4-core partitions and class numbers
5-dissections and sign patterns of Ramanujan's parameter and its companion
In 1998, Michael Hirschhorn discovered the 5-dissection formulas of the Rogers-Ramanujan continued fraction and its reciprocal. We obtain the 5-dissections for functions and , which are essentially Ramanujan’s parameter and its companion. Additionally, 5-dissections of the reciprocals of these two functions are derived. These 5-dissection formulas imply that the coefficients in their series expansions have periodic sign patterns with few exceptions.
6 nombres dont les sommes deux à deux sont des carrés
"Almost every" algebraic number-field has a large class-number
?ber die Anzahl der Zahlen a2 + b2 in einer arithmetischen Reihe gro?er Differenz.
?ber singul?re Invarianten elliptischer Funktionenk?rper.
(Corrigenda) Über die Lösbarkeit gewisser diophantischer Gleichungen dritten Grades.
...-Expansions, Linear Recurrences, and the Sum-of-Digits Function.
"Factorisatio numerorum" in arithmetical semigroups
...-Invariants and ...-Transforms.
ℚ-linear relations of special values of the Estermann zeta function
Łukasiewicz language and diagonals of formal series. (Langage de Łukasiewicz et diagonales de séries formelles.)
(Non)Automaticity of number theoretic functions
Denote by Liouville’s function concerning the parity of the number of prime divisors of . Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that is not –automatic for any . This yields that is transcendental over for any prime . Similar results are proven (or reproven) for many common number–theoretic functions, including , , , , , and others.
(Non-)weakly mixing operators and hypercyclicity sets
We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space , any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for or , . Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.
...-Sätze für gewisse multiplikative arithmetische Funktionen
...-Sätze für zwei arithmetische Funktionen
'The mother of all continued fractions'
We give the relationship between regular continued fractions and Lehner fractions, using a procedure known as insertion}. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show how these (and other) continued fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known as the full...
?-transforms of rational function measures on Zs.
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