Exponential sums over primes in short intervals, arithmetic progressions and zeros of the Riemann zeta function

We outline some ideas which give a connection between zeros of the Riemann zeta function and exponential sums over primes in short intervals and arithmetic progressions, with the goal of such techniques to establish conditional relations between zeros of various zeta functions. The ideas build on some work of Jutila which combines the circle method with Voroni summation to establish short interval moment estimates for the Riemann zeta function. Comments, ideas and future discussions are encouraged.