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We establish the equivalence of conjectures concerning the pair correlation of zeros of L-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston and Montgomery [‘Pair correlation of zeros and primes in short intervals’, Analytic number theory and Diophantine problems (Stillwater, 1984), Progress in Mathematics 70 (1987) 183–203] and Montgomery and Soundararajan [‘Primes in short intervals’, Comm. Math. Phys. 252 (2004) 589–617] for the Riemann zeta-function to other L-functions in the Selberg class. Our approach is based on the statistics of the zeros because the analogue of the Hardy–Littlewood conjecture for the auto-correlation of the arithmetic functions we consider is not available in general. One of our main findings is that the variances of sums of these arithmetic functions over primes in short intervals have a different form when the degree of the associated L-functions is 2 or higher to that which holds when the degree is 1 (for example, the Riemann zeta-function). Specifically, when the degree is 2 or higher, there are two regimes in which the variances take qualitatively different forms, whilst in the degree-1 case there is a single regime.