The Riemann hypothesis is perhaps the most significant open problem in number theory. It asserts that the zeros of the Riemann zeta function have real part 1/2. In this talk we report on progress towards an equicharacteristic analogue of the Riemann hypothesis for curves as conjectured by David Goss. Previously this conjecture was only known for curves with class number one. We give a new shorter proof of the equicharacteristic Riemann hypothesis for the affine line (originally proven by Wan, Diaz-Vargas, Poonen, and Sheats). This proof uses the Dwork trace formula. We then report on an approach using the Anderson trace formula together with a local-to-global patching argument to deduce the equicharacteristic Riemann hypothesis for all curves whose compactification is ordinary. This shows that Goss’ conjecture holds for `almost all’ curves. Parts of this are joint with James Upton.