No abstract

The next element in the 3n + 1 sequence is defined to be (3n + 1)/2 if n is odd or n/2 otherwise. The Collatz conjecture states that no matter what initial value of n is chosen, the sequence always reaches 1 (where it goes into the repeating sequence (1, 2, 1, 2, 1, 2, . . .)). The only known Collatz cycle is (1, 2). Let c be an odd integer not divisible by 3. Similar cycles exist for the more general 3n + c sequence. The 3n + c cycles are commonly grouped according to their length and number of odd elements. The smallest odd element in one of these cycles is greater than the smallest odd elements of the other cycles in the group. A parity vector corresponding to a cycle consists of 0's for the even elements and 1's for the odd elements. A parity vector generated by the ceiling function is used to determine this smallest odd element. Similarly, the largest odd element in one of these cycles is less than the largest odd elements of the other cycles in the group. A parity vector generated by the floor function is used to determine this largest odd element. This smallest odd element and largest odd element appear to be in the same cycle. This means that the parity vector generated by the floor function can be rotated to match the parity vector generated by the ceiling function. Two linear congruences are involved in this rotation. The natural numbers generated by one of these congruences appear to be uniformly distributed (after sorting).

The time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

Lehman proved that the sum of certain Mertens function values is 1. Functions involving the sum of the signs of these Mertens function values are considered here. Specifically, the upper bounds of these functions involving the number of Mertens function values equal to zero are determined. Franel and Landau derived an arithmetic statement involving the Farey sequence that is equivalent to the Riemann hypothesis. Since there is a relationship between the Mertens function and the Riemann hypothesis, there should be a relationship between the Mertens function and the Farey sequence. Functions of subsets of the fractions in Farey sequences that are analogous to the Mertens function are introduced. Results analogous to Lehman’s theorem are the defining property of these functions. A relationship between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem is postulated.

The Mertens function is the summatory Mobius function but the Mertens function can be generated recursively without using this definition. This recursive definition is the basis of autocorrelations that can be done on sequences of Mertens function values. Fourier transforms of the autocorrelations result in the energy spectral density. A likely upper bound of the absolute value of the Mertens function is determined.