Richard Warlimont

Über die Häufigkeit großer Differenzen konsekutiver Primzahlen

The following theorem is going to be proved. Letpm be them-th prime and putdm:=pm+1−pm. LetN(σ,T), 1/2≤σ≤1,T≥3. denote the number of zeros ϱ=β+iγ of the Riemann zeta function which fulfill β≥σ and |γ|≤T. Letc≥2 andh≥0 be constants such thatN(σ,T)≪Tc(1−σ) (logT)h holds true uniformly in 1/2≤σ≤1. Let ε>0 be given. Then there is some constantK>0 such that

On the Distribution of the Degrees of Prime Element Factors in Additive Arithmetical Semigroups

Abstract. Results of Arnold Knopfmacher about the distribution of the degrees of irreducible factors in the canonical decomposition of monic polynomials in ?q[X] are generalized to additive arithmetical semigroups (G,∂) satisfying a weak condition called axiom ?

Euler-Summierbarkeit konform-äquivalenter Reihen

AbstractLet Φ be a conformal one to one mapping of the unit disc with Φ (1)=1. Given the convergent series Σan putf(z):=Σanzn, |z|<1, and letan(Φ) be defined by the relation

Sieving by large prime factors

f(\Phi (w)) = \sum\limits_{n = 0}^\infty {a_n } (\Phi )w^n ,\left| w \right|< 1.

ber die kleinste natrliche Zahl maximaler Ordnung modm

The series Σan(Φ) is called the conformal equivalent of Σan with respect to Φ and need not converge as was discovered byTurán. We prove that it is nevertheless summable (E;q),q>0.

On squarefree numbers in arithmetic progressions

We study the distribution of elements in an additive arithmetical semigroup (G, ∂) (as introduced by John Knopfmacher) in whose canonical decomposition the degrees of the prime elements belong to a given union of residue classes modk.

Eine Bemerkung zu einem Ergebnis von N. G. de Bruijn

LetQ be a set of primes with density <1. An asymptotic is proved for the number of positive integers ≦x which do not have a prime divisor which is >y and belongs toQ.

Eine Bemerkung ber quadratfreie Zahlen ?1 (mod k )

Let to every elementx of a finite setM be associated some nonempty subsetM (x) ofM in such a way that the implicationy∈M(x)⇒x∈M(y) is fulfilled. We prove two upper estimations for the least number of setsM(x) which are necessary to coverM. Several applications to number theory are presented.