By John Knopfmacher

Reference textual content offering assurance of the most recent advances within the box of quantity concept, with an emphasis on summary leading quantity theorems, mean-value theorems of multiplicative services, and the conventional distribution of additive components.

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1, and for the purpose of reference to that theorem it should be emphasized that the coefficients of the series /*(y) are here denoted by /(n), instead of by f*(n)(n = 0, 1, 2, . ). First consider the unitary-divisor function dt such that d f ( a ) is the total number of divisors d of a 6 Q for which d and a/d are coprime. 3: Averages and densities ... 1, df(y) = [ Z ( y } } 2 / Z ( y 2 ) . 5) PROPOSITION. The average-value of the unitary-divisor function df for elements of degree N in Q is asymptotically AN/Z(q-2) as N -» oo.

1) CANONICAL PRODUCT LEMMA. (i) If f is a multiplicative function on Q then f*(y] = II 1 + f(p}yd(p) + f(p2}y28(p) + ••• + f(pr)yrd(p) + •• Hence, if f is a PIM-function, then f*(y] = n m>0 r where cr = f ( p ) {p&P}. 4: Asymptotic moments of ... where a = f ( p ) {p € P}. 2 below. 2) LEMMA. Let f denote a PIM-function on Q such that f(pT) ^ 0 for some prime-power pr £ Q. Suppose that f ( p r ) + O(tr) as r —> oo (p G P), for a constant t satisfying I < t < q0 , where q0 = min{|p| : p 6 P} and m is the least positive integer such that f ( p m ) ^ 0.

1 but, since our treatment of the latter category uses facts about it which may be less familiar to some readers, we begin with a direct discussion of J-q alone. 1) THEOREM. The total number Fq(N) of non-isomorphic modules of cardinal qN in J-q is equal to P0(q-l}qN + O (q^N) as N -> oo, where Po(y) = II^Li (^~~yr}~1 is the classical "partition" generating function. PROOF. 1: Asymptotic enumeration of ... - _-. ; monic prime polynomials p * oo r=l where Zq(y] = JJ < f 1 — ys(p'j : monic prime polynomials p € is the generating function of the semigroup Qq.