THE SERIES OF RECIPROCALS OF PRIMES AND POLYNOMIALS WITH PRIME EXPONENTS

A famous theorem of Weierstrass states that every continuous function on the closed unit interval [0, 1] can be uniformly approximated on the interval by poly-nomials. A natural question arises as to whether it suffices to use only polynomials with some of the possible exponents. There is a very beautiful theorem that gives a complete answer to this question. Theorem (The Müntz-Szasz Theorem). If {pn} is an increasing sequence of pos-itive numbers, then n=1 1 pn diverges if and only if every continuous function can be uniformly approximated on [0, 1] by functions in the linear span of the collection {xpn} ∪ {1}. A very nice discussion of the proof of this theorem can be found in [4, p. 313]. The purpose of this note is twofold. We first point out that the following is an immediate consequence of the Müntz–Szasz Theorem and Euler’s famous theorem that the sum of the reciprocals of the primes diverges: Every continuous function on [0, 1] can be uniformly approximated by polynomials with prime exponents. Our second purpose takes a little more space to accomplish. We offer a descrip-tion of a proof of Euler’s theorem that appears to be easier to check and to remember than other presentations that we have seen (although it is not very different from other proofs). Theorem (Euler’s Theorem). If pj denotes the jth prime number, then the series j=1 1 pj diverges. Proof. If the series converged, there would exist an M such that j=