| Title:
|
Lattice points in super spheres (English) |
| Author:
|
Krätzel, Ekkehard |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
373-391 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $p<k<2p-4$. (English) |
| Keyword:
|
lattice points |
| Keyword:
|
exponential sums |
| MSC:
|
11P21 |
| idZBL:
|
Zbl 0993.11050 |
| idMR:
|
MR1732659 |
| . |
| Date available:
|
2009-01-08T18:53:20Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119094 |
| . |
| Reference:
|
[1] Copson E.T.: Asymptotic Expansions.Cambridge University Press, Cambridge, 1965. Zbl 1096.41001, MR 0168979 |
| Reference:
|
[2] Hoeppner S., Krätzel E.: The number of lattice points inside and on the surface $|t_1|^k+|t_2|^k+\ldots +|t_n|^k=x$.Math. Nachr. 163 (1993), 257-268. MR 1235070 |
| Reference:
|
[3] Krätzel E.: Lattice Points.DVW, Berlin, 1988 and Kluwer, Dordrecht-Boston-London, 1988. MR 0998378 |
| Reference:
|
[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. MR 1227866 |
| Reference:
|
[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. |
| Reference:
|
[6] Schmidt-Röh R.: Ein additives Gitterpunktproblem.Doctoral Thesis, FSU Jena, 1989. |
| Reference:
|
[7] Schnabel L.: Über eine Verallgemeinerung des Kreisproblems.Wiss. Z. FSU Jena, Math.-Naturwiss. R. 31 (1982), 667-681. Zbl 0497.10038, MR 0682557 |
| Reference:
|
[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. MR 0112883 |
| . |
