- be able to recognize and prove simple properties of number theoretic functions
- know the prime counting function and related asymptotic results
- know important theorems from the theory of prime numbers
- be able to compute the continued fraction expansion of reals and of convergents
- be able to learn properties of numbers by means of their continued fraction expansion
- know the relation between continued fractions and the approximation of reals by means of rationals
- know important approximation theorems in analytic number theory
- know the concept of transcendental numbers and to be able to construct transcendental numbers
- know the principles of Diophantine approximation
- know the concepts of uniform distribution modulo 1, discrepancy and its application to numerical integration
- know important theorem’s from the theory of uniform distribution like Weyl’s criterion and Kronecker’s theorem and to be able to prove them
- know the concept of normal numbers and being aware, that almost all numbers are normal
- understand the structure of the group of prime residues, primitive roots and their applications
- know the concept of quadratic remainders and important related theorems like the law of quadratic reciprocity according to Gauß
- know the principles of elementary number theory in general integral domains and to be able to work with these principles
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Arithmetic functions, Dirichlet series, multiplicativity, prime number counting function, distribution of prime numbers, prime number theorem, Bertrand's postulate, continued fraction algorithm and convergents, periodic continued fractions and quadratic irrationalities, approximation theorems of Dirichlet and Hurwitz, Pell's equation, algebraic and transcendental numbers, Diophantine approximation, best approximation, uniform distribution modulo 1, Weyl's criterion and numerical integration, discrepancy, Kronecker's theorem, normal numbers, structure of the prime residue class group, quadratic residues, elementary number theory in general integral domains
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