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Questions tagged [polymath5]

For questions arising from topics discussed during the Polymath5 project, which tried to solve the Erdős discrepancy problem.

2
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1answer
252 views

Homogeneous van der Waerden

The Erdős Discrepancy Problem is whether in any two-coloring of the naturals for any $C$ there is a sequence $d, 2d, \ldots nd$ such that the difference of red and blue numbers in it is more than $C$. ...
1
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0answers
168 views

Difficulty understanding equivalent statement of Erdős Discrepancy Problem

Recently I watched a famous youtube video of talk given by Terry Tao on Erdős Discrepancy Problem https://www.youtube.com/watch?v=QauoO0j9Y9Y. I never heard of this problem before his announcement of ...
20
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2answers
2k 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 ...
12
votes
0answers
992 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 ...
13
votes
0answers
480 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 ...
28
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6answers
12k 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 ...
13
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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 are the precise asymptotics of $\sum_{k=1}^n 1/(k d(k))...
20
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3answers
991 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 ...
9
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2answers
1k 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
526 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 stage,...
92
votes
6answers
15k 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 ...
120
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46answers
17k views

Experimental mathematics leading to major advances

I would like to ask about examples where experimentation by computers have led to major mathematical advances. A new look Now as the question is five years old and there are certainly more examples ...
28
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3answers
2k 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
283 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 ...
27
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5answers
4k 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 $|\mu(1)+\mu(2)+\dots+\...