We discuss various independent aspects of sharp Lieb–Thirring inequalities. First, we present an argument of Stubbe which shows that Riesz means of order two and higher approach their semiclassical limit monotonically, thus leading to an alternative proof of sharp Lieb–Thirring inequalities. Next, we discuss the number of negative eigenvalues of Schrödinger operators with radial potentials, following Glaser, Grosse, and Martin. This leads, on the one hand, to a sharp CLR inequality for radial potentials in dimension 4 and, on the other hand, to a counterexample to the Lieb–Thirring conjecture with exponent zero in sufficiently high dimensions. Next, we discuss briefly an approach that disproves the Lieb–Thirring conjecture in a certain range of positive exponents. Finally, we discuss the Lieb–Thirring inequality with exponent one in its dual formulation, also known as kinetic energy inequality, in which it enters in many applications.