Advanced Number Theory Notes #1 to #8

For the past year or so I have been writing technical memos for myself on topics in advanced number theory, i.e., topics going beyond the areas usually classified as elementary number theory (due to the fact that they exclude complex analytic techniques). These notes represent my particular take on these topics and are written in such a way that they will be easy for me to review in future if I need to use them for research work, for example. I have been doing them as notes on social media but will continue them here as blogs from now on. For easy reference, the following are links to the preceding advanced number theory notes I have made on social media. The notes I will post in this blog will follow on sequentially from these:

Advanced Number Theory Note #1: The abelian group structure of arithmetical functions under Dirichlet convolution

Advanced Number Theory Note #2: Derivation and applications of some important asymptotic formulas

Advanced Number Theory Note #3: Partial sums of Dirichlet convolutions, with some applications

Advanced Number Theory Note #4: Reformulations of the prime number theorem using Chebyshev’s functions

Advanced Number Theory Note #5: Proof of an important ‘Tauberian theorem’ and some applications

Advanced Number Theory Note #6: Reformulation of the prime number theorem using the Möbius function

Advanced Number Theory Note #7: Non-vanishing of the Dirichlet L(1, χ) function for real non-principal χ

Advanced Number Theory Note #8: Dirichlet’s theorem on primes in arithmetic progressions