The evaluation of Knopfmacher's curious limit

We begin with a function expressed as a certain infinite product. It is a twice-mutated variation of another product that has its origins in counting irreducible factors of univariate polynomials over Galois fields. Knopfmacher's limit is taken as we approach 1 from below in this product. We derive and execute an algorithm that finds a good approximation to this limit using moderate computational resources. We also investigate the coefficients of the power series for the logarithm of this product; these will be shown to exhibit size fluctuations that render the straightforward power series inadequate for the purpose of estimating the limit in question. Statement of the problem The problem was posed by Arnold Knopfmacher in a note to the Usenet news group comp.soft-sys.math.mathematica, January 1999 [Knopfmacher 1999a] (herein we make small changes in notation). We wish to obtain a numerical estimate (say 8 decimal digits) of the limit as x tends to 1 from below of the function p@xD = ‰k=2 ¶ I1xm@kD ÅÅÅÅÅÅÅÅ ÅÅÅÅÅ k+1 M ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ 1-x where m@kD = k k ÅÅÅÅÅÅÅÅÅÅ d@kD and d@kD is the smallest prime factor of k . There are similar formulas in [Knopfmacher and Warlimont 1995] analyzing probabilities related to numbers of irreducible factors of distinct degrees in univariate polynomials over Galois fields. According to [Knopfmacher 1999b] his coauthor studied a limiting case of such a formula. It gave rise to a limit seemingly harder than the one above. Knopfmacher extracted the present problem as a simpler case to try first, and subsequently posted it to the news group. Thus this example might be described as a double mutation of a problem that is of independent interest elsewhere in the realm of number theory. It later turned out that the original problem was the more tractable but by then the mutation had acquired a life of its own. In this report we show how to compute a good approximation to the above limit using a blend of theory and the computational capabilities of Mathematica [Wolfram 1999] (Mathematica (TM) is a registered trademark of Wolfram Research, Incorporated). In Mathematica this function may be written as d@k_D := Divisors@kD@@2DD m@k_D := k − k d@kD p@x_D := ‰k=2 ∞ I1 − xm@kD k+1 M 1 − x Crude bounds First we deduce crude bounds for the limit. Among other things this will demonstrate the existence of a lim inf and lim sup. Proposition 1: (a) A lim inf for p@xD as x Ø 1 is given by ‰ EulerGamma . (b) A lim sup is given by 2 ‰ EulerGamma . Proof: For 0 < x < 1 note that  k=2 ¶ ikjjjj1xb k ÅÅÅÅÅ 2 r ÅÅÅÅÅÅÅÅÅÅ k+1 y{zzzz ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ ÅÅ 1-x < p@xD < ‰k=2 ¶ I1xk-1 ÅÅÅÅÅÅÅÅÅÅ k+1 M ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅ 1-x Everything is positive so the inequalities are preserved on taking logarithms: