Study and research guide on Euler–Mascheroni constant

You can start with a book, for instance: Julian Havil, Gamma: exploring Euler's constant, 2009. It starts at a very elementary level, with a lot of history, but progresses to a lot of results, identities, that are useful to dive into before exploring more technical lands, such as Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, 2013.

Apparently, Stefan Krämer, Die Eulersche Konstante $\gamma$ und verwandte Zahlen. Diplomarbeit, 2005, Universität Göttingen is quite cited too, and I could not find an electronic version yet. Yet, he has a webpage on Euler's Constant $\gamma=0.577...$ Its Mathematics and History, a work (German version) with over 300 A4-pages and 1250 items of bibliography. And he says:

If you feel the project is interesting you can write to: Email: [email protected]

For some starting points for rational approximations of irrational numbers, continued fractions and Diophantine approximation:

  • some classic theorems: Liouville, Thue–Siegel–Roth, Hurwitz, Borel
  • Irrationality and transcendance
  • Continued fractions, Yann Bugeaud
  • The book suggested by @Will Jagy, Neverending Fractions, looks like a must read.

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Research

Analytic Number Theory

Euler Mascheroni Constant

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