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19
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
0answers
853 views

A question about Mobius inversion

I don't know how precise I can make this question. I want to know whether there is a theorem that says that a certain phenomenon always happens, but I think the best I can do in order to pin down the ...
12
votes
0answers
457 views

Making a character small at a reciprocal

The following question emerged from thinking about the Erdős discrepancy problem. I don't know whether an answer would be directly helpful, but it might, and in any case I find the question quite ...
26
votes
6answers
6k views

How many surjections are there from a set of size n?

It is well-known that the number of surjections from a set of size n to a set of size m is quite a bit harder to calculate than the number of functions or the number of injections. (Of course, for ...
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) ...
18
votes
3answers
861 views

The probability for a sequence to have small partial sums

The question Let $a_1,a_2,\dots,a_n$ be a sequence whose entries are +1 or -1. Let t be a parameter. My question is to give an estimate for the number of such sequences so that $|a_1+a_2+\dots ...
8
votes
2answers
858 views

Analytic continuation of Dirichlet series with completely multiplicative coefficients of modulus 1

A sequence $a_n \in \mathbb{T}$ ($n \geq 1$) satisfying $a_{mn} = a_m a_n$ for all $m, n \geq 1$ defines a Dirichlet series $f(s) = \sum_n a_n n^{-s}$, absolutely convergent for $\Re s > 1$. If in ...
10
votes
1answer
488 views

Is this a well-known probabilistic model?

While I was thinking about the Erdős discrepancy problem, the following random walk model arose rather naturally. You fix a positive integer k, and you take a random step of 1 or -1 at each ...
73
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 ...
53
votes
33answers
8k views

Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances. Motivation I am aware about a few such cases and I think it will be useful to gather ...
27
votes
3answers
1k views

Improving a sequence of 1s and -1s

Suppose you take a $\pm 1$ sequence and you want to "improve it" by taking pointwise limits of translates. What properties can you guarantee to get in the limit? Two examples illustrate what I think ...
4
votes
2answers
271 views

Finding the codomain of a monoid homomorphism

We have a monoid M, and a function $f: M \rightarrow \{0, 1\}$. We're promised that this function factors through a (surjective) homomorphism to a finite abelian group (considered as a commutative ...
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 ...