@phdthesis{Stumpf2022,
  author    = {Stumpf, Pascal},
  title     = {On coverings and reduced residues in combinatorial number theory},
  doi       = {10.25972/OPUS-29350},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-293504},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {Our starting point is the Jacobsthal function \(j(m)\), defined for each positive integer \(m\) as the smallest number such that every \(j(m)\) consecutive integers contain at least one integer relatively prime to \(m\). It has turned out that improving on upper bounds for \(j(m)\) would also lead to advances in understanding the distribution of prime numbers among arithmetic progressions. If \(P_r\) denotes the product of the first \(r\) prime numbers, then a conjecture of Montgomery states that \(j(P_r)\) can be bounded from above by \(r (\log r)^2\) up to some constant factor. However, the until now very promising sieve methods seem to have reached a limit here, and the main goal of this work is to develop other combinatorial methods in hope of coming a bit closer to prove the conjecture of Montgomery. Alongside, we solve a problem of Recam{\´a}n about the maximum possible length among arithmetic progressions in the least (positive) reduced residue system modulo \(m\). Lastly, we turn towards three additive representation functions as introduced by Erdős, S{\´a}rk{\"o}zy and S{\´o}s who studied their surprising different monotonicity behavior. By an alternative approach, we answer a question of S{\´a}rk{\"o}zy and demostrate that another conjecture does not hold.},
  subject      = {Kombinatorische Zahlentheorie},
  language  = {en}
}