Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Erdos Discrepancy Problem   According to the Erd\H{o}s discrepancy conjecture, for any infinite $\pm 1$
sequence, there exists a homogeneous arithmetic progression of unbounded
discrepancy. In other words, for any $\pm 1$ sequence $(x_1,x_2,...)$ and a
discrepancy $C$, there exist integers $m$ and $d$ such that $|\sum_{i=1}^m x_{i
\cdot d}| > C$. This is an $80$-year-old open problem and recent development
proved that this conjecture is true for discrepancies up to $2$. Paul Erd\H{o}s
also conjectured that this property of unbounded discrepancy even holds for the
restricted case of completely multiplicative sequences (CMSs), namely sequences
$(x_1,x_2,...)$ where $x_{a \cdot b} = x_{a} \cdot x_{b}$ for any $a,b \geq 1$.
The longest CMS with discrepancy $2$ has been proven to be of size $246$. In
this paper, we prove that any completely multiplicative sequence of size
$127,646$ or more has discrepancy at least $4$, proving the Erd\H{o}s
discrepancy conjecture for CMSs of discrepancies up to $3$. In addition, we
prove that this bound is tight and increases the size of the longest known
sequence of discrepancy $3$ from $17,000$ to $127,645$. Finally, we provide
inductive construction rules as well as streamlining methods to improve the
lower bounds for sequences of higher discrepancies.

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The categories of this paper are: hep-ph
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin