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We show under the Generalized Riemann Hypothesis that for every non-constant integer-valued polynomial f , for every δ > 0 , and almost every prime q in [ Q , 2 Q ] , the number of primes from the interval [ x , x + x 1 2 + δ ] that are values of f modulo  q is the expected one, provided Q is not more than x 2 3 − 𝜖 . We obtain this via a variant of the classical truncated Perron's formula for the partial sums of the coefficients of a Dirichlet series. [ABSTRACT FROM AUTHOR]

We prove that, for T ≥ 2 000, T ≥ Q ≥ 10, and σ ≥ 0.52, we have Σq≤Q q/φ(q) xmod*q ΣN(σ,T,X)≤20(56Q5T3)1-σ log5-2σ(Q2T)+32Q2log2(Q2T), where Xmod* q denotes a sum over all primitive Dirichlet characters X to the modulus q. Furthermore, we have N(σ,T,1)≤2T log(1+9.8/2T(3T)8(1-σ)/3log5-2σ(T))+103(logT)2. [ABSTRACT FROM AUTHOR]