Acta Mathematica Academiae Scientiarum Hungaricae 67. (1995)

1995 / 1-2. szám - Manstavičius, E.: Functional approach in the divisor distribution problems

FUNCTIONAL APPROACH IN THE DIVISOR DISTRIBUTION PROBLEMS 3 space of continuous functions on the interval [0,1] endowed with the supre­­mum distance p(v). We recall that the Strassen set 1C agrees with the set of absolutely continuous functions g such that g(0) = 0 and Now we return to arithmetical functions. Let where h(pa) E R, m,a E N, and p stands for a prime number. As usual, pa II m denotes that pa divides m but pa+1 does not. We put where Lv = logmax{i;,e} and LkU — L(Lk-\U ), к ^ 1. Let q,q' be two consecutive primes belonging to the set S(h) = {p : h(p) ф 0}. For a fixed natural number m, joining the points (0,0) and (D(q),hq(m) E S(h), by straight hnes in the coordinate plane we get the graph of the function defined by H(m,t) = (hq(m) - A(g))D(q') - D(q)+ (hg,(m) - A(q')) t - D(q) D(q') - D(q) when D(q) ^ t < D(q'). Let further Gk(m,t) = ß{k)~xH(m, D(k)t) when 0 5i t ^ 1 and к ^ ко > qo = min {q : q E S(h)}. Thus, Gk{m,-) E C. Observe that we use natural subscripts к for the sequence Gk(m,t) instead of primes q for convenience only. Supposing that each m ^ n is taken at random with equal probability 1/n, we insert the sequence Gk(m, •), к ^ ко, in the context described above. Formally speaking one can take = N, Tn = 2N, and Pn = vn where vn(A) - n_1#{m ^ n,m E A}, A C N. Our main result is the following theorem. Theorem 1. Let D(p) —» oo and f\g'(t))2dtí l. Jo hk(m) = Mp“)> pa\\m ptik A(u) = D(u) = XI - -), and ß(u) = y/2D(u)L2D(u), S P ^ P P рЪи pSu h(p) = о Acta Mathematica Hungarica 67, 1995

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