ACTA ARITHMETICA XLIII (1984) It. R. HALL(Hork) and 6. TENENBAUM (Nancy) 11. h~troduetion~ Since the work of ErdGs-Kac, Erdos-Wintrier, Kubilius, Delange, Wirsing, Elliott and Hslbz, the limiting distribution of additive and multiplicative functions is fairly well1 known. However the local behaviour of these functions still presents difficulties. In this paper our aim is to gain some insight into this beha~our, on short intervals. Given any (real-valued) arithmetical function f we introduce two further functions, which reflect the local irregularity of the values of f. Here 7c = k(n) is s function of uz, possibly constant; the case when k behaves like a power of logn is important. Our methods would apply to many additive or multipli- cative functions but it seemed preferable to us to fix our attention on the simplest case, when f (p) is constant for primes p. So we consider, typically co(uz) := card@:pln,p prime), z(m) := card{d: din, d ~2+3. The starting point of this study was a double conjecture of Erdos, proved in [I], on the functions z$ (n) and z ; (n). The first part of the con- jecture was that for each fixed k there holds In [4] it was shown that this remains valid if k = k($)+00 in such a way %hat k < (log~)'~~"-1xp ( - 6 (x) %(1ogloga)) where [(x) is any function tending to infinity. The exponent log4-1 is sharp. On seeing this result Erdos conjectured that when