18
votes
2answers
1k views

The behavior of a certain greedy algorithm for Erdős Discrepancy Problem

Let $N$ be a positive integer. We want to find a completely multiplicative functions f(n) with values $\pm 1$ for $n \le N$ such that the discrepancy $$D=\max_{n \le N} |\{\sum_{i=1}^nf(i)\}|$$ is ...
11
votes
3answers
2k views

An elementary number theoretic infinite series

For a positive integer k, let d(k) be the number of divisors of k. So d(1)=1, d(p) =2 if p is a prime, d(6)=4, and d(12)=6. What is the precise asymptotics of SUM_{k=1}^n 1/(kd(k)) Background: 1) ...
70
votes
6answers
8k views

Why does the Riemann zeta function have non-trivial zeros?

This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...
23
votes
5answers
2k views

Partial sums of multiplicative functions

It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that ...