Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New estimates for some functions defined over primes   In this paper we first establish new explicit estimates for Chebyshev's
$\vartheta$-function. Applying these new estimates, we derive new upper and
lower bounds for some functions defined over the prime numbers, for instance
the prime counting function $\pi(x)$, which improve the currently best ones.
Furthermore, we use the obtained estimates for the prime counting function to
give two new results concerning the existence of prime numbers in short
intervals.

20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg