Let be a continuous function and be the set of all positive integers such that there exists such that
Prove that is infinite and evaluate the limit
Let denote the factorial of a real number; that is . Evaluate the limit:
It holds that
where denotes the -th harmonic number and the Euler – Mascheroni constant.
Let . If:
- for all
then prove that is linear.
For convenience, identify with here. Then note that for any such function , also a solution for any point on the unit circle. Also is a solution. Note that and hence we can wlog assume that . So is a point on the unit circle with distance to . Hence , so w.l.o.g. assume that . But then for any , both and have the same distance to and . So supposing , all lie on the perpendicular bisector between these points and in particular and are collinear which clearly is absurd. Hence for all which proves the claim.
Let denote the zeta function. Prove that