A Conjecture in Statistical Group theory

Let n and k be positive integers, and write g(n) for the number of (isomorphism classes of) groups of order n. Let F(x, k) be the number of integers n not exceeding x for which g(n)=k. Then the following inequality holds for all values of k except for at most a finite number of exceptions: As x approaches infinity,


F(x, k) >> x/(log log x)^c(k)



for a computable constant c(k), where the constant implied by the inequality >> may also depend on k.

We have proved this conjecture for all values of the set T in the paper entitled, "Groups of Cubefree Order" listed in my first post of this Blog (December 26, 2011). We have also, in unpublished work, established this conjecture for some integers k that are fifth-power-free but not cubefree. By hand-calculations, the first value of k for which we have not been able to establish the inequality above is k=71.

Keith Dennis has conjectured that all but finitely-many integers are in the set S of first-elements of ordered pairs recursively-generated by the following three rules:

(A) (1, 2) is in the set

(B) If (a, b) is in the set, then so is (b, a+b);

(C) If (a, b) and (C, d) are in the set, then so is (ac, bd).

In my paper "Local Distribution Results for the group-counting function at positive integers", Congressus Numerantium 50 (December) 1985, 107-110, I have listed the elements of this set S not exceeding 200:

7, 11, 19, 29, 31, 47, 49, 53, 67, 71, 73, 79, 87, 91, 103, 119, 127, 131, 137, 139, 141, 143, 146, 147, 151, 155, 179, 191, 193.

The set T above properly contains this set S. In my paper. When the group-counting function assumes a prescribed integer value at squarefree integers frequently, but not extremely frequently, Acta Arith. (1992) 1-12 , I show that indeed, the inequality above holds for k in S, even if we only consider squarefree integers n.

Selected Publications

Selected Publications
C. Spiro, For the local distribution of the group-counting function, orders divisible by fifth powers can be neglected, submitted December 2011.

C. Spiro-Silverman, Groups of cubefree order, Acta Arith., LXXI 3. (1995), 209-227.

Divisibility of the -fold iterated divisor function of n into n, ACTA ARITHMETICA LXVIII.4 (1994), 307-339

C. Spiro, Additive Uniqueness Sets for Arithmetic Functions, J. Number Theory 42.2, October (1992), 232-246
C. Spiro-Silverman, When the group-counting function assumes a prescribed integer value at squarefree integers frequently, but not extremely frequently, Acta Arith. 61 (1992), 1-12.
With P. Erdös, A. Granville, and C. Pomerance, On the normal behavior of the iterates of some Arithmetic Functions, Analytic Number Theory: Procedings of a Conference in Honor of Paul T. Bateman, Birkhauser Boston, Inc. 1990

C. Spiro, The probability that the number of groups of squarefree order is two
more than a fixed prime, Proc. London Math. Soc. 60 (1990), 444–470.

C. Spiro, Local distribution results for the group-counting function at positive integers, Congressus Numerantium 50 (1985) 107-110

C. Spiro, An iteration problem involving the divisor function, Acta Arith. 46 (1986),
215–225. http://matwbn.icm.edu.pl/ksiazki/aa/aa46/aa4633.pdf

C. Spi ro, How often is the number of divisors of n a divisor of n?, J. Number
Theory 21 (1985), 81–100.X.

C. Spiro, Extensions of some formula of A. Selberg, Internat. J. Math. Math. Sci. 8
(1985), 283–302.

C.. Spiro, How Often Does the Number of Divisors of an Integer Divide Its Successor?, J. London Math. Soc. (1985) s2-31 (1): 30-40.

C. Spiro, The Frequency With Which an Integral-Valued, Prime-Independent, Multiplicative or Additive Function of n Divides a Polynomial Function of n, PhD Dissertation, University of Illinois at Urbana-Champaign, 1981. 188 pp.