This is a fascinating question that combines elementary concepts (divisors, standard deviation) with deep results in analytic number theory. Here is a breakdown of the answers to your questions, explaining the reasoning in detail.

Summary of Answers

• Question 1: Does the average primeness tend to zero? Yes. The limit is indeed zero. The intuitive reason is twofold:
  1. Prime numbers, where f(p) = 1, become increasingly rare. Their contribution to the average tends to z...

Excellent question. You've connected the dots perfectly. The short answer is yes, you are correct.

The most general and accurate way to characterize the optimality of the Wilcoxon-Mann-Whitney (WMW) test is to say that it is the most powerful (or more precisely, the locally most powerful rank) test for detecting alternatives under the proportional odds (PO) model.

Let's break down why this is the case, building on the points you've already made.

1. Formalizing the Proportio...

Excellent question. You've correctly identified a key challenge in frequentist inference after multiple imputation (MI) and are asking about the standard, and most accepted, solution.

Yes, a generalization for multi-parameter tests (often called "chunk tests" or omnibus tests) exists, and it's a crucial part of the MI inference framework developed by Rubin and his colleagues. The resulting test is an F-test, which is a natural generalization of the t-test.

Summary of the Single-Parameter C...