Dr. Saban Alaca

Associate Professor, School of Mathematics and Statistics, Carleton University

Graduate Textbook

Introductory Algebraic Number Theory (with Kenneth S. Williams), Cambridge University Press

Journal Publications

Representations by certain octonary quadratic forms whose coefficients are 1, 2, 3, and 6, International Journal of Number Theory, Accepted 2013. (with Y. Kesicioglu)

Some product-to-sum identities, Journal of Combinatorics and Number Theory 4 (2012), 35-52. (with L. Pehlivan and K. S. Williams)

Some arithmetic identities involving divisor functions, Functiones et Approximatio 46 (2012), 261--271. (with F. Uygul and K. S. Williams)

Restricted Eisenstein Series and Certain Convolution Sums, Journal of Combinatorics and Number Theory 3 (2011), 1-14. (with A. Alaca, F. Uygul, and K. S. Williams)

The number of representations of a positive integer by certain octonary quadratic forms, Functiones et Approximatio 43 (2010), 45-54. (with K. S. Williams)

Sextenary quadratic forms and an identity of Klein and Fricke, International Journal of Number Theory 6 (2010), 169-183. (with A. Alaca and K. S. Williams)

Fourteen octonary quadratic forms, International Journal of Number Theory 6 (2010), 37-50. (with A. Alaca and K. S. Williams)

Representations by sextenary quadratic forms whose coefficients are 1, 2 and 4, Acta Arithmetica 141 (2010), 289-309. (with A. Alaca, F. Uygul and K. S. Williams)

Sums of 4k squares: a polynomial approach, Journal of Combinatorics and Number Theory 1(2) (2009), 33-52. (with A. Alaca and K. S. Williams)

Some new theta function identities with applications to sextenary quadratic forms, Journal of Combinatorics and Number Theory 1 (2009), 89-98. (with A. Alaca and K. S. Williams)

Some infinite products of Ramanujan type, Canadian Mathematical Bulletin 52 (2009), 481-492. (with A. Alaca and K. S. Williams)

Evaluation of the sums \(\sum_{m\equiv a \hspace{-1mm} \pmod 4} \sigma(m) \sigma(n-m)\), Czechoslovak Mathematical Journal 59 (2009), 847-859. (with A. Alaca and K. S. Williams)

Some identities involving theta functions, Journal of Number Theory 129 (2009), 1404-1431. (with A. Alaca and K. S. Williams)

The number of representations of a positive integer by certain quaternary quadratic forms, International Journal of Number Theory 5 (2009), 13-40. (with A. Alaca, M. F. Lemire and K. S. Williams)

Liouville's sextenary quadratic forms \(x^2 + y^2 + z^2 + t^2 + 2u^2 + 2v^2\), \(x^2 + y^2 + 2z^2 + 2t^2 + 2u^2 + 2v^2\) and \(x^2 + 2y^2 + 2z^2 + 2t^2 + 2u^2 + 4v^2\), Far East Journal of Mathematical Sciences 30 (2008), 547-556.(with A. Alaca and K. S. Williams)

Seven octonary quadratic forms, Acta Arithmetica 135 (2008), 339-350. (with A. Alaca and K. S. Williams)

Berndt's curious formula, International Journal of Number Theory 4 (2008), 677-689.(with A. Alaca and K. S. Williams)

Theta function identities and representations by certain quaternary quadratic forms, Int. J. Number Theory 4 (2008), 219-239. (with A. Alaca, M. F. Lemire and K. S. Williams)

Theta function identities and representations by certain quaternary quadratic forms II, International Mathematical Forum 3 (2008), 539-579. (with A. Alaca, S. M. F. Lemire and K. S. Williams)

Arithmetic progressions and binary quadratic forms, American Mathematical Monthly 115 (2008), 252-254. (with A. Alaca and K. S. Williams)

The convolution sum \(\displaystyle \sum_{m \lt n/16} \hspace{-2mm} \sigma(m) \sigma(n-16m)\), Canadian Mathematical Bulletin 51 (2008), 3-14. (with A. Alaca and K. S. Williams)

Jacobi's identity and representation of integers by certain quaternary quadratic forms, Int. J. Modern Math. 2 (2007), 143-176. (with A. Alaca, M. F. Lemire and K. S. Williams)

On the quaternary forms \(x^2 + y^2 +z^2 + 5t^2\), \(x^2 + y^2 + 5z^2 + 5t^2\) and \(x^2 + 5y^2 + 5z^2 + 5t^2\), JP J. Algebra Number Theory Appl. 9 (2007), 37-53. (with A. Alaca and K. S. Williams)

The simplest proof of Jacobi's six squares theorem, Far East J. Math. Sci. 27 (2007), 187-192. (with A. Alaca and K. S. Williams)

Evaluation of the convolution sums \(\displaystyle \hspace{-2mm} \sum_{l+24m=n} \sigma(l) \sigma(m)\) and \(\displaystyle \sum_{3l+8m=n} \hspace{-2mm} \sigma(l) \sigma(m)\), Math. J. Okayama Univ. 49 (2007), 93-111. (with A. Alaca and K. S. Williams)

Nineteen quaternary quadratic forms, Acta Arithmetica 130 (2007), 277-310. (with A. Alaca, M. F. Lemire and K. S. Williams)

Evaluation of the convolution sums \(\displaystyle \sum_{l+ 6m =n} \hspace{-2mm} \sigma(l) \sigma(m)\) and \(\displaystyle \sum_{2l+ 3m =n} \hspace{-2mm} \sigma(l) \sigma(m)\), J. Number Theory 124 (2007), 491-510. (with K. S. Williams)

Lambert series and Liouville's identities, Dissertationes Math. 445 (2007), 1-72. (with A. Alaca, E. McAfee and K. S. Williams)

An infinite class of identities, Bull. Austral. Math. Soc. 75 (2007), 239-246. (with A. Alaca and K. S. Williams)

Nonexistence of a composition law, Math Mag. 80 (2007), 142-144. (with K. S. Williams)

Evaluation of the convolution sums \(\displaystyle \sum_{l+18m=n} \hspace{-2mm} \sigma(l)\sigma(m)\) and \(\displaystyle \sum_{2l+9m=n} \hspace{-2mm} \sigma(l)\sigma(m)\), International Mathematical Forum 2 (2007), 45-68. (with A. Alaca and K. S. Williams)

Congruences for Brewer sums, Finite Fields and Their Applications, 13 (2007), no. 1, 1-19.

Evaluation of the convolution sums \(\displaystyle \sum_{l+12m=n} \hspace{-2mm} \sigma(l)\sigma(m)\) and \(\displaystyle \sum_{3l+4m=n} \hspace{-2mm} \sigma(l)\sigma(m)\), Advances in Theoretical and Applied Mathemeatics 1 (2006), 27-48. (with A. Alaca and K. S. Williams)

On the two-dimensional theta functions of the Borweins, Acta Arith. 124 (2006), 177-195. (with A. Alaca and K. S. Williams)

Explicit decomposition of a rational prime in a cubic field, Int. J. Math. Math. Sci. 2006 (2006), 1-11. (with B. K. Spearman and K. S. Williams)

The factorization of 2 in cubic fields with index 2, Far East J. Math. Sci. 14 (2004), 273-282. (with K. S. Williams)

An integral basis and the discriminant of a quintic field defined by a trinomial x5 + ax + b, JP Journal of Algebra, Number Theory and Applications, 4(2004), no. 2, pp. 261--299, (with Ayse Alaca)

p-Integral bases of a quartic field defined by a trinomial x4 + ax + b, Far East J. Math. Sci. 12 (2004), 137-168. (with K. S. Williams).

On Voronoi's method for finding an integral basis of a cubic field, Util. Math. 65 (2004), 163-166. (with K. S. Williams).

A simple method for finding an integral basis of a quartic field defined by a trinomial x4+ax+b, JP J. Algebra Number Theory Appl. 3 (2003), 477-505. (with K. S. Williams).

p-integral bases of algebraic number fields, Utilitas Mathematica 56(1999), pp. 97-106.

p-integral bases of a cubic field. American Mathematical Society, 126(1998), No. 7, pp. 1949-1953.

Updated: January 2013