Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lower bounds for the variance of sequences in arithmetic progressions:
  primes and divisor functions   We develop a general method for lower bounding the variance of sequences in
arithmetic progressions mod $q$, summed over all $q \leq Q$, building on
previous work of Liu, Perelli, Hooley, and others. The proofs lower bound the
variance by the minor arc contribution in the circle method, which we lower
bound by comparing with suitable auxiliary exponential sums that are easier to
understand.
  As an application, we prove a lower bound of $(1-\epsilon) QN\log(Q^2/N)$ for
the variance of the von Mangoldt function $(\Lambda(n))_{n=1}^{N}$, on the
range $\sqrt{N} (\log N)^C \leq Q \leq N$. Previously such a result was only
available assuming the Riemann Hypothesis. We also prove a lower bound
$\gg_{k,\delta} Q N (\log N)^{k^2 - 1}$ for the variance of the divisor
functions $d_k(n)$, valid on the range $N^{1/2+\delta} \leq Q \leq N$, for any
natural number $k \geq 2$.

26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th