Ekkehard Krätzel

Curriculum Vitae

• 1963 Ph.D. Friedrich Schiller - University Jena, Thesis: Höhere Thetafunktionen (Higher theta functions)
• 1965 Habilitation Friedrich Schiller - University Jena, Thesis: Theorie und Anwendung der L-Transformation (Theory and application of the L-transformation)
• 1969 - 1992 Full Professor at the Friedrich Schiller - University Jena
• 1993 Visiting Professor at the Albert Ludwigs - Universität Freiburg
• 1991, 1993 - 1996 Visiting Professor at the University of Vienna
• Since 1996 Honorary Professor at the University of Vienna

Scientific interests

• My main field of work is number theory, more specifically
  • Analytic functions in number theory, especially higher theta functions, higher Dedekind functions, functions related to convex bodies. Gauss sums and Dedekind sums. [N 1,2,10]
  • Arithmetic functions: Estimates for growth and mean values, especially investigations on asymmetric divisor functions, powerful numbers and on the number of non-isomorphic finite abelian groups. [N 4,5,6,7,8,9,11,12,13,14,15,16,21,22,29]
  • Estimates for the number of lattice points in large compact domains. Estimates for double exponential sums. [N 3,17,18,19,20,23,24,25,26,27,28,30,31,32,33,34,35,36]
• My second field of work is analysis, more specifically: The theory of integral transformations. Asymptotic methods in analysis, especially asymptotic expansions in many variables and the saddle point method [A 1-7].
• Last not least there is one paper on probability theory I am very fond of [P1].

Books

• Zahlentheorie (Number Theory), VEB Deutscher Verlag der Wissenschaften, Berlin, 1981.
• Lattice Points, Kluwer Academic Publishers, Dordrecht, VEB Deutscher Verlag der Wissenschaften, Berlin, 1988.
• Analytische Funktionen in der Zahlentheorie (Analytic Functions in Number Theory), B.G.Teubner, Stuttgart/Leipzig/Wiesbaden, 2000.

Co-editor of:

• H. Minkowski. Ausgewählte Arbeiten zur Zahlentheorie und zur Geometrie. Mit D. Hilberts Gedächtnisrede auf H. Minkowski. Hrsg. und mit einem Anhang versehen von E. Krätzel und B. Weißbach (H. Minkowski. Selected Papers on Number Theory and Geometry. With D. Hilbert's commemorative address in honor of H. Minkowski. Edited and with a foreword and appendices by E. Krätzel and B. Weißbach), BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1989.

Selected papers on number theory

[N1] Höhere Thetafunktionen I,II (Higher theta functions I,II), Math. Nachr. 30 (1965), 17-32 and 33-46.
[N2] Kubische und biquadratische Gaußsche Summen (Cubic and biquadratic Gauss sums), J. Reine Angew. Math. 228 (1967), 159-165.
[N3] Eine Verallgemeinerung des Kreisproblems (A generalization of the circle problem), Arch. Math. 18 (1967), 181-187.
[N4] Ein Teilerproblem (A divisor problem), J. Reine Angew. Math. 235 (1969), 150-174.
[N5] Mittlere Darstellung natürlicher Zahlen als Differenz zweier k-ter Potenzen (Average representation of positive integers as a difference of two k-th powers), Acta Arith. 16 (1969), 111-121.
[N6] Die maximale Ordnung der Anzahl der wesentlich verschiedenen Abelschen Gruppen n-ter Ordnung (The maximum order of the number of essentially different abelian groups of order n), Q. J. Math. Oxford 21 (1970), 273-275.
[N7] Zahlen k-ter Art (Numbers of k-th type), Am. J. Math. 94 (1972), 309-328.
[N8] Mittlere Darstellungen natürlicher Zahlen als Summe von n k-ten Potenzen (Average representation of positive integers as a sum of n k-th powers), Czechosl. Math. J. 23 (1973), 57-73.
[N9] Die Werteverteilung der Anzahl der nicht-isomorphen Abelschen Gruppen endlicher Ordnung in kurzen Intervallen (The value distribution of the number of non-isomorphic abelian groups of finite order in short intervals), Math. Nachr. 98 (1980), 135-144.
[N10] Dedekindsche Funktionen und Summen I,II (Dedekind functions and sums I,II), Period. Math. Hung. 12 (1981), 112-113 and 163-179.
[N11] Die Werteverteilung der Anzahl der nicht-isomorphen Abelschen Gruppen endlicher Ordnung und ein verwandtes zahlentheoretisches Problem (The value distribution of the number of nonisomorphic abelian groups of finite order and a related number-theoretic problem), Publ. Inst. Math., Nouv. Ser. 31(45) (1982), 93-101.
[N12] Divisor problems and powerful numbers, Math. Nachr. 114 (1983), 97-104.
[N13] Zweifache Exponentialsumen und dreidimensionale Gitterpunktprobleme (Double exponential sums and three-dimensional lattice problems), Elementary and analytic theory of numbers, Banach Cent. Publ. 17 (1985), 337-369.
[N14] On the average number of direct factors of a finite Abelian group, Acta Arith. 51 (1988), 369-379.
[N15] The distribution of powerful integers of type 4, Acta Arith. 52 (1989), 141-145.
[N16] The distribution of values of a(n), Arch. Math. 57 (1991), 47-52.
[N17] with W. G. Nowak, Lattice points in large convex bodies II, Acta Arith. 62 (1992), 285-295.
[N18] Estimates in the general divisor problem, Abh. Math. Semin. Univ. Hamburg 62 (1992), 191-206.
[N19] Theorie der Gitterpunkte (Theory of lattice points), Elem. Math. 47 (1992), 6-18.
[N20] with S. Hoeppner, The number of lattice points inside and on the surface |t₁|^k +|t₂|^k +···+|t_n|^k=x, Math. Nachr. 163 (1993), 257-268.
[N21] with D. Wolke, Über die Anzahl der abelschen Gruppen gegebener Ordnung (On the number of Abelian groups of a given order), Analysis 14 (1994), 257-266.
[N22] On the distribution of square-full and cube-full numbers, Monatsh. Math. 120 (1995), 105-119.
[N23] Double exponential sums, Analysis 16 (1996), 109-123.
[N24] Lattice points in large convex bodies: Analytic foundations, in: W. G. Nowak, J. Schoißengeier (Eds.), Proc. Conf. Analytic and Elementary Number Theory, Vienna 1996, pp. 149-164.
[N25] Weighted lattice points in threedimensional convex bodies and the number of lattice points in parts of elliptic paraboloids, J. Reine Angew. Math. 485 (1997), 11-23.
[N26] Lattice points in super spheres, Commentat. Math. Univ. Carolinae 40 (1999), 373-391.
[N27] Lattice points in three-dimensional convex bodies, Math. Nachr. 212 (2000), 77-90.
[N28] with W. G. Nowak, Primitive lattice points in a thin strip along the boundary of a large convex planar domain, Acta Arith. 99 (2001), 331-341.
[N29] A sum formula related to ellipsoids with applications to lattice point theory, Abh. Math. Semin. Univ. Hamburg 71 (2001), 143-159.
[N30] Lattice points in three-dimensional convex bodies with points of Gaussian curvature zero at the boundary, Monatsh. Math. 137 (2002), 197-211.
[N31] Lattice points in some special three-dimensional convex bodies with point of Gaussian curvature zero at the boundary, Commentat. Math. Univ. Carolinae 43 (2002), 755-771.
[N32] Lattice points in planar convex domains, Monatsh. Math. 143 (2004), 145-162.
[N33] Estimates of Weyl's exponential sums for very small values of the variable, Arch. Math. 83 (2004), 328-339.
[N34] Effektive Abschätzungen für den Gitterrest gewisser ebener und dreidimensionaler Bereiche (Effective estimates for the lattice point discrepancy of certain planar and three-dimensional domains), Monatsh. Math. 146 (2005), 21-35.
[N35] with A. Ivic, M. Kühleitner, W. G. Nowak, Lattice points in large regions and related arithmetic functions: recent developments in a very classic topic. In: W. Schwarz, J. Steuding (eds.), Elementary and analytic number theory. Proceedings of the ELAZ conference, Mainz, Germany, May 24-28, 2004. Franz Steiner Verlag, Stuttgart, 2006. Schriften der Wissenschaftlichen Gesellschaft an der Johann-Wolfgang-Goethe-Universität Frankfurt am Main 20, pp. 89-128.
[N36] with W. G. Nowak, Eine explizite Abschätzung für die Gitter-Diskrepanz von Rotationsellipsoiden (An effective estimate for the lattice point discrepancy of spheroids), Monatsh. Math. 152 (2007), 45-61.

Selected papers on analysis

[A1] Eine Verallgemeinerung der Laplace- und Meijer-Transformation (A generalization of the Laplace- and the Meijer-transformation), Wiss. Z. Friedrich Schiller - Univ. Jena, Math.-Naturw. Reihe 14 (1965), 369-381.
[A2] Die Faltung der L-Transformation (The convolution of the L-transformation), Wiss. Z. Friedrich Schiller - Univ. Jena, Math.-Naturw. Reihe 14 (1965), 383-390.
[A3] Übertragung der Post-Widderschen Umkehrformel der Laplace-Transformation auf die L-Transformation (Transference of the Post-Widder inversion formula for the Laplace transformation to the L-transformation), Math. Nachr. 35 (1967), 295-304.
[A4] Differentialgleichungen vom Fokker-Planckschen Typ (Differential equations of Fokker-Planck type), Math. Nachr. 35 (1967), 137-147.
[A5] Differentiationssätze der L-Transformation und Differentialgleichungen nach dem Operator (d⁄dt)t^1⁄n-ν (t^1-1⁄nd⁄dt)^n-1 t^ν+1-2⁄n (Differentiation theorems of the L-transformation and differential equations with respect to the operator (d⁄dt)t^1⁄n-ν (t^1-1⁄nd⁄dt)^n-1 t^ν+1-2⁄n), Math. Nachr. 35 (1967), 105-114.
[A6] Asymptotische Entwicklung von Funktionen und asymptotische Lösung von Funktionalgleichungen in mehreren Veränderlichen (Asymptotic expansion of functions and asymptotic solution of functional equations in many variables), Math. Nachr. 78 (1977), 249-261.
[A7] with H. Menzer, Asymptotische Entwicklung von Integralen mit zwei Parametern (Asymptotic expansion of integrals with two parameters), Acta Math. Acad. Sci. Hung. 35 (1980), 311-333.

Finally, a paper on probability theory

[P1] Zweidimensionale Todesprozesse in stetiger Zeit (Two-dimensional death processes with continous time), Math. Nachr. 85 (1978), 315-342.