| Title:
|
Primitive lattice points inside an ellipse (English) |
| Author:
|
Nowak, Werner Georg |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
55 |
| Issue:
|
2 |
| Year:
|
2005 |
| Pages:
|
519-530 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $Q(u, v)$ be a positive definite binary quadratic form with arbitrary real coefficients. For large real $x$, one may ask for the number $B(x)$ of primitive lattice points (integer points $(m, n)$ with $\gcd (M,n) =1$) in the ellipse disc $Q(u, v)\le x$, in particular, for the remainder term $R(x)$ in the asymptotics for $B(x)$. While upper bounds for $R(x)$ depend on zero-free regions of the zeta-function, and thus, in most published results, on the Riemann Hypothesis, the present paper deals with a lower estimate. It is proved that the absolute value or $R(x)$ is, in integral mean, at least a positive constant $c$ time $x^{1/4}$. Furthermore, it is shown how to find an explicit value for $c$, for each specific given form $Q$. (English) |
| Keyword:
|
primitive lattice points |
| Keyword:
|
lattice point discrepancy |
| Keyword:
|
planar domains |
| MSC:
|
11E45 |
| MSC:
|
11P21 |
| idZBL:
|
Zbl 1081.11064 |
| idMR:
|
MR2137159 |
| . |
| Date available:
|
2009-09-24T11:25:16Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127999 |
| . |
| Reference:
|
[1] P. Bleher: On the distribution of the number of lattice points inside a family of convex ovals.Duke Math. J. 67 (1992), 461–481. Zbl 0762.11031, MR 1181309 |
| Reference:
|
[2] J. B. Conrey: More than two fifth of the zeros of the Riemann zeta-function are on the critical line.J. Reine Angew. Math. 399 (1989), 1–26. MR 1004130 |
| Reference:
|
[3] H. Davenport and H. Heilbronn: On the zeros of certain Dirichlet series I.J. London Math. Soc. 11 (1936), 181–185. MR 1574345 |
| Reference:
|
[4] H. Davenport and H. Heilbronn: On the zeros of certain Dirichlet series II.J. London Math. Soc. 11 (1936), 307–312. MR 1574931 |
| Reference:
|
[5] I. S. Gradshteyn and I. M. Ryzhik: Table of Integrals, Series, and Products, 5th ed..A. Jeffrey (ed.), Academic Press, San Diego, 1994. MR 1243179 |
| Reference:
|
[6] M. N. Huxley: Exponential sums and lattice points II.Proc. London Math. Soc. 66 (1993), 279–301. Zbl 0820.11060, MR 1199067 |
| Reference:
|
[7] M. N. Huxley: Area, Lattice Points, and Exponential Sums. LMS Monographs, New Ser. Vol. 13.Clarendon Press, Oxford, 1996. MR 1420620 |
| Reference:
|
[8] M. N. Huxley: Exponential sums and lattice points III.Proc. London Math. Soc. 87 (2003), 591–609. Zbl 1065.11079, MR 2005876 |
| Reference:
|
[9] M. N. Huxley and W. G. Nowak: Primitive lattice points in convex planar domains.Acta Arithm. 76 (1996), 271–283. MR 1397317, 10.4064/aa-76-3-271-283 |
| Reference:
|
[10] A. Ivić: The Riemann zeta-function.Wiley & Sons, New York, 1985. MR 0792089 |
| Reference:
|
[11] E. Krätzel: Lattice Points.Kluwer Academic Publishers, Berlin, 1988. MR 0998378 |
| Reference:
|
[12] E. Krätzel: Analytische Funktionen in der Zahlentheorie.Teubner, Wiesbaden, 2000. MR 1889901 |
| Reference:
|
[13] N. Levinson: More than one third of the zeros of Riemann’s zeta-function are on $\sigma =\frac{1}{2}$.Adv. Math. 13 (1974), 383–436. MR 0564081, 10.1016/0001-8708(74)90074-7 |
| Reference:
|
[14] W. Müller: Lattice points in convex planar domains: Power moments with an application to primitive lattice points.In: Proc. Number Theory Conf., Vienna 1996, W. G. Nowak, J. Schoißengeier (eds.), , Vienna, 1996, pp. 189–199. |
| Reference:
|
[15] W. G. Nowak: An $\Omega $-estimate for the lattice rest of a convex planar domain.Proc.Roy. Soc. Edinburgh, Sect. A 100 (1985), 295–299. Zbl 0582.10033, MR 0807708, 10.1017/S0308210500013834 |
| Reference:
|
[16] W. G. Nowak: On the mean lattice point discrepancy of a convex disc.Arch. Math. (Basel) 78 (2002), 241–248. Zbl 1013.11065, MR 1888708, 10.1007/s00013-002-8242-0 |
| Reference:
|
[17] J. Pintz: On the distribution of square-free numbers.J. London Math. Soc. 28 (1983), 401–405. Zbl 0532.10025, MR 0724708 |
| Reference:
|
[18] H. S. A. Potter: Approximate equations for the Epstein zeta-function.Proc. London Math. Soc. 36 (1934), 501–515. Zbl 0008.30001 |
| Reference:
|
[19] A. Selberg: On the Zeros of Riemann’s zeta-function.Skr. Norske Vid. Akad., Oslo, 1943. Zbl 0028.11101, MR 0010712 |
| Reference:
|
[20] E. C. Titchmarsh: The Theory of the Riemann zeta-function, 2nd ed.Clarendon Press, Oxford, 1986. Zbl 0601.10026, MR 0882550 |
| Reference:
|
[21] M. Voronin: On the zeros of zeta-functions of quadratic forms.Trudy Mat. Inst. Steklova 142 (1976), 135–147. Zbl 0432.10009, MR 0562285 |
| Reference:
|
[22] : Wolfram Research, Inc., Mathematica 4.1., Champaign, 2001. |
| Reference:
|
[23] J. Wu: On the primitive circle problem.Monatsh. Math. 135 (2002), 69–81. Zbl 0994.11035, MR 1894296, 10.1007/s006050200006 |
| Reference:
|
[24] W. Zhai, X. D. Cao: On the number of coprime integer pairs within a circle.Acta Arithm. 90 (1999), 1–16. MR 1708625 |
| Reference:
|
[25] W. Zhai: On primitive lattice points in planar domains.Acta Arithm. 109 (2003), 1–26. Zbl 1027.11075, MR 1980849, 10.4064/aa109-1-1 |
| . |
