli numbers
Properties and functional equations
Orthogonality relations
Representation in modern software and computational use cases
Euler Numbers
Definition and properties
Generating function
Relation with secant and hyperbolic secant functions
Applications and relevance
Use in combinatorial identities
Connection with Euler's formulae for trigonometric functions
Comparison and contrast with Bernoulli numbers
Partitions and Partition Functions
Definition and basic properties
Partitions of integers
Notation and examples
Ferrers diagrams and their use in visualizing partitions
Generating functions for partitions
Euler’s product formula
Partition generating functions in relation to modular forms
Analytical properties
Partition function as a function of integers
Asymptotic formulas and growth rates
Ramanujan's Partition Congruences
Modulo results for partition function
Ramanujan's famous congruences
Applications in combinatorics and number theory
Hook-length formula in the context of the combinatorial representation of partitions
Use in partition identities and their relationship with hypergeometric series
Partition theory and its impact on other areas of mathematics
Connections with the theory of symmetric functions
Role in algebraic topology and representation theory
9. Modular Forms and Elliptic Curves
First Page
11. Historical and Philosophical Contexts