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New estimates for some functions defined over primes
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abstract
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.
Forward citations
Cited by 2 Pith papers
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Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$
Replacing the n-th prime by the sum of logarithms of the first n primes makes analogues of Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures provable theorems.
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Sierpinski's Hypothesis H1
Sierpiński's 1958 matrix conjecture—each row of the n×n grid of the first n² numbers contains a prime—is verified for n≤4,553,432,387, with unconditional partial results for larger n.
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