He was born into a family of Swedish nobility. His grandfather, Nils Samuel von Koch (1801-1881), was the Attorney-General ("Justitiekansler") of Sweden. His father, Richert Vogt von Koch (1838-1913) was a Lieutenant-Colonel in the Royal Horse Guards of Sweden.
von Koch wrote several papers on number theory. One of his results was a 1901 theorem proving that the Riemann hypothesis is equivalent to a strengthened form of the prime number theorem.
He described the Koch curve in a 1906 paper entitled "Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane" [1].