Jürgen Spilker

Charakterisierung der additiven, fast-geraden Funktionen

Let f be an additive number-theoretical function and q≥1. We give necessary and sufficient conditions for f to satisfy. By means of this characterization we show that f is if and only if the above condition is satisfied and f has a mean value; in this case the Ramanujan expansion of f converges pointwise to f.

Ramanujan expansions of bounded arithmetic functions

Using a functional analytic idea, I will show that every bounded arithmetic function possesses a Ramanujan expansion which is pointwise absolutely convergent.

Elliptische Fixpunkte symplektischer Matrizen

ZusammenfassungU. Christian hat in [1] eine notwendige und hinreichende Bedingung an eine reelle symplektische Matrix dafür aufgestellt, daß der zugeordnete Homöomorphismus der Siegelschen Halbebene einen Fixpunkt besitzt. Sein Beweis benutzt entscheidend die Reduktionstheorie symplektischer Matrizen. Hier wird ein einfacher analytisch-geometrischer Beweis für dieses Kriterium dargestellt. Er beruht auf dem Fixpunktsatz von Schauder und einer Art Konvexitätseigenschaft symplektischer Kugeln.

Eine bemerkung zur charakterisierung der fast-periodischen multipliκατιven zahlentheoretischen funktionen mit Von Null verschiedenem mittelwert

Denote by j4 q the vector space of B q-almostperiodic arithmetical functions. H. Daboussi £2 J , £jJ and P.D.T.A. Elliott 118], [l<3 characterized the set of multiplicative functions in jl^ with mean-value M(f) £ O by a statement concerning the convergence of certain series extended over the values f(p^) of f at prime-powers ( see (1.2^), (1.2g) and (1.2^) The authors give a simpler proof for the implication: If f £ ~4r ^ (q i 1) is multiplicative with mean-value M(f) φ O, then the series mentioned above are convergent. The proof consists in reducing the assertion to the special case q = 2, for which a simple proof is available by Daboussi Delange [5J ( see also [2^3 ) . AMS classification: Ιο Η 45 , lo Κ 35 .

Untergruppen der GL2, welche durch homogene lineare Kongruenzen definiert sind

We generalize a result of Rankin [1]: Let R be a p — adic valuation ring or one of its factor rings and let G be GL2(R) or SL2(R). Then for M∈ M2(R), GM:={s∈G| trace MS=O} is a group iff trace M=0 and det M satisfies a simple condition (det M=O in most cases). We give similar conditions for several homogeneous linear equations defining subgroups of G.

The Lucas property for linear recurrences of second order

Here the integers ni are the coefficients of the p-adic representation of n. E. Lucas and later authors applied this concept to binomial coefficients. Recently, H. Zhong and T. Cai proved a necessary and sufficient condition for the Lucas property modulo p involving the sequences F (an + b), where F (n) is the Fibonacci sequence and a, b are two fixed positive integers. In this note two necessary and sufficient conditions for the Lucas property modulo p are proven, which are applicable to all functions f(n) = g(an+b), when g(n) satisfies a linear recurrence formula of order two. This generalizes the result of H. Zhong and T. Cai. We extend our result to Carmichael numbers p. An application to the difference of exponential functions is presented.

Arithmetical Functions of the Form f([g(n)])

AbstractTwo results on composed functions

Arithmetical functions : an introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties

f([g{\text{(}}n{\text{)])}}