# On the Riemann hypothesis and the difference between primes

@article{Dudek2014OnTR,
title={On the Riemann hypothesis and the difference between primes},
journal={International Journal of Number Theory},
year={2014},
volume={11},
pages={771-778}
}
We prove some results concerning the distribution of primes assuming the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval $(x-\frac{4}{\pi} \sqrt{x}... Primes and prime ideals in short intervals • Mathematics • 2016 We prove the analog of Cram\'er's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the Primes in explicit short intervals on RH • Mathematics • 2015 On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers. A Conditional Explicit Result for the Prime Number Theorem in Short Intervals • Mathematics • 2019 We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis. Fourier optimization and prime gaps • Mathematics Commentarii Mathematici Helvetici • 2019 We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between Explicit interval estimates for prime numbers • Mathematics Math. Comput. • 2022 Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of$(\Delta, x_0)$such that for all$x \geq x_0$there exists at least one prime in the The Prime Gaps Between Successive Primes to Ensure that there is Atleast One Prime Between Their Squares Assuming the Truth of the Riemann Hypothesis Based on Dudek’s proof that assumed the truth of the Riemann’s hypothesis, that there exists a prime between {x – (4/pi)( x^ 1/2)(log x)} and x, we determine the size of prime gaps that must exist Explicit Estimates in the Theory of Prime Numbers It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that On the error term in the explicit formula of Riemann--von Mangoldt • Mathematics • 2021 We provide an explicit$O(x/T)\$ error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramar\'e (2016) explicit. We also include applications to primes between