P. T. Bateman, S. Chowla, and P. Erdős :
“Remarks on the size of \( L(1,\chi) \) ,”
Publ. Math. Debrecen
1
(1950 ),
pp. 165–182 .
MR
0037322
Zbl
0036.30702
article

Abstract
People
BibTeX

In this paper we consider the value of the Dirichlet \( L(s,\chi) \) functions at \( s = 1,\chi \) being a non-principal residue-character and \( L(s,\chi) \) being defined for \( \mathfrak{R}(s) > 0 \) by
\[ L(s,\chi) = \sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}. \]

@article {key0037322m,
AUTHOR = {Bateman, P. T. and Chowla, S. and Erd\H{o}s,
P.},
TITLE = {Remarks on the size of \$L(1,\chi)\$},
JOURNAL = {Publ. Math. Debrecen},
FJOURNAL = {Publicationes Mathematicae Debrecen},
VOLUME = {1},
YEAR = {1950},
PAGES = {165--182},
URL = {https://www.renyi.hu/~p_erdos/1950-10.pdf},
NOTE = {MR:0037322. Zbl:0036.30702.},
ISSN = {0033-3883},
}
P. Ungar, V. Thebault, P. Bateman, P. Erdos, and F. J. Dyson :
“Advanced problems and solutions: Problems for
solution: 4385–4389 ,”
Amer. Math. Monthly
57 : 3
(1950 ),
pp. 188–189 .
MR
1527514
article

People
BibTeX
@article {key1527514m,
AUTHOR = {Ungar, P. and Thebault, Victor and Bateman,
Paul and Erdos, Paul and Dyson, F. J.},
TITLE = {Advanced problems and solutions: {P}roblems
for solution: 4385--4389},
JOURNAL = {Amer. Math. Monthly},
FJOURNAL = {The American Mathematical Monthly},
VOLUME = {57},
NUMBER = {3},
YEAR = {1950},
PAGES = {188--189},
DOI = {10.2307/2304436},
NOTE = {MR:1527514.},
ISSN = {0002-9890},
CODEN = {AMMYAE},
}
R. L. Gupta, P. T. Bateman, P. Erdos, V. Thebault, and H. F. Sandham :
“Advanced problems and solutions: Problems for
solutions: 4390–4394 ,”
Amer. Math. Monthly
57 : 4
(1950 ),
pp. 265 .
MR
1527543
article

People
BibTeX
@article {key1527543m,
AUTHOR = {Gupta, Roshan Lal and Bateman, P. T.
and Erdos, Paul and Thebault, Victor
and Sandham, H. F.},
TITLE = {Advanced problems and solutions: {P}roblems
for solutions: 4390--4394},
JOURNAL = {Amer. Math. Monthly},
FJOURNAL = {The American Mathematical Monthly},
VOLUME = {57},
NUMBER = {4},
YEAR = {1950},
PAGES = {265},
DOI = {10.2307/2305951},
NOTE = {MR:1527543.},
ISSN = {0002-9890},
CODEN = {AMMYAE},
}
P. Bateman and P. Erdős :
“Geometrical extrema suggested by a lemma of Besicovitch ,”
Am. Math. Mon.
58 : 5
(May 1951 ),
pp. 306–314 .
MR
0041466
Zbl
0043.16202
article

Abstract
People
BibTeX

In [1945] Besicovitch needed as a lemma a result of the following type

Given a set \( \Gamma \) of coplanar circles, the center of no one of them being in the interior of another, and \( U \) the circle (or a circle) of \( \Gamma \) whose radius does not exceed the radius of any other circle of \( \Gamma \) , then the number of circles meeting \( U \) does not exceed 18.

Besicovitch proved the weaker theorem obtained from this one by replacing 18 by 21. In this paper we shall prove Theorem 1 as it stands.

@article {key0041466m,
AUTHOR = {Bateman, Paul and Erd\H{o}s, Paul},
TITLE = {Geometrical extrema suggested by a lemma
of {B}esicovitch},
JOURNAL = {Am. Math. Mon.},
FJOURNAL = {The American Mathematical Monthly},
VOLUME = {58},
NUMBER = {5},
MONTH = {May},
YEAR = {1951},
PAGES = {306--314},
DOI = {10.2307/2307717},
NOTE = {MR:0041466. Zbl:0043.16202.},
ISSN = {0002-9890},
}
P. T. Bateman and P. Erdős :
“Partitions into primes ,”
Publ. Math. Debrecen
4
(1956 ),
pp. 198–200 .
Dedicated to the memory of Tibor Szele.
MR
0079013
Zbl
0073.03101
article

Abstract
People
BibTeX

Let \( P(n) \) denote the number of partitions of the integer \( n \) into primes (1 is not counted as prime), repetitions being allowed. That is, \( P(n) \) is the number of ways \( n \) can be expressed in the form
\[ n_1p_1 + n_2p_2 + \cdots ,\]
where \( p_j \) denotes the \( j \) th prime number and \( n_1,n_2,\dots \) are arbitrary non-negative integers. The purpose of this note is to prove that
\[ P(n+1) \geq P(n) \quad (n=1,2,3,\dots). \]

@article {key0079013m,
AUTHOR = {Bateman, P. T. and Erd\H{o}s, P.},
TITLE = {Partitions into primes},
JOURNAL = {Publ. Math. Debrecen},
FJOURNAL = {Publicationes Mathematicae Debrecen},
VOLUME = {4},
YEAR = {1956},
PAGES = {198--200},
URL = {https://www.renyi.hu/~p_erdos/1956-15.pdf},
NOTE = {Dedicated to the memory of Tibor Szele.
MR:0079013. Zbl:0073.03101.},
ISSN = {0033-3883},
}
P. T. Bateman and P. Erdős :
“Monotonicity of partition functions ,”
Mathematika
3 : 1
(1956 ),
pp. 1–14 .
MR
0080121
Zbl
0074.03502
article

Abstract
People
BibTeX

Let \( A \) be an arbitrary set of positive integers (finite or infinite) other than the empty set or the set consisting of the single element unity. Let \( p(n) = p_A(n) \) denote the number of partitions of the integer \( n \) into parts taken from the set \( A \) , repetitions being allowed. In other words, \( p(n) \) is the number of ways \( n \) can be expressed in the form
\[ n_1a_1 + n_2a_2 + \cdots ,\]
where \( a_1,a_2,\dots \) are the distinct elements of \( A \) and \( n_1 \) , \( n_2,\dots \) are arbitrary non-negative integers. In this paper we shall prove that \( p(n) \) is a strictly increasing function of \( n \) for sufficiently large \( n \) if and only if \( A \) has the following property (which we shall subsequently call property \( P_1 \) ): \( A \) contains more than one element, and if we remove any single element from \( A \) , the remaining elements have greatest common divisor unity.

@article {key0080121m,
AUTHOR = {Bateman, P. T. and Erd\H{o}s, P.},
TITLE = {Monotonicity of partition functions},
JOURNAL = {Mathematika},
FJOURNAL = {Mathematika. A Journal of Pure and Applied
Mathematics},
VOLUME = {3},
NUMBER = {1},
YEAR = {1956},
PAGES = {1--14},
DOI = {10.1112/S002557930000084X},
NOTE = {MR:0080121. Zbl:0074.03502.},
ISSN = {0025-5793},
}
P. Erdos and P. T. Bateman :
“Advanced problems and solutions: Solutions: 4785 ,”
Amer. Math. Monthly
66 : 2
(1959 ),
pp. 151–153 .
MR
1530231
article

People
BibTeX
@article {key1530231m,
AUTHOR = {Erdos, Paul and Bateman, P. T.},
TITLE = {Advanced problems and solutions: {S}olutions:
4785},
JOURNAL = {Amer. Math. Monthly},
FJOURNAL = {The American Mathematical Monthly},
VOLUME = {66},
NUMBER = {2},
YEAR = {1959},
PAGES = {151--153},
DOI = {10.2307/2310035},
NOTE = {MR:1530231.},
ISSN = {0002-9890},
CODEN = {AMMYAE},
}
P. Erdos and P. T. Bateman :
“Problems and solutions: Solutions of advanced
problems: 5724 ,”
Amer. Math. Monthly
78 : 4
(1971 ),
pp. 412 .
MR
1536305
article

People
BibTeX
@article {key1536305m,
AUTHOR = {Erdos, Paul and Bateman, P. T.},
TITLE = {Problems and solutions: {S}olutions
of advanced problems: 5724},
JOURNAL = {Amer. Math. Monthly},
FJOURNAL = {The American Mathematical Monthly},
VOLUME = {78},
NUMBER = {4},
YEAR = {1971},
PAGES = {412},
URL = {http://www.jstor.org/stable/2316924},
NOTE = {MR:1536305.},
ISSN = {0002-9890},
CODEN = {AMMYAE},
}
P. T. Bateman, P. Erdős, C. Pomerance, and E. G. Straus :
“The arithmetic mean of the divisors of an integer ,”
pp. 197–220
in
Analytic number theory
(Philadelphia, PA, 12–15 May 1980 ).
Edited by M. I. Knopp .
Lecture Notes in Mathematics 899 .
Springer (Berlin ),
1981 .
Dedicated to Emil Grosswald on the occasion of his sixty-eighth birthday.
MR
654528
Zbl
0478.10027
incollection

People
BibTeX
@incollection {key654528m,
AUTHOR = {Bateman, Paul T. and Erd\H{o}s, Paul
and Pomerance, Carl and Straus, E. G.},
TITLE = {The arithmetic mean of the divisors
of an integer},
BOOKTITLE = {Analytic number theory},
EDITOR = {Knopp, M. I.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {899},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1981},
PAGES = {197--220},
DOI = {10.1007/BFb0096462},
NOTE = {(Philadelphia, PA, 12--15 May 1980).
Dedicated to Emil Grosswald on the occasion
of his sixty-eighth birthday. MR:654528.
Zbl:0478.10027.},
ISSN = {0075-8434},
ISBN = {9783540111733},
}
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},
}
P. T. Bateman :
“Some personal memories of Paul Erdős ,”
pp. 9–12
in
Paul Erdős and his mathematics
(Budapest, 4–11 July 1999 ),
vol. I .
Edited by G. Halász, L. Lovász, M. Simonovits, and V. T. Sós .
Bolyai Society Mathematical Studies 11 .
Springer (Berlin ),
2002 .
MR
1954676
Zbl
1031.01503
incollection

People
BibTeX
@incollection {key1954676m,
AUTHOR = {Bateman, P. T.},
TITLE = {Some personal memories of {P}aul {E}rd\H{o}s},
BOOKTITLE = {Paul {E}rd\H{o}s and his mathematics},
EDITOR = {Hal\'asz, G\'abor and Lov\'asz, L\'aszl\'o
and Simonovits, Mikl\'os and S\'os,
Vera T.},
VOLUME = {I},
SERIES = {Bolyai Society Mathematical Studies},
NUMBER = {11},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2002},
PAGES = {9--12},
NOTE = {(Budapest, 4--11 July 1999). MR:1954676.
Zbl:1031.01503.},
ISSN = {1217-4696},
ISBN = {9783540422365},
}