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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
|
2014
|
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
|
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

Determination of $∑_{k=1}^{p-1} g^{ak²}$ modulo p

64%
Williams K., Hardy K.
Acta Arithmetica
|
1992
|
tom 61
|
nr 2
161-172
4
Content available remote

On the number of representations of n by ax² + bxy + cy²

64%
Sun Z.-H., Williams K.
Acta Arithmetica
|
2006
|
tom 122
|
nr 2
101-171
5
Content available remote

Some new evaluations of the Legendre symbol ((a+b√q)/p)

64%
Pehlivan L., Williams K.
Acta Arithmetica
|
2015
|
tom 170
|
nr 4
361-380
6
Content available remote

Ramanujan identities and Euler products for a type of Dirichlet series

64%
Sun Z.-H., Williams K.
Acta Arithmetica
|
2006
|
tom 122
|
nr 4
349-393
7
Content available remote

The genera representing a positive integer

64%
Kaplan P., Williams K.
Acta Arithmetica
|
2002
|
tom 102
|
nr 4
353-361
8
Content available remote

A restricted Epstein zeta function and the evaluation of some definite integrals

64%
Muzaffar H., Williams K.
Acta Arithmetica
|
2002
|
tom 104
|
nr 1
23-66
9
Content available remote

Sums of twelve squares

64%
Huard J., Williams K.
Acta Arithmetica
|
2003
|
tom 109
|
nr 2
195-204
10
Content available remote

On the number of representations of a positive integer by a binary quadratic form

64%
Kaplan P., Williams K.
Acta Arithmetica
|
2004
|
tom 114
|
nr 1
87-98
11
Content available remote

The Chowla-Selberg formula for genera

52%
Huard J., Kaplan P., Williams K.
Acta Arithmetica
|
1995
|
tom 73
|
nr 3
271-301
12
Content available remote

Representations by sextenary quadratic forms whose coefficients are 1, 2 and 4

52%
Alaca A., Alaca Ş., Uygul F., Williams K.
Acta Arithmetica
|
2010
|
tom 141
|
nr 3
289-309
13
Content available remote

On Tornheim's double series

52%
Huard J., Williams K., Nan-Yue Z.
Acta Arithmetica
|
1996
|
tom 75
|
nr 2
105-117
14
Content available remote

On the two-dimensional theta functions of the Borweins

52%
Alaca A., Alaca Ş., Williams K.
Acta Arithmetica
|
2006
|
tom 124
|
nr 2
177-195
15
Content available remote

Nineteen quaternary quadratic forms

52%
Alaca A., Alaca Ş., Lemire M., Williams K.
Acta Arithmetica
|
2007
|
tom 130
|
nr 3
277-310
16
Content available remote

Pascal's triangle (mod 9)

52%
Huard J., Spearman B., Williams K.
Acta Arithmetica
|
1996-1997
|
tom 78
|
nr 4
331-349
17
Content available remote

Predictive criteria for the representation of primes by binary quadratic forms

52%
Muskat J., Spearman B., Williams K.
Acta Arithmetica
|
1995
|
tom 70
|
nr 3
215-278
18
Content available remote

Seven octonary quadratic forms

52%
Alaca A., Alaca Ş., Williams K.
Acta Arithmetica
|
2008
|
tom 135
|
nr 4
339-350
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