New estimates for some functions defined over primes
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 $π(x)$, 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.
14 pages, v2: minor changes (for instance, a smaller interval in Theorem 4.1 is obtained)
11N05 (Primary), 11A41 (Secondary), Distribution of primes, Mathematics - Number Theory, prime counting function, FOS: Mathematics, Asymptotic results on arithmetic functions, Number Theory (math.NT), Chebyshev's \(\vartheta\)-function, Primes
11N05 (Primary), 11A41 (Secondary), Distribution of primes, Mathematics - Number Theory, prime counting function, FOS: Mathematics, Asymptotic results on arithmetic functions, Number Theory (math.NT), Chebyshev's \(\vartheta\)-function, Primes
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