Koecher studied mathematics and physics at the Georg-August-Universität in Göttingen. In 1951, he received his doctorate under Max Deuring with his work (Koecher 1953) on Dirichlet series with functional equation where he introduced Koecher–Maass series. He qualified in 1954 at the Westfälische Wilhelms University in Münster. From 1962 to 1970, Koecher was department chair at the University of Munich. He retired in 1989.

His main research area was the theory of Jordan algebras, where he introduced the Kantor–Koecher–Tits construction and the Koecher–Vinberg theorem. He discovered the Koecher boundedness principle in the theory of Siegel modular forms.

• Koecher, Max (1953), "Über Dirichlet-Reihen mit Funktionalgleichung", Journal für die reine und angewandte Mathematik, 192: 1–23, ISSN 0075-4102, MR 0057907
• Krieg, A.; Petersson, H. P. (1993), "Max Koecher zum Gedächtnis", Jahresbericht der Deutschen Mathematiker-Vereinigung, 95 (1): 1–27, ISSN 0012-0456, MR 1203926
• Petersson, Holger P. (1994), "Max Koecher's work on Jordan algebras", in Kaup, Wilhelm; McCrimmon, Kevin; Petersson, Holger P. (eds.), Jordan algebras (Oberwolfach, 1992), Berlin: de Gruyter, pp. 187–195, ISBN 978-3-11-014251-8, MR 1293319

• Max Koecher at the Mathematics Genealogy Project
• Max Koecher on Wikimedia Commons