Problems for Solution: 4518-4522

The irrationality of a number theoretical series

“…. For k = 0, 1 it follows from a general result by Erdős and Straus [3], that S k is irrational, whereas for k = 2 the same was shown by Erdős and Kac [2]. In [1], Erdős posed the question whether S k is irrational for all k. We will prove the following theorem.…”

On the Q-linear independence of the sums ∑n=1∞σk(

“…This was proved by J.B. Kelly [2] and R. Breusch [3] respectively for k = 1 and 2. The irrationality for k = 3 was established independently by Schlage-Puchta [8] and by Friedlander, Luca and Stoiciu [4].…”

“…For k 1 let σ k (n) denote the sum of the k-th powers of the positive divisors of n. In 1953 Erdős and Kac [3], [5,Problem 14] conjectured that the sum…”