Abstract Being incoherent is often viewed as a paradigm kind of irrationality. Numerous authors attempt to explain the distinct-seeming failure incoherence by positing set requirements structural rationality. I argue that notion coherence are meant capture very slippery, and intuitive judgments – in particular, charge distinct, blatant irrationality imperfectly correlated with respecting canon ...

We outline a proof of an analogue of Khintchine’s Theorem in R, where the ordinary height is replaced by a distance function satisfying an irrationality condition as well as certain decay and symmetry conditions.

We prove sharp irrationality measures for a q-analogue ofπ and related q-series, and indicate open problems on linear and algebraic independence of the series that might be viewed as q-analogues of some classical mathematical constants.

Here is an immediate proof in the following cases: 1. a(n) = n: 2. a(n) = d(n) =number of divisors of n; 3. a(n) = w(n) =number of distinct prime divisors of n: 4. a(n) = D(n) =number of total prime divisors of n (that is. counted with repetitions): 5. a(n) = dJ(n) =the Euler function of n: 6. a( n) = cr( n) =the sum of the divisors of n; 7. a(n) = Pn =the nth prime: 8. a(n) = 71(n) =the number...