  Fluctuations are important near phase transitions, where they can be
difficult to describe quantitatively. Superconductivity in mesoscopic rings is
particularly intriguing because the critical temperature is an oscillatory
function of magnetic field. There is an exact theory for thermal fluctuations
in one-dimensional superconducting rings, which are therefore expected to be an
excellent model system. We measure the susceptibility of many rings, one ring
at a time, using a scanning SQUID that can isolate magnetic signals from seven
orders of magnitude larger background applied flux. We find that the
fluctuation theory describes the results and that a single parameter
characterizes the ways in which the fluctuations are especially important at
magnetic fields where the critical temperature is suppressed.
