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- W2551907830 abstract "It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes $n^3$ and $(n+1)^3$ for all $n geq exp(exp(33.3))$. To prove this, we first derive an explicit version of the Riemann--von Mangoldt explicit formula. We then assume the Riemann hypothesis and show that there will be a prime in the interval $(x-4/ pi sqrt{x} log x, x]$ for all $x > 2$. Moreover, we show that the constant $4/pi$ can be reduced to $(1+epsilon)$ for all sufficiently large values of $x$. Using explicit results on primes in arithmetic progressions, we prove two new results in additive number theory. First, we prove that every integer greater than 2 can be written as the sum of a prime and a square-free number. We then work similarly to prove that every integer greater than 10 and not congruent to $1$ modulo $4$ can be written as the sum of the square of a prime and a square-free number. Finally, we provide new explicit results on an arcane inequality of Ramanujan." @default.
- W2551907830 created "2016-11-30" @default.
- W2551907830 creator A5064963455 @default.
- W2551907830 date "2016-11-22" @default.
- W2551907830 modified "2023-10-17" @default.
- W2551907830 title "Explicit Estimates in the Theory of Prime Numbers" @default.
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- W2551907830 doi "https://doi.org/10.25911/5d7638521bcc4" @default.
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