A new short proof of Narananʹs theorem, explaining Lotkaʹs law and Zipfʹs law

Narananʹs important theorem, published in Nature in 1970, states that if the number of journals grows exponentially and if the number of articles in each journal grows exponentially (at the same rate for each journal), then the system satisfies Lotkaʹs law and a formula for the Lotkaʹs exponent is given in function of the growth rates of the journals and the articles. This brief communication re-proves this result by showing that the system satisfies Zipfʹs law, which is equivalent with Lotkaʹs law. The proof is short and algebraic and does not use infinitesimal arguments.