Distribution of alternative power sums and Euler polynomials modulo a prime

For a fixed integer s≥2s≥2, we estimate exponential sums with alternative power sums As(n)=∑i=0n(−1)iis individually and on average, where As(n)As(n) is computed modulo pp. Our estimates imply that, for any ϵ>0ϵ>0, the sets {As(n):n