Bounds on the Number of Threshold Functions

It has been conjectured [1] that the number Rn of threshold functions of n arguments has the limiting form: Limn¿¿ log2 Rn/n² = const. Bounds previously obtained [2], [3] show that such a constant would have to lie between ¿ and one. In the present note this constant is shown to have a lower bound of ¿.¹ The result is extended to the number Rn^m of threshold functions defined on m minterms of n arguments and suggests the more general form in the limit of large n, m/n. {logm/n Rn^m/n} = const. with the same limits for the constant, providing that the minterms are spread out in a certain sense.