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1
Content available remote Ternary quadratic forms ax² + by² + cz² representing all positive integers 8k + 4
100%
Williams K.
Acta Arithmetica
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2014
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tom 166
|
nr 4
391-396
EN
Under the assumption that the ternary form x² + 2y² + 5z² + xz represents all odd positive integers, we prove that a ternary quadratic form ax² + by² + cz² (a,b,c ∈ ℕ) represents all positive integers n ≡ 4(mod 8) if and only if it represents the eight integers 4,12,20,28,52,60,140 and 308.
2
Content available remote Quadratic forms and a product-to-sum formula
100%
Williams K.
Acta Arithmetica
|
2013
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tom 158
|
nr 1
79-97
EN
Let q ∈ ℂ satisfy |q| < 1. If $f(q) = ∑_{n=0}^{∞} f_n q^n$ we write $[f(q)]_n = f_n$. We prove a general product-to-sum formula which includes known formulae such as $[q∏_{k=1}^{∞} (1-q^{2k})^3 (1-q^{6k})^3 _n = ∑_{{(x_1,x_2)∈ ℤ^2 \atop x_1^2+3x_2^2=n}} 1/2(x_1^2-3x_2^2)$ and $[q∏_{k=1}^{∞}(1-q^{4k})^6]_n = ∑_{{(x_1,x_2)∈ ℤ^2 \atop x_1^2+4x_2^2=n}} 1/2(x_1^2-4x_2^2).
3
Content available remote The genera representing a positive integer
63%
Kaplan P. , Williams K.
|
|
tom 102
|
nr 4
353-361
4
Content available remote A restricted Epstein zeta function and the evaluation of some definite integrals
63%
Muzaffar H. , Williams K.
|
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tom 104
|
nr 1
23-66
5
Content available remote Sums of twelve squares
63%
Huard J. , Williams K.
Acta Arithmetica
|
2003
|
tom 109
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nr 2
195-204
6
Content available remote Determination of $∑_{k=1}^{p-1} g^{ak²}$ modulo p
63%
Williams K. , Hardy K.
Acta Arithmetica
|
1992
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tom 61
|
nr 2
161-172
7
Content available remote On the number of representations of a positive integer by a binary quadratic form
63%
Kaplan P. , Williams K.
Acta Arithmetica
|
2004
|
tom 114
|
nr 1
87-98
8
Content available remote Representations by sextenary quadratic forms whose coefficients are 1, 2 and 4
51%
Alaca A. , Alaca Ş. , Uygul F. , Williams K.
Acta Arithmetica
|
2010
|
tom 141
|
nr 3
289-309
9
Content available remote The Chowla-Selberg formula for genera
51%
Huard J. , Kaplan P. , Williams K.
|
|
nr 3
271-301
10
Content available remote On the two-dimensional theta functions of the Borweins
51%
Alaca A. , Alaca Ş. , Williams K.
Acta Arithmetica
|
2006
|
tom 124
|
nr 2
177-195
11
Content available remote Predictive criteria for the representation of primes by binary quadratic forms
51%
Muskat J. , Spearman B. , Williams K.
|
|
nr 3
215-278
12
Content available remote On Tornheim's double series
51%
Huard J. , Williams K. , Nan-Yue Z.
Acta Arithmetica
|
1996
|
tom 75
|
nr 2
105-117
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