Abstract
We give asymptotic upper and lower bounds for the number of squarefree $d$ ($0\lt d\leq X$) such that the equation $x^2-dy^2=-1$ is solvable. These estimates, as usual, can equivalently be interpreted in terms of real quadratic fields with a fundamental unit with norm $-1$ and give strong evidence in the direction of a conjecture due to P. Stevenhagen.
-
[Bl] V. Blomer, On the negative Pell equation, 2006.
@misc{Bl,
author={Blomer, V.},
TITLE={On the negative Pell equation},
NOTE={preprint},
YEAR={2006},
} -
[Br] J. Brüdern, Einf" uhrung in die Analytische Zahlentheorie, Berlin: Springer-Lehrbuch, 1995.
@book{Br,
author={Brüdern, J.},
TITLE={Einf\" uhrung in die Analytische Zahlentheorie},
PUBLISHER={Springer-Lehrbuch},
ADDRESS={Berlin},
YEAR={1995},
ZBLNUMBER={0830.11001},
} -
[CoLe]
H. Cohen and H. W. Lenstra Jr., "Heuristics on class groups of number fields," in Number Theory, New York: Springer-Verlag, 1984, vol. 1068, pp. 33-62.
@incollection {CoLe, MRKEY = {756082},
AUTHOR = {Cohen, Harvey and Lenstra, Jr., H. W.},
TITLE = {Heuristics on class groups of number fields},
BOOKTITLE = {Number Theory},
VENUE={{N}oordwijkerhout, 1983},
SERIES = {Lecture Notes in Math.},
VOLUME = {1068},
PAGES = {33--62},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1984},
MRCLASS = {11R29 (13C13)},
MRNUMBER = {85j:11144},
MRREVIEWER = {F. J. van der Linden},
DOI = {10.1007/BFb0099440},
ZBLNUMBER = {0558.12002},
} -
[Cohn] H. Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, New York: Springer-Verlag, 1978.
@book {Cohn, MRKEY = {506156},
AUTHOR = {Cohn, Harvey},
TITLE = {A Classical Invitation to Algebraic Numbers and Class Fields},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1978},
PAGES = {xiii+328},
ISBN = {0-387-90345-3},
MRCLASS = {12-01 (12Axx)},
MRNUMBER = {80c:12001},
MRREVIEWER = {Ezra Brown},
ZBLNUMBER = {0395.12001},
} -
[Com] L. Comtet, Advanced Combinatorics, enlarged ed., Dordrecht: D. Reidel Publishing Co., 1974.
@book {Com, MRKEY = {0460128},
AUTHOR = {Comtet, Louis},
TITLE = {Advanced Combinatorics},
EDITION = {enlarged},
PUBLISHER = {D. Reidel Publishing Co.},
ADDRESS = {Dordrecht},
YEAR = {1974},
PAGES = {xi+343},
ISBN = {90-277-0441-4},
MRCLASS = {05-02},
MRNUMBER = {57 \#124},
ZBLNUMBER = {0283.05001},
} -
[Dav] H. Davenport, Multiplicative Number Theory, Second ed., New York: Springer-Verlag, 1980, vol. 74.
@book {Dav, MRKEY = {606931},
AUTHOR = {Davenport, Harold},
TITLE = {Multiplicative Number Theory},
SERIES = {Grad. Texts Math.},
VOLUME = {74},
EDITION = {Second},
NOTE = {Revised by Hugh L. Montgomery},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1980},
PAGES = {xiii+177},
ISBN = {0-387-90533-2},
MRCLASS = {10-01 (10-02 10Hxx)},
MRNUMBER = {82m:10001},
MRREVIEWER = {H.-E. Richert},
ZBLNUMBER = {0453.10002},
} -
[Dieu] J. A. Dieudonné, La Géométrie des Groupes Classiques, New York: Springer-Verlag, 1971.
@book {Dieu, MRKEY = {0310083},
AUTHOR = {Dieudonn{é},
Jean A.},
TITLE = {La Géométrie des Groupes Classiques},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1971},
PAGES = {viii+129},
MRCLASS = {20G15 (15A63 15A66)},
MRNUMBER = {46 \#9186},
MRREVIEWER = {J. Burlak},
ZBLNUMBER = {0221.20056},
} -
[Dir] L. P. G. Dirichlet, Vorlesungen über Zahlentheorie, New York: Chelsea Publishing Co., 1968.
@book {Dir, MRKEY = {0237283},
AUTHOR = {Dirichlet, P. G. Lejeune},
TITLE = {Vorlesungen über {Z}ahlentheorie},
PUBLISHER = {Chelsea Publishing Co.},
ADDRESS = {New York},
YEAR = {1968},
PAGES = {xvii+657},
MRCLASS = {01.60 (10.00)},
MRNUMBER = {38 \#5573},
} -
[Est] T. Estermann, Introduction to Modern Prime Number Theory, Cambridge: Cambridge, at the Univ. Press, 1952, vol. 41.
@book {Est, MRKEY = {0047692},
AUTHOR = {Estermann, T.},
TITLE = {Introduction to Modern Prime Number Theory},
SERIES = {Cambridge Tracts Math. and Math. Phys.},
VOLUME={41},
PUBLISHER = {Cambridge, at the Univ. Press},
ADDRESS={Cambridge},
YEAR = {1952},
PAGES = {x+75},
MRCLASS = {10.0X},
MRNUMBER = {13,915b},
MRREVIEWER = {N. G. de Bruijn},
ZBLNUMBER = {0049.03103},
} -
[FoKl2]
&. Fouvry and J. Klüners, "Cohen-Lenstra heuristics of quadratic number fields," in Algorithmic Number Theory, New York: Springer-Verlag, 2006, vol. 4076, pp. 40-55.
@incollection {FoKl2, MRKEY = {2282914},
AUTHOR = {Fouvry, {É}tienne and Kl{ü}ners, J{ü}rgen},
TITLE = {Cohen-{L}enstra heuristics of quadratic number fields},
BOOKTITLE = {Algorithmic Number Theory},
SERIES = {Lecture Notes in Comput. Sci.},
VOLUME = {4076},
PAGES = {40--55},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2006},
MRCLASS = {11R29 (11R11 11Y40)},
MRNUMBER = {2008f:11125},
MRREVIEWER = {Michael J. Jacobson, Jr.},
DOI = {10.1007/11792086_4},
ZBLNUMBER={1143.11352},
} -
[new]
&. Fouvry and J. Klüners, "The parity of the period of the continued fraction of $\sqrt{d}$," PLMS, vol. 101, pp. 337-391, 2010.
@article{new,
author = {Fouvry, {É}tienne and Kl{ü}ners, J{ü}rgen},
TITLE={The parity of the period of the continued fraction of $\sqrt{d}$},
JOURNAL={PLMS},
VOLUME={101},
YEAR={2010},
PAGES={337--391},
URL={http://plms.oxfordjournals.org/content/101/2/337.abstract},
} -
[FoKl1]
&. Fouvry and J. Klüners, "On the 4-rank of class groups of quadratic number fields," Invent. Math., vol. 167, iss. 3, pp. 455-513, 2007.
@article {FoKl1, MRKEY = {2276261},
AUTHOR = {Fouvry, {É}tienne and Kl{ü}ners, J{ü}rgen},
TITLE = {On the 4-rank of class groups of quadratic number fields},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {167},
YEAR = {2007},
NUMBER = {3},
PAGES = {455--513},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11R29 (11R11 11R45)},
MRNUMBER = {2007k:11187},
MRREVIEWER = {J. Browkin},
DOI = {10.1007/s00222-006-0021-2},
ZBLNUMBER={1126.11062},
} -
[FrIw]
J. Friedlander and H. Iwaniec, "The polynomial $X^2+Y^4$ captures its primes," Ann. of Math., vol. 148, iss. 3, pp. 945-1040, 1998.
@article {FrIw, MRKEY = {1670065},
AUTHOR = {Friedlander, John and Iwaniec, Henryk},
TITLE = {The polynomial {$X\sp 2+Y\sp 4$} captures its primes},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {148},
YEAR = {1998},
NUMBER = {3},
PAGES = {945--1040},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {11N05 (11N32 11N35 11N36)},
MRNUMBER = {2000c:11150a},
MRREVIEWER = {Andrew Granville},
DOI = {10.2307/121034},
ZBLNUMBER = {0926.11068},
} -
[Ge]
F. Gerth III, "The $4$-class ranks of quadratic fields," Invent. Math., vol. 77, iss. 3, pp. 489-515, 1984.
@article {Ge, MRKEY = {759260},
AUTHOR = {Gerth, III, Frank},
TITLE = {The {$4$}-class ranks of quadratic fields},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {77},
YEAR = {1984},
NUMBER = {3},
PAGES = {489--515},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11R11 (11R29)},
MRNUMBER = {85j:11137},
MRREVIEWER = {Kenz{ô} Komatsu},
DOI = {10.1007/BF01388835},
ZBLNUMBER = {0533.12004},
} -
[Gol]
L. J. Goldstein, "A generalization of the Siegel-Walfisz theorem," Trans. Amer. Math. Soc., vol. 149, pp. 417-429, 1970.
@article {Gol, MRKEY = {0274416},
AUTHOR = {Goldstein, Larry Joel},
TITLE = {A generalization of the {S}iegel-{W}alfisz theorem},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {149},
YEAR = {1970},
PAGES = {417--429},
ISSN = {0002-9947},
MRCLASS = {10.65},
MRNUMBER = {43 \#181},
MRREVIEWER = {H. L. Montgomery},
DOI = {10.2307/1995404},
ZBLNUMBER = {0201.05701},
} -
[HaRa] G. H. Hardy and S. Ramanujan, "The normal number of prime factors of a number $n$," Quart. J. Math., vol. 48, pp. 76-92, 1920.
@article{HaRa,
author={Hardy, G. H. and Ramanujan, S.},
TITLE={The normal number of prime factors of a number $n$},
JOURNAL={Quart. J. Math.},
VOLUME={48},
PAGES={76--92},
YEAR={1920},
NOTE={see also {\it Collected Works of G. H. Hardy},
Vol. II, Oxford Univ. Press, 1967, 100--113},
} -
[Has] H. Hasse, Number Theory, New York: Springer-Verlag, 1980, vol. 229.
@book {Has, MRKEY = {562104},
AUTHOR = {Hasse, Helmut},
TITLE = {Number Theory},
SERIES = {Grundl. Math. Wissen.},
VOLUME= {229},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1980},
PAGES = {xvii+638},
ISBN = {3-540-08275-1},
MRCLASS = {12-01 (10-01)},
MRNUMBER = {81c:12001b},
MRREVIEWER = {M. Rabindranathan},
ZBLNUMBER = {0423.12002},
} -
[HB3]
D. R. Heath-Brown, "The size of Selmer groups for the congruent number problem. II," Invent. Math., vol. 118, iss. 2, pp. 331-370, 1994.
@article {HB3, MRKEY = {1292115},
AUTHOR = {Heath-Brown, D. R.},
TITLE = {The size of {S}elmer groups for the congruent number problem. {II}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {118},
YEAR = {1994},
NUMBER = {2},
PAGES = {331--370},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11G40 (11G05)},
MRNUMBER = {95h:11064},
MRREVIEWER = {Fernando Q. Gouv{ê}a},
DOI = {10.1007/BF01231536},
ZBLNUMBER = {0815.11032},
} -
[HB4] D. R. Heath-Brown, "A mean value estimate for real character sums," Acta Arith., vol. 72, iss. 3, pp. 235-275, 1995.
@article {HB4, MRKEY = {1347489},
AUTHOR = {Heath-Brown, D. R.},
TITLE = {A mean value estimate for real character sums},
JOURNAL = {Acta Arith.},
FJOURNAL = {Acta Arithmetica},
VOLUME = {72},
YEAR = {1995},
NUMBER = {3},
PAGES = {235--275},
ISSN = {0065-1036},
CODEN = {AARIA9},
MRCLASS = {11L40 (11M06 11M26)},
MRNUMBER = {96h:11081},
MRREVIEWER = {Matti Jutila},
ZBLNUMBER = {0828.11040},
} -
[HB5] D. R. Heath-Brown, "Kummer’s conjecture for cubic Gauss sums," Israel J. Math., vol. 120, iss. , part A, pp. 97-124, 2000.
@article {HB5, MRKEY = {1815372},
AUTHOR = {Heath-Brown, D. R.},
TITLE = {Kummer's conjecture for cubic {G}auss sums},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {120},
YEAR = {2000},
NUMBER = {, part A},
PAGES = {97--124},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {11L05 (11L40)},
MRNUMBER = {2001m:11134},
MRREVIEWER = {Matti Jutila},
ZBLNUMBER = {0989.11042},
} -
[Hec]
E. Hecke, "Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen," Math. Z., vol. 6, iss. 1-2, pp. 11-51, 1920.
@article {Hec, MRKEY = {1544392},
AUTHOR = {Hecke, Erich},
TITLE = {Eine neue {A}rt von {Z}etafunktionen und ihre {B}eziehungen zur {V}erteilung der {P}rimzahlen},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {6},
YEAR = {1920},
NUMBER = {1-2},
PAGES = {11--51},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {Contributed Item},
MRNUMBER = {1544392},
DOI = {10.1007/BF01202991},
JFMNUMBER={47.0152.01},
} -
[Hec2] E. Hecke, Lectures on the Theory of Algebraic Numbers, New York: Springer-Verlag, 1981, vol. 77.
@book {Hec2, MRKEY = {638719},
AUTHOR = {Hecke, Erich},
TITLE = {Lectures on the Theory of Algebraic Numbers},
SERIES = {Grad. Texts Math.},
VOLUME = {77},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1981},
PAGES = {xii+239},
ISBN = {0-387-90595-2},
MRCLASS = {12-01 (01A75)},
MRNUMBER = {83m:12001},
ZBLNUMBER = {0504.12001},
} -
[Hei]
H. Heilbronn, "On the averages of some arithmetical functions of two variables," Mathematika, vol. 5, pp. 1-7, 1958.
@article {Hei, MRKEY = {0097362},
AUTHOR = {Heilbronn, H.},
TITLE = {On the averages of some arithmetical functions of two variables},
JOURNAL = {Mathematika},
FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
VOLUME = {5},
YEAR = {1958},
PAGES = {1--7},
ISSN = {0025-5793},
MRCLASS = {10.00},
MRNUMBER = {20 \#3831},
MRREVIEWER = {W. H. Mills},
DOI = {10.1112/S0025579300001273},
ZBLNUMBER = {0125.02604},
} -
[Hoo]
C. Hooley, "On the Pellian equation and the class number of indefinite binary quadratic forms," J. Reine Angew. Math., vol. 353, pp. 98-131, 1984.
@article {Hoo, MRKEY = {765829},
AUTHOR = {Hooley, Christopher},
TITLE = {On the {P}ellian equation and the class number of indefinite binary quadratic forms},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {353},
YEAR = {1984},
PAGES = {98--131},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {11E41 (11E16 11P55)},
MRNUMBER = {86d:11032},
MRREVIEWER = {O. H. K{ö}rner},
DOI = {10.1515/crll.1984.353.98},
ZBLNUMBER = {0539.10019},
} -
[IrRo] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Second ed., New York: Springer-Verlag, 1990, vol. 84.
@book {IrRo, MRKEY = {1070716},
AUTHOR = {Ireland, Kenneth and Rosen, Michael},
TITLE = {A Classical Introduction to Modern Number Theory},
SERIES = {Grad. Texts Math.},
VOLUME = {84},
EDITION = {Second},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1990},
PAGES = {xiv+389},
ISBN = {0-387-97329-X},
MRCLASS = {11-01 (11-02)},
MRNUMBER = {92e:11001},
MRREVIEWER = {Glenn Stevens},
ZBLNUMBER = {0712.11001},
} -
[IwKo] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53.
@book {IwKo, MRKEY = {2061214},
AUTHOR = {Iwaniec, Henryk and Kowalski, Emmanuel},
TITLE = {Analytic Number Theory},
SERIES = {Amer. Math. Soc. Colloq. Publ.},
VOLUME = {53},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2004},
PAGES = {xii+615},
ISBN = {0-8218-3633-1},
MRCLASS = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
MRNUMBER = {2005h:11005},
MRREVIEWER = {K. Soundararajan},
ZBLNUMBER = {1059.11001},
} -
[Jan] G. J. Janusz, Algebraic Number Fields, Second ed., Providence, RI: Amer. Math. Soc., 1996, vol. 7.
@book {Jan, MRKEY = {1362545},
AUTHOR = {Janusz, Gerald J.},
TITLE = {Algebraic Number Fields},
SERIES = {Grad. Stud. Math.s},
VOLUME = {7},
EDITION = {Second},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1996},
PAGES = {x+276},
ISBN = {0-8218-0429-4},
MRCLASS = {11Rxx (11-02 11R37)},
MRNUMBER = {96j:11137},
MRREVIEWER = {W. Narkiewicz},
ZBLNUMBER = {0854.11001},
} -
[KaLa]
M. Karoubi and T. Lambre, "Sur la $K$-théorie du foncteur norme," J. Algebra, vol. 321, iss. 10, pp. 2754-2781, 2009.
@article {KaLa, MRKEY = {2512625},
AUTHOR = {Karoubi, Max and Lambre, Thierry},
TITLE = {Sur la {$K$}-théorie du foncteur norme},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {321},
YEAR = {2009},
NUMBER = {10},
PAGES = {2754--2781},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {11R70 (19F99)},
MRNUMBER = {2010g:11199},
MRREVIEWER = {J{ü}rgen Ritter},
DOI = {10.1016/j.jalgebra.2008.09.044},
ZBLNUMBER = {1178.19003},
} -
@misc{Lem,
author={Lemmermeyer, F.},
TITLE={The $4$-class group of real quadratic number fields},
NOTE={preprint},
URL={http://www.rzuser.uni-heidelberg.de/~hb3/rank4.ps},
ZBLNUMBER={0634.12008},
} -
[Lou] S. Louboutin, "Groupes des classes d’idéaux triviaux," Acta Arith., vol. 54, iss. 1, pp. 61-74, 1989.
@article {Lou, MRKEY = {1024418},
AUTHOR = {Louboutin, St{é}phane},
TITLE = {Groupes des classes d'idéaux triviaux},
JOURNAL = {Acta Arith.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
VOLUME = {54},
YEAR = {1989},
NUMBER = {1},
PAGES = {61--74},
ISSN = {0065-1036},
CODEN = {AARIA9},
MRCLASS = {11R11 (11R29)},
MRNUMBER = {91a:11051},
MRREVIEWER = {F. Diaz y Diaz},
} -
[Mit] T. Mitsui, "Generalized prime number theorem," Jap. J. Math., vol. 26, pp. 1-42, 1956.
@article {Mit, MRKEY = {0092814},
AUTHOR = {Mitsui, Takayoshi},
TITLE = {Generalized prime number theorem},
JOURNAL = {Jap. J. Math.},
VOLUME = {26},
YEAR = {1956},
PAGES = {1--42},
MRCLASS = {10.1X},
MRNUMBER = {19,1161g},
MRREVIEWER = {S. Chowla},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0126.27503},
} -
[Na] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Second ed., New York: Springer-Verlag, 1990.
@book {Na, MRKEY = {1055830},
AUTHOR = {Narkiewicz, W{\l}adys{\l}aw},
TITLE = {Elementary and Analytic Theory of Algebraic Numbers},
EDITION = {Second},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1990},
PAGES = {xiv+746},
ISBN = {3-540-51250-0},
MRCLASS = {11Rxx (11-01)},
MRNUMBER = {91h:11107},
MRREVIEWER = {Dorian Goldfeld},
ZBLNUMBER = {0717.11045},
} -
[ReRe] L. Rédei and H. Reichardt, "Die Anzahl der durch $4$ teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers," J. Reine Angew. Math., vol. 170, pp. 69-74, 1933.
@article{ReRe,
author={Rédei, L. and Reichardt, H.},
TITLE={Die Anzahl der durch $4$ teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers},
JOURNAL={J. Reine Angew. Math.},
VOLUME={170},
PAGES={69--74},
YEAR={1933},
ZBLNUMBER={0009.29401},
} -
[Re1] L. Rédei, "Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlk" orper," J. Reine Angew. Math., vol. 171, pp. 55-60, 1934.
@article{Re1,
author={Rédei, L.},
TITLE={Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlk\" orper},
JOURNAL={J. Reine Angew. Math.},
VOLUME={171},
PAGES={55--60},
YEAR={1934},
ZBLNUMBER={0009.05101},
JFMNUMBER={60.0125.02},
} -
[Re2] L. Rédei, "Eine obere Schranke der Anzahl der durch vier teilbaren invarianten der absoluten Klassengruppe im quadratischen Zahlk" orper," J. Reine Angew. Math., vol. 171, pp. 61-64, 1934.
@article{Re2,
author={Rédei, L.},
TITLE={Eine obere Schranke der Anzahl der durch vier teilbaren invarianten der absoluten Klassengruppe im quadratischen Zahlk\" orper},
JOURNAL={J. Reine Angew. Math.},
VOLUME={171},
PAGES={61--64},
YEAR={1934},
ZBLNUMBER={0010.33801},
} -
[Re3] L. Rédei, "Über die Grundeinheit und die durch 8 teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper," J. Reine Angew. Math., vol. 171, pp. 131-148, 1934.
@article{Re3,
author={Rédei, L.},
TITLE={Über die Grundeinheit und die durch 8 teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper},
JOURNAL={J. Reine Angew. Math.},
VOLUME={171},
PAGES={131--148},
YEAR={1934},
ZBLNUMBER={0010.33802},
} -
[Ri]
G. J. Rieger, "Über die Anzahl der als Summe von zwei Quadraten darstellbaren und in einer primen Restklasse gelegenen Zahlen unterhalb einer positiven Schranke. II," J. Reine Angew. Math., vol. 217, pp. 200-216, 1965.
@article {Ri, MRKEY = {0174533},
AUTHOR = {Rieger, G. J.},
TITLE = {Über die {A}nzahl der als {S}umme von zwei {Q}uadraten darstellbaren und in einer primen {R}estklasse gelegenen {Z}ahlen unterhalb einer positiven {S}chranke. {II}},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {217},
YEAR = {1965},
PAGES = {200--216},
ISSN = {0075-4102},
MRCLASS = {10.43 (10.45)},
MRNUMBER = {30 \#4734},
MRREVIEWER = {R. D. James},
DOI = {10.1515/crll.1965.217.200},
ZBLNUMBER = {0141.04305},
} -
[Serre] J. Serre, Local Fields, New York: Springer-Verlag, 1979.
@book {Serre, MRKEY = {554237},
AUTHOR = {Serre, Jean-Pierre},
TITLE = {Local Fields},
SERIES = {Grad. Texts Math.},
NUMBER = {67},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1979},
PAGES = {viii+241},
ISBN = {0-387-90424-7},
MRCLASS = {12Bxx},
MRNUMBER = {82e:12016},
ZBLNUMBER = {0423.12016},
} -
[Scholz]
A. Scholz, "Über die Lösbarkeit der Gleichung $t^2-Du^2=-4$," Math. Z., vol. 39, iss. 1, pp. 95-111, 1935.
@article {Scholz, MRKEY = {1545490},
AUTHOR = {Scholz, Arnold},
TITLE = {Über die {L}ösbarkeit der {G}leichung {$t\sp 2-Du\sp 2=-4$}},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {39},
YEAR = {1935},
NUMBER = {1},
PAGES = {95--111},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {Contributed Item},
MRNUMBER = {1545490},
DOI = {10.1007/BF01201346},
} -
[Sh]
P. Shiu, "A Brun-Titchmarsh theorem for multiplicative functions," J. Reine Angew. Math., vol. 313, pp. 161-170, 1980.
@article {Sh, MRKEY = {552470},
AUTHOR = {Shiu, P.},
TITLE = {A {B}run-{T}itchmarsh theorem for multiplicative functions},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {313},
YEAR = {1980},
PAGES = {161--170},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {10H25},
MRNUMBER = {81h:10065},
MRREVIEWER = {A. I. Vinogradov},
DOI = {10.1515/crll.1980.313.161},
ZBLNUMBER = {0412.10030},
} -
[St]
P. Stevenhagen, "The number of real quadratic fields having units of negative norm," Experiment. Math., vol. 2, iss. 2, pp. 121-136, 1993.
@article {St, MRKEY = {1259426},
AUTHOR = {Stevenhagen, Peter},
TITLE = {The number of real quadratic fields having units of negative norm},
JOURNAL = {Experiment. Math.},
FJOURNAL = {Experimental Mathematics},
VOLUME = {2},
YEAR = {1993},
NUMBER = {2},
PAGES = {121--136},
ISSN = {1058-6458},
MRCLASS = {11R11 (11R27 11Y40)},
MRNUMBER = {94k:11120},
MRREVIEWER = {Duncan A. Buell},
URL = {http://projecteuclid.org/getRecord?id=euclid.em/1048516217},
ZBLNUMBER = {0792.11041},
} -
[We]
A. Weil, Number Theory : An Approach Through History, From Hammurapi to Legendre, Boston, MA: Birkhäuser, 1984.
@book {We, MRKEY = {734177},
AUTHOR = {Weil, Andr{é}},
TITLE = {Number Theory{\rm :} An Approach Through History, From Hammurapi to Legendre},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1984},
PAGES = {xxi+375},
ISBN = {0-8176-3141-0},
MRCLASS = {01A05 (11-03)},
MRNUMBER = {85c:01004},
MRREVIEWER = {Ezra Brown},
DOI = {10.1007/978-0-8176-4571-7},
ZBLNUMBER = {0531.10001},
}