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We study the arithmetic (real) function, with f 'essentially bounded'. Inparticular, we obtain non-trivial bounds, through f 'correlations', for the'Selberg integral' and the 'symmetry integral' of f in almost all shortintervals $[x-h,x+h]$, $N\le x\le 2N$, beyond the 'classical' level, up to avery high level of distribution (for $h$ not too small). This time we go beyondLarge Sieve inequality. Precisely, our method applies Weil bound forKloosterman sums.

On the symmetry of arithmetical functions in almost all short intervals,IV