A generalization of the sum of divisors function involves a recursive definition. This leads to variants of superabundant numbers, colossally abundant numbers, and Gronwall's theorem (relevant to the Riemann hypothesis).

The Mobius function is commonly used to define Euler's totient function and the Mangoldt function. Similarly, the summatory Mobius function (the Mertens function) can be used to define the summatory totient function and the summatory Mangoldt function (the second Chebyshev function).

In this paper, we present the definition, some properties, and solve a problem on special primes. These properties help in providing us with a better understanding of the problem posed related to special primes on the open problem garden website. The problem involves finding all the primes q, given a prime p such that q≡1(mod p) and 2^((q−1)/p)≡1(mod q). We prove that a prime number q is a special prime of p if and only if the order of 2 in U(q) divides q−1p. Also, we prove that a prime number q is not a special prime for any prime number if 2 is a generator of the group U(q) and that there exist infinitely many special primes for any given prime number.

The article derives the probability for lethal recessive alleles in the case of recessive disadvantage or advantage. It is shown that the recessive advantage of a lethal gene can be detected by the ratio of heterozygotes and homozygotes. This demonstrates that the higher IQ of certain ethnic groups cannot be explained by the recessive advantage of lethal genes. The article shows that lethal genes can survive in the population if some lineages of families have much more children than the average.

Let us assume that defence mechanisms are so strong that the average outcome of a hacking attack is unsuccessful. How to calculate the costs arising from false positives and false negatives in intruder detection? Is it better for the hacker to make fewer but more effective attacks rather than several but less effective attacks? How to calculate the difference between these alternative strategies?

Congestion pricing has received lots of attention in the scientific discussion. Congestion pricing means that the operator increases prices at the time of congestion and the traffic demand is expected to decrease. In a certain sense, shadow prices are an optimal way of congestion pricing: users are charged shadow prices, i.e., the expectations of future losses because of blocked connections. The shadow prices can be calculated exactly from Howard’s equation, but this method is difficult. The paper presents simple approximations to the solution of Howard’s equation and a way to derive more exact approximations. If users do not react by lowering their demand, they will receive higher bills to pay. Many users do not react to increased prices but would want to know how the congestion pricing mechanism affects the bills. The distribution of the price of a connection follows from knowing the shadow prices and the probability of a congestion state. There is another interesting distribution. The network produces profit to the operator, or equivalently, blocked connections produce a cost to the operator. The average cost rate can be calculated from Howard’s equation, but the costs have some distribution. The distribution gives the risk that the actual costs exceed the average costs, and the operator should include this risk to the prices. The main result of this paper shows how to calculate the distribution of the costs in the future for congestion pricing by shadow prices and for congestion pricing with a more simple pricing scheme that produces the same average costs.

The paper describes a method of solving some stochastic processes using generating functions. A general theorem of generating functions of a particular type is derived. A generating function of this type is applied to a stochastic process yielding polynomial time algorithms for certain partitions. The method is generalized to a stochastic process describing a rather general linear transform. Finally, the main idea of the method is used in deriving a theoretical polynomial time algorithm to the knapsack problem.

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.