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This is an expository article that results from a talk given to second yearstudents at Oldenburg university. The aim of the talk was to show whatbeautiful and unexpected results may be obtained if one plays with daringanalogies in a way that is usually not done in undergraduate education(unfortunately): We start from the fact that the sum of diagonal entries of asymmetric matrix equals the sum of its eigenvalues. We then guess an analogousformula where the matrix is replaced by a function of two real variables andsums are replaced by integrals in a systematic way. We show that this is indeeda worthwhile process: In a special case it yields that the sum of inversesquares of the positive integers is pi^2/6. Finally, an outline of the proof ofthe guessed formula is given, and further applications, for example to theconnection between billiards and the frequencies of a drum, are explained.

Ueber Eigenwerte, Integrale und pi^2/6: Die Idee der Spurformel (On eigenvalues, integrals and pi^2/6: The idea of the trace formula)