[1] P. T. Bateman :
“On the representations of a number as the sum of three squares Trans. Am. Math. Soc.
71 : 1
(1951 ),
pp. 70–101 .
This is Bateman’s 1946 PhD thesis .
MR
0042438 Zbl
0043.04603 article
BibTeX
@article {key0042438m,
AUTHOR = {Bateman, Paul T.},
TITLE = {On the representations of a number as
the sum of three squares},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {71},
NUMBER = {1},
YEAR = {1951},
PAGES = {70--101},
DOI = {10.2307/1990859},
NOTE = {This is Bateman's
1946 PhD thesis. MR:0042438. Zbl:0043.04603.},
ISSN = {0002-9947},
}
[2] P. T. Bateman and E. Grosswald :
“On Epstein’s zeta function Acta Arith.
9 : 4
(1964 ),
pp. 365–373 .
Dedicated to Professor L. J. Mordell on the occasion of his seventy-fifth birthday.
MR
0179141 Zbl
0128.27004 article
Abstract
People
BibTeX
@article {key0179141m,
AUTHOR = {Bateman, P. T. and Grosswald, E.},
TITLE = {On {E}pstein's zeta function},
JOURNAL = {Acta Arith.},
FJOURNAL = {Acta Arithmetica. Polska Akademia Nauk.
Instytut Matematyczny},
VOLUME = {9},
NUMBER = {4},
YEAR = {1964},
PAGES = {365--373},
URL = {http://matwbn.icm.edu.pl/ksiazki/aa/aa9/aa9136.pdf},
NOTE = {Dedicated to Professor L.~J. Mordell
on the occasion of his seventy-fifth
birthday. MR:0179141. Zbl:0128.27004.},
ISSN = {0065-1036},
}
[3] P. T. Bateman :
“Multiplicative arithmetic functions and the representation of integers as sums of squares ,”
pp. 9–13
in
Proceedings of the 1972 number theory conference
(Boulder, CO, 14–18 August 1972 ).
University of Colorado ,
1972 .
MR
0389830 Zbl
0323.10019 incollection
BibTeX
@incollection {key0389830m,
AUTHOR = {Bateman, Paul T.},
TITLE = {Multiplicative arithmetic functions
and the representation of integers as
sums of squares},
BOOKTITLE = {Proceedings of the 1972 number theory
conference},
PUBLISHER = {University of Colorado},
YEAR = {1972},
PAGES = {9--13},
NOTE = {(Boulder, CO, 14--18 August 1972). MR:0389830.
Zbl:0323.10019.},
}
[4] P. T. Bateman and E. Grosswald :
“Positive integers expressible as a sum of three squares in essentially only one way J. Number Theory
19 : 3
(December 1984 ),
pp. 301–308 .
Dedicated to the memory of Ernst Straus (1922–1983) [misprinted on paper as “Strass”].
MR
769785 Zbl
0558.10038 article
Abstract
People
BibTeX
The set \( S \) consisting of those positive integers \( n \) which are uniquely expressible in the form
\[ n = a^2 + b^2 + c^2, \quad a \geq b \geq c \geq 0 ,\]
is considered. Since \( n \in S \) if and only if \( 4n \in S \) , we may restrict attention to those \( n \) not divisible by 4. Classical formulas and the theorem that there are only finitely many imaginary quadratic fields with given class number imply that there are only finitely many \( n \in S \) with \( n = 0 \pmod{4} \) . More specifically, from the existing knowledge of all the imaginary quadratic fields with odd discriminant and class number 1 or 2 it is readily deduced that there are precisely twelve positive integers \( n \) such that \( n \in S \) and \( n \in 3 \pmod8 \) . To determine those \( n \in S \) such that \( n \equiv 1,2 \) , \( 5,6 \pmod8 \) requires the determination of the imaginary quadratic fields with even discriminant and class number 1, 2, or 4. While the latter information is known empirically, it has not been proved that the known list of 33 such fields is complete. If it is complete, then our arguments show that there are exactly 21 positive integers \( n \) such that \( n \in S \) and \( n \equiv 1,2 \) , \( 5,6 \pmod8 \) .
@article {key769785m,
AUTHOR = {Bateman, Paul T. and Grosswald, Emil},
TITLE = {Positive integers expressible as a sum
of three squares in essentially only
one way},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {19},
NUMBER = {3},
MONTH = {December},
YEAR = {1984},
PAGES = {301--308},
DOI = {10.1016/0022-314X(84)90074-X},
NOTE = {Dedicated to the memory of Ernst Straus
(1922--1983) [misprinted on paper as
``Strass'']. MR:769785. Zbl:0558.10038.},
ISSN = {0022-314X},
CODEN = {JNUTA9},
}
[5] P. T. Bateman :
“Integers expressible in a given number of ways as a sum of two squares ,”
pp. 37–45
in
A tribute to Emil Grosswald: Number theory and related analysis .
Edited by M. I. Knopp and M. Sheingorn .
Contemporary Mathematics 143 .
American Mathematical Society (Providence, RI ),
1993 .
Dedicated to the memory of Elizabeth and Emil Grosswald.
MR
1210510 Zbl
0790.11032 incollection
Abstract
People
BibTeX
For a fixed positive integer \( m \) , we give an asymptotic formula for the number of postive integers \( n \) not exceeding \( x \) having the property that there are exactly \( m \) pairs of integers \( u_1 \) , \( u_2 \) with
\[ n = u_1^2 + u_2^2\( , \) u_1 \geq u_2 \geq 0 .\]
@incollection {key1210510m,
AUTHOR = {Bateman, Paul T.},
TITLE = {Integers expressible in a given number
of ways as a sum of two squares},
BOOKTITLE = {A tribute to {E}mil {G}rosswald: {N}umber
theory and related analysis},
EDITOR = {Knopp, Marvin Isadore and Sheingorn,
Mark},
SERIES = {Contemporary Mathematics},
NUMBER = {143},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1993},
PAGES = {37--45},
NOTE = {Dedicated to the memory of Elizabeth
and Emil Grosswald. MR:1210510. Zbl:0790.11032.},
ISSN = {0271-4132},
ISBN = {9780821851555},
}
[6] P. T. Bateman, A. J. Hildebrand, and G. B. Purdy :
“Sums of distinct squares Acta Arith.
67 : 4
(1994 ),
pp. 349–380 .
MR
1301824 Zbl
0815.11048 article
Abstract
People
BibTeX
Throughout this paper we shall suppose that \( s \) is an integer \( \geq 5 \) . Then order of magnitude considerations show that every sufficiently large integer is expressible as a sum of \( s \) distinct non-zero squares. In fact, E. M. Wright [1933] proved that, if \( s \geq 5 \) , then for large \( n \) we can essentially prescribe the ratios of the squares in expressing \( n \) as a sum of \( s \) squares. Thus, for each \( s \geq 5 \) there exists a largest integer \( N(s) \) which is not expressible as a sum of \( s \) distinct non-zero squares. In this paper we shall obtain asymptotic estimates for \( N(s) \) .
@article {key1301824m,
AUTHOR = {Bateman, Paul T. and Hildebrand, Adolf
J. and Purdy, George B.},
TITLE = {Sums of distinct squares},
JOURNAL = {Acta Arith.},
FJOURNAL = {Acta Arithmetica},
VOLUME = {67},
NUMBER = {4},
YEAR = {1994},
PAGES = {349--380},
URL = {http://matwbn.icm.edu.pl/ksiazki/aa/aa67/aa6745.pdf},
NOTE = {MR:1301824. Zbl:0815.11048.},
ISSN = {0065-1036},
CODEN = {AARIA9},
}
[7] P. T. Bateman :
“The asymptotic formula for the number of representations of an integer as a sum of five squares 129–139
in
Analytic number theory: Proceedings of a conference in honor of Heini Halberstam
(Allerton Park, IL, 16–20 May 1995 ),
vol. 1 .
Edited by B. C. Berndt, H. G. Diamond, and A. J. Hildebrand .
Progress in Mathematics 138 .
Birkhäuser (Boston ),
1996 .
Dedicated to Heini and Doreen Halberstam.
MR
1399334 Zbl
0857.11019 incollection
Abstract
People
BibTeX
In this paper we remark that the asymptotic formula for the number of representations of an integer as a sum of five squares can be derived by a simple elementary argument from Jacobi’s formula for the number of representations of an integer as a sum of four squares. A by-product of our argument is an asymptotic formula for the sum
\[ \sum_{|j| < n^{1/2}} \sigma(n - j^2), \]
where \( \sigma \) denotes the sum-of-divisors function.
@incollection {key1399334m,
AUTHOR = {Bateman, Paul T.},
TITLE = {The asymptotic formula for the number
of representations of an integer as
a sum of five squares},
BOOKTITLE = {Analytic number theory: {P}roceedings
of a conference in honor of {H}eini
{H}alberstam},
EDITOR = {Berndt, Bruce C. and Diamond, Harold
G. and Hildebrand, Adolf J.},
VOLUME = {1},
SERIES = {Progress in Mathematics},
NUMBER = {138},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston},
YEAR = {1996},
PAGES = {129--139},
DOI = {10.1007/978-1-4612-4086-0_6},
NOTE = {(Allerton Park, IL, 16--20 May 1995).
Dedicated to Heini and Doreen Halberstam.
MR:1399334. Zbl:0857.11019.},
ISSN = {0743-1643},
ISBN = {9781461286455},
}
[8] P. T. Bateman and M. I. Knopp :
“Some new old-fashioned modular identities Ramanujan J.
2 : 1–2
(1998 ),
pp. 247–269 .
Dedicated to the memory of Paul Erdős.
MR
1642881 Zbl
0909.11018 article
Abstract
People
BibTeX
This paper uses modular functions on the theta group to derive an exact formula for the sum
\[ \sum_{|j|\leq n^{1/2}}\sigma(n - j^2) \]
in terms of the singular series for the number of representations of an integer as a sum of five squares. (Here \( \sigma(k) \) denotes the sum of the divisors of \( k \) if \( k \) is a positive integer and \( \sigma(0) = -1/24 \) .)
Several related identities are derived and discussed.
Two devices are used in the proofs. The first device establishes the equality of two expressions, neither of which is a modular form, by showing that the square of their difference is a modular form. The second device shows that a certain modular function is identically zero by noting that it has more zeros than poles in a fundamental region.
@article {key1642881m,
AUTHOR = {Bateman, Paul T. and Knopp, Marvin I.},
TITLE = {Some new old-fashioned modular identities},
JOURNAL = {Ramanujan J.},
FJOURNAL = {Ramanujan Journal},
VOLUME = {2},
NUMBER = {1--2},
YEAR = {1998},
PAGES = {247--269},
DOI = {10.1023/A:1009782529605},
NOTE = {Dedicated to the memory of Paul Erd\H{o}s.
MR:1642881. Zbl:0909.11018.},
ISSN = {1382-4090},
CODEN = {RAJOF9},
}
[9] P. T. Bateman, B. A. Datskovsky, and M. I. Knopp :
“Sums of squares and the preservation of modularity under congruence restrictions 59–71
in
Symbolic computation, number theory, special functions, physics and combinatorics
(Gainesville, FL, 11–13 November 1999 ).
Edited by F. G. Garvan and M. E. H. Ismail .
Developments in Mathematics 4 .
Kluwer Academic (Dordrecht ),
2001 .
MR
1880079 Zbl
1040.11018 incollection
Abstract
People
BibTeX
If \( s \) is a fixed positive integer and \( n \) is any nonnegative integer, let \( r_s(8n + s) \) be the number of solutions of the equation
\[ x_1^2 + x_2^2 + \cdots + x_2^s = 8n + s \]
in integers \( x_1, x_2, \dots, x_s \) , and let \( r_s^*(8n + s) \) be the number of solutions of the same equation in odd integers. Alternatively, \( r_s^*(8n + s) \) is the number of ways of expressing \( n \) as a sum of triangular numbers, i.e., the number of solutions of the equation
\[ \frac{y_1(y_1 - 1)}{2} + \frac{y_2(y_2 - 1)}{2} + \cdots +\frac{y_s(y_s - 1)}{2} = n \]
in integers \( y_1, y_2,\dots, y_s \) . It is known that for \( 1 \leq s \leq 7 \) there exists a positive constant \( c_s \) such that
\begin{equation*}\tag{*} r_s(8n + s) = c_sr_s^*(8n + s) \end{equation*}
for all nonnegative integers \( n \) . In this paper we prove that if \( s > 7 \) , then no constant \( c_s \) exists such that (*) holds, even for all sufficiently large \( n \) . The proof uses the theory of modular forms of weight \( s/2 \) and appropriate multiplier system on the group \( \Gamma_0(64) \) and the so-called principle of the preservation of modularity under congruence restrictions.
@incollection {key1880079m,
AUTHOR = {Bateman, Paul T. and Datskovsky, Boris
A. and Knopp, Marvin I.},
TITLE = {Sums of squares and the preservation
of modularity under congruence restrictions},
BOOKTITLE = {Symbolic computation, number theory,
special functions, physics and combinatorics},
EDITOR = {Garvan, Frank G. and Ismail, Mourad
E. H.},
SERIES = {Developments in Mathematics},
NUMBER = {4},
PUBLISHER = {Kluwer Academic},
ADDRESS = {Dordrecht},
YEAR = {2001},
PAGES = {59--71},
DOI = {10.1007/978-1-4613-0257-5_4},
NOTE = {(Gainesville, FL, 11--13 November 1999).
MR:1880079. Zbl:1040.11018.},
ISSN = {1389-2177},
}
