Number Theory

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