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**85**

votes

**42**answers

12k views

### Experimental Mathematics

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 ...

**0**

votes

**0**answers

127 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**

votes

**2**answers

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 ...

**25**

votes

**5**answers

3k 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+\...

**79**

votes

**6**answers

11k 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 ...

**20**

votes

**3**answers

914 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 ...

**12**

votes

**3**answers

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) ...

**11**

votes

**0**answers

921 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

**0**answers

466 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 ...

**27**

votes

**6**answers

9k 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 ...

**9**

votes

**2**answers

931 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 ...

**28**

votes

**3**answers

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 ...

**10**

votes

**1**answer

507 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,...

**4**

votes

**2**answers

275 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 ...