Lattice points in super spheres

Ekkehard Krätzel

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 2, page 373-391
  • ISSN: 0010-2628

Abstract

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In this article we consider the number R k , p ( x ) of lattice points in p -dimensional super spheres with even power k ≥ 4 . We give an asymptotic expansion of the d -fold anti-derivative of R k , p ( x ) for sufficiently large d . From this we deduce a new estimation for the error term in the asymptotic representation of R k , p ( x ) for p < k < 2 p - 4 .

How to cite

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Krätzel, Ekkehard. "Lattice points in super spheres." Commentationes Mathematicae Universitatis Carolinae 40.2 (1999): 373-391. <http://eudml.org/doc/248389>.

@article{Krätzel1999,
abstract = {In this article we consider the number $R_\{k,p\}(x)$ of lattice points in $p$-dimensional super spheres with even power $k \ge 4$. We give an asymptotic expansion of the $d$-fold anti-derivative of $R_\{k,p\}(x)$ for sufficiently large $d$. From this we deduce a new estimation for the error term in the asymptotic representation of $R_\{k,p\}(x)$ for $p<k<2p-4$.},
author = {Krätzel, Ekkehard},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {lattice points; exponential sums; lattice points; exponential sums; super spheres},
language = {eng},
number = {2},
pages = {373-391},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Lattice points in super spheres},
url = {http://eudml.org/doc/248389},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Krätzel, Ekkehard
TI - Lattice points in super spheres
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 2
SP - 373
EP - 391
AB - In this article we consider the number $R_{k,p}(x)$ of lattice points in $p$-dimensional super spheres with even power $k \ge 4$. We give an asymptotic expansion of the $d$-fold anti-derivative of $R_{k,p}(x)$ for sufficiently large $d$. From this we deduce a new estimation for the error term in the asymptotic representation of $R_{k,p}(x)$ for $p<k<2p-4$.
LA - eng
KW - lattice points; exponential sums; lattice points; exponential sums; super spheres
UR - http://eudml.org/doc/248389
ER -

References

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  1. Copson E.T., Asymptotic Expansions, Cambridge University Press, Cambridge, 1965. Zbl1096.41001MR0168979
  2. Hoeppner S., Krätzel E., The number of lattice points inside and on the surface | t 1 | k + | t 2 | k + ... + | t n | k = x , Math. Nachr. 163 (1993), 257-268. (1993) MR1235070
  3. Krätzel E., Lattice Points, DVW, Berlin, 1988 and Kluwer, Dordrecht-Boston-London, 1988. MR0998378
  4. Kuba G., On the sums of two k-th powers of numbers in residue classes II, Abh. Math. Sem. Hamburg 63 (1993), 87-95. (1993) MR1227866
  5. Müller W., Nowak W.G., Lattice points in planar domains: Applications of Huxley's Discrete Hardy-Littlewood-Method, Numbertheoretic analysis, Vienna 1988-1989, Springer Lecture Notes 1452 (eds. E. Hlawka and R.F. Tichy) (1990), pp.139-164. 
  6. Schmidt-Röh R., Ein additives Gitterpunktproblem, Doctoral Thesis, FSU Jena, 1989. 
  7. Schnabel L., Über eine Verallgemeinerung des Kreisproblems, Wiss. Z. FSU Jena, Math.-Naturwiss. R. 31 (1982), 667-681. (1982) Zbl0497.10038MR0682557
  8. Wild R.E., On the number of lattice points in x t + y t = n t / 2 , Pacific J. Math. 8 (1958), 929-940. (1958) MR0112883

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