Edit Profile Williams, Kenneth S. Compute Distance To: Compute Author ID: williams.kenneth-s Published as: Wiliams, Kenneth S.; Williams, K. S.; Williams, Kenneth S.; Williams, kenneth S. Homepage: http://people.math.carleton.ca/~williams/index.html External Links: MGP · dblp · GND Documents Indexed: 379 Publications since 1964, including 13 Books Reviewing Activity: 94 Reviews all top 5 Co-Authors 111 single-authored 80 Spearman, Blair Kenneth 43 Alaca, Şaban 32 Alaca, Ayşe 22 Hudson, Richard H. 21 Kaplan, Pierre 21 Leonard, Philip A. 19 Hardy, Kenneth 16 Huard, James G. 10 Zhang, Nanyue 7 Pehlivan, Lerna 6 Buell, Duncan A. 6 Friesen, Christian 6 Lemire, Mathieu F. 5 Doyle, Greg 5 Muskat, Joseph B. 4 Lavallee, Melisa J. 4 McAfee, Erin 4 Uygul, Faruk 3 Aygin, Zafer Selçuk 3 Buck, Nicholas 3 Eloff, Daniel 3 Evans, Ronald J. 3 McCann, Kevin S. 3 Muzaffar, Habib Bin 3 Ou, Zhiming M. 3 Richman, David Harry 3 Silvester, Alan K. 3 Yang, Qiduan 2 Berndt, Bruce Carl 2 Chalk, John H. H. 2 Cheng, Nathalie 2 Halter-Koch, Franz 2 Holtz, N. M. 2 Jacobson, David 2 Mortimer, Brian C. 2 Smith, Lones 2 Sun, Zhihong 2 Urbanowicz, Jerzy 2 van der Poorten, Alfred J. 2 Yamamoto, Yoshihiko 1 Ayoub, Raymond G. 1 Currie, James D. 1 Dlab, Vlastimil H. 1 Dobrowolski, Edward 1 Driver, Eric D. 1 Dunkl, Charles F. 1 Fox, Glenn J. 1 Gandhi, J. M. 1 Godin, Shawn 1 Gupta, Rajiv 1 Howe, Lawrence J. 1 Ishii, Noburo 1 Kim, Byungchan 1 Kimura, Noriaki 1 Liu, Dachuan 1 Richman, David Ross 1 Spearman, Laura Y. 1 Watanabe, Aya 1 Williams, Hugh Cowie all top 5 Serials 33 Acta Arithmetica 27 Canadian Mathematical Bulletin 24 Proceedings of the American Mathematical Society 21 Far East Journal of Mathematical Sciences 16 Journal of Number Theory 15 American Mathematical Monthly 15 Pacific Journal of Mathematics 15 International Journal of Number Theory 14 Rocky Mountain Journal of Mathematics 11 Mathematics Magazine 8 Mathematics of Computation 7 Canadian Journal of Mathematics 7 International Journal of Mathematics and Mathematical Sciences 7 JP Journal of Algebra, Number Theory and Applications 6 Journal of Combinatorics and Number Theory 5 Archiv der Mathematik 4 Indian Journal of Pure & Applied Mathematics 4 Journal für die Reine und Angewandte Mathematik 4 Utilitas Mathematica 4 The Ramanujan Journal 4 Integers 3 Bulletin of the Australian Mathematical Society 3 Duke Mathematical Journal 3 Journal of the London Mathematical Society. Second Series 3 Kodai Mathematical Journal 3 Mathematical Journal of Okayama University 3 Proceedings of the Edinburgh Mathematical Society. Series II 3 Proceedings of the Cambridge Philosophical Society 3 International Mathematical Forum 3 2 Czechoslovak Mathematical Journal 2 The Fibonacci Quarterly 2 Functiones et Approximatio. Commentarii Mathematici 2 Manuscripta Mathematica 2 Mathematica Scandinavica 2 Mathematika 2 Monatshefte für Mathematik 2 Osaka Journal of Mathematics 2 Tamkang Journal of Mathematics 2 Transactions of the American Mathematical Society 2 L’Enseignement Mathématique. 2e Série 2 Journal de Théorie des Nombres de Bordeaux 2 Far East Journal of Mathematical Sciences 2 Analysis (München) 2 Scientiae Mathematicae Japonicae 2 Missouri Journal of Mathematical Sciences 2 International Journal of Modern Mathematics 1 Communications in Algebra 1 Israel Journal of Mathematics 1 Journal of Mathematical Analysis and Applications 1 The Mathematical Gazette 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Arkiv för Matematik 1 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1 Bulletin of the Calcutta Mathematical Society 1 Colloquium Mathematicum 1 Dissertationes Mathematicae 1 Glasgow Mathematical Journal 1 Illinois Journal of Mathematics 1 The Journal of the Indian Mathematical Society. New Series 1 Journal of Natural Sciences and Mathematics 1 Kyungpook Mathematical Journal 1 Mathematica Slovaca 1 The Mathematics Student 1 Proceedings of the Japan Academy. Series A 1 Publicationes Mathematicae 1 Tokyo Journal of Mathematics 1 Yokohama Mathematical Journal 1 European Journal of Combinatorics 1 Analysis 1 Acta Mathematica Hungarica 1 Kobe Journal of Mathematics 1 Aequationes Mathematicae 1 Atti della Accademia Nazionale dei Lincei. Serie Ottava. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 1 Notices of the American Mathematical Society 1 New Zealand Journal of Mathematics 1 Georgian Mathematical Journal 1 Skrifter. Det Kongelige Norske Videnskabers Selskab 1 Taiwanese Journal of Mathematics 1 The Asian Journal of Mathematics 1 The Bulletin of the Hong Kong Mathematical Society 1 Bollettino della Unione Matematica Italiana. Series V. B 1 CRM Proceedings & Lecture Notes 1 Fields Institute Communications 1 London Mathematical Society Student Texts 1 Canadian Mathematical Society Series of Monographs and Advanced Texts 1 International Journal of Contemporary Mathematical Sciences 1 International Journal of Algebra 1 Advances in Theoretical and Applied Mathematics 1 The College Mathematics Journal all top 5 Fields 350 Number theory (11-XX) 14 Field theory and polynomials (12-XX) 9 Special functions (33-XX) 6 Combinatorics (05-XX) 3 General and overarching topics; collections (00-XX) 3 Commutative algebra (13-XX) 3 Sequences, series, summability (40-XX) 2 History and biography (01-XX) 2 Algebraic geometry (14-XX) 2 Real functions (26-XX) 2 Functions of a complex variable (30-XX) 2 Mathematics education (97-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Associative rings and algebras (16-XX) 1 Group theory and generalizations (20-XX) 1 Approximations and expansions (41-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Geometry (51-XX) Publications by Year all cited Publications top 5 cited Publications Citations contained in zbMATH 240 Publications have been cited 1,532 times in 920 Documents Cited by ▼ Year ▼ Gauss and Jacobi sums. Zbl 0906.11001Berndt, Bruce C.; Evans, Ronald J.; Williams, Kenneth S. 266 1998 Elementary evaluation of certain convolution sums involving divisor functions. Zbl 1062.11005Huard, James G.; Ou, Zhiming M.; Spearman, Blair K.; Williams, Kenneth S. 43 2002 A simple norm inequality. Zbl 0178.16202Dunkl, Charles F.; Williams, K. S. 40 1964 Number theory in the spirit of Liouville. Zbl 1227.11002Williams, Kenneth S. 37 2011 Evaluation of the convolution sums \(\sum _{l+6m=n}\sigma (l)\sigma (m)\) and \(\sum _{2l+3m=n}\sigma (l)\sigma (m)\). Zbl 1169.11002Alaca, Şaban; Williams, Kenneth S. 32 2007 Introductory algebraic number theory. Zbl 1035.11001Alaca, Şaban; Williams, Kenneth S. 32 2004 Nineteen quaternary quadratic forms. Zbl 1131.11025Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 30 2007 The convolution sum \(\sum_{m<n/8} \sigma(m) \sigma(n-8m)\). Zbl 1130.11006Williams, Kenneth S. 29 2006 Integers of biquadratic fields. Zbl 0205.35401Williams, Kenneth S. 29 1970 The convolution sum \(\sum_{m<n/16}\sigma(m)\sigma(n-16m)\). Zbl 1132.11305Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 24 2008 Evaluation of the convolution sums \(\sum_{l+18m=n}\sigma(l)\sigma(m)\) and \(\sum_{2l+9m=n}\sigma(l)\sigma(m)\). Zbl 1163.11003Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 24 2007 The convolution sum \(\sum_{m<n/9}\sigma(m)\sigma(n-9m)\). Zbl 1082.11003Williams, Kenneth S. 24 2005 The number of representations of a positive integer by certain octonary quadratic forms. Zbl 1213.11087Alaca, Şaban; Williams, Kenneth S. 23 2010 Evaluation of the convolution sums \(\sum_{l+12m=n}\sigma(l)\sigma(m)\) and \(\sum_{3l+4m=n}\sigma(l)\sigma(m)\). Zbl 1154.11003Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 23 2006 On the two-dimensional theta functions of the Borweins. Zbl 1127.11035Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 22 2006 Note on cubics over \(GF(2^n)\) and \(GF(3^n)\). Zbl 0321.12029Williams, Kenneth S. 21 1975 On Tornheim’s double series. Zbl 0858.40008Huard, James G.; Williams, Kenneth S.; Zhang, Nanyue 20 1996 Some results on the generalized Stieltjes constants. Zbl 0808.11054Zhang, Nanyue; Williams, Kenneth S. 19 1994 Evaluation of two convolution sums involving the sum of divisors function. Zbl 1163.11301Lemire, Mathieu; Williams, Kenneth S. 18 2006 Theta functions identities and representations by certain quaternary quadratic forms. Zbl 1163.11029Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 17 2008 Evaluation of the convolution sums \(\sum_{l+24m=n}\sigma(l)\sigma(m)\) and \(\sum_{3l+8m=n}\sigma(l)\sigma(m)\). Zbl 1132.11304Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 17 2007 Integral bases for quartic fields with quadratic subfields. Zbl 0826.11048Huard, James G.; Spearman, Blair K.; Williams, Kenneth S. 17 1995 Fourier series of a class of eta quotients. Zbl 1252.11043Williams, Kenneth S. 13 2012 The number of representations of a positive integer by certain quaternary quadratic forms. Zbl 1221.11097Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 12 2009 Calculation of the class numbers of imaginary cyclic quartic fields. Zbl 0624.12004Hardy, Kenneth; Hudson, R. H.; Richman, D.; Williams, Kenneth S.; Holtz, N. M. 12 1987 Mertens’ theorem for arithmetic progressions. Zbl 0286.10022Williams, Kenneth S. 12 1974 On the number of representations of \(n\) by \(ax^2+bxy+cy^2\). Zbl 1095.11024Sun, Zhihong; Williams, Kenneth S. 11 2006 Jacobi’s identity and representations of integers by certain quaternary quadratic forms. Zbl 1209.11040Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 10 2007 Some series representations of \(\zeta(2n+1)\). Zbl 0808.11053Zhang, Nanyue; Williams, Kenneth S. 10 1993 Fourteen octonary quadratic forms. Zbl 1228.11049Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 9 2010 Evaluation of some convolutions sums involving the sum of divisors functions. Zbl 1094.11004Cheng, Nathalie; Williams, Kenneth S. 9 2005 Values of the Riemann zeta functions and integrals involving \(\log (2\sinh {\theta\over 2})\) and \(\log (2\sin {\theta\over 2})\). Zbl 0828.11041Zhang, Nan-Yue; Williams, Kenneth S. 9 1995 Binomial coefficients and Jacobi sums. Zbl 0551.12015Hudson, Richard H.; Williams, Kenneth S. 9 1984 Pascal’s triangle (mod 8). Zbl 0889.11007Huard, James G.; Spearman, Blair K.; Williams, Kenneth S. 8 1998 Evaluation of two trigonometric sums. Zbl 0820.11010Williams, Kenneth S.; Zhang, Nan-Yue 8 1994 The integers of a cyclic quartic field. Zbl 0707.11078Hudson, R. H.; Williams, K. S. 8 1990 Pell’s equations \(X^ 2-mY^ 2=-1,-4\) and continued fractions. Zbl 0596.10013Kaplan, Pierre; Williams, Kenneth S. 8 1986 An Artin character and representations of primes by binary quadratic forms. II. Zbl 0485.12003Halter-Koch, Franz; Kaplan, Pierre; Williams, Kenneth S. 8 1982 The cyclotomic numbers of order seven. Zbl 0311.12008Leonard, Philip A.; Williams, Kenneth S. 8 1975 Quartics over GF(2\(^n\)). Zbl 0274.12013Leonard, Philip A.; Williams, Kenneth S. 8 1973 Note on Salie’s sum. Zbl 0224.10039Williams, Kenneth S. 8 1971 Theta function identities and representations by certain quaternary quadratic forms. II. Zbl 1209.11041Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 7 2008 Convolution sums involving the divisor function. Zbl 1156.11301Cheng, Nathalie; Williams, Kenneth S. 7 2004 Congruences for \(L\)-functions. Zbl 0972.11001Urbanowicz, Jerzy; Williams, Kenneth S. 7 2000 Some Lambert series expansions of products of theta functions. Zbl 0938.11017Williams, Kenneth S. 7 1999 The Chowla-Selberg formula for genera. Zbl 0855.11018Huard, James G.; Kaplan, Pierre; Williams, Kenneth S. 7 1995 Application of the Hurwitz zeta function to the evaluation of certain integrals. Zbl 0791.11045Zhang, Nanyue; Williams, Kenneth S. 7 1993 Special values of the Lerch zeta function and the evaluation of certain integrals. Zbl 0785.11046Williams, Kenneth S.; Zhang, Nanyue 7 1993 Determination of all imaginary cyclic quartic fields with class number 2. Zbl 0678.12003Hardy, Kenneth; Hudson, Richard H.; Richman, David; Williams, Kenneth S. 7 1989 Evaluation of character sums connected with elliptic curves. Zbl 0369.10025Williams, Kenneth S. 7 1979 On \[ \sum_{n=1}^ \infty (1/n^{2k}) \]. Zbl 0224.40008Williams, Kenneth S. 7 1971 Some arithmetic identities involving divisor functions. Zbl 1318.11001Alaca, Şaban; Uygul, Faruk; Williams, Kenneth S. 6 2012 Seven octonary quadratic forms. Zbl 1171.11021Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 6 2008 On a double series of Chan and Ong. Zbl 1163.11318Williams, Kenneth S. 6 2006 The simplest arithmetic proof of Jacobi’s four squares theorem. Zbl 0958.11029Spearman, Blair K.; Williams, Kenneth S. 6 2000 A deterministic algorithm for solving \(n=fu^ 2+gv^ 2\) in coprime integers u and v. Zbl 0713.11088Hardy, Kenneth; Muskat, Joseph B.; Williams, Kenneth S. 6 1990 Some new residuacity criteria. Zbl 0456.10001Hudson, Richard H.; Williams, Kenneth S. 6 1980 The quadratic and quartic character of certain quadratic units. II. Zbl 0386.12002Leonard, Philip A.; Williams, Kenneth S. 6 1979 On the imaginary bicyclic biquadratic fields with class-number 2. Zbl 0379.12002Buell, D. A.; Williams, H. C.; Williams, K. S. 6 1977 The quadratic and quartic character of certain quadratic units. I. Zbl 0335.12006Leonard, Philip A.; Williams, Kenneth S. 6 1977 A rational octic reciprocity law. Zbl 0311.10004Williams, Kenneth S. 6 1976 On Euler’s criterion for cubic nonresidues. Zbl 0273.10002Williams, Kenneth S. 6 1975 The septic character of \(2, 3, 5\) and \(7\). Zbl 0265.10004Leonard, Philip A.; Williams, Kenneth S. 6 1974 On two conjectures of Chowla. Zbl 0186.09301Williams, Kenneth S. 6 1969 A “four integers” theorem and a “five integers” theorem. Zbl 1327.11024Williams, Kenneth S. 5 2015 On the quaternary forms \(x^2+y^2+z^2+5t^2\), \(x^2+y^2+5z^2+5t^2\) and \(x^2+5y^2+5z^2+5t^2\). Zbl 1196.11058Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 5 2007 Dihedral quintic fields with a power basis. Zbl 1161.11393Lavallee, Melissa J.; Spearman, Blair K.; Williams, Kenneth S.; Yang, Qiduan 5 2005 Values of the Dedekind eta function at quadratic irrationalities. Zbl 0936.11026van der Poorten, Alfred; Williams, Kenneth S. 5 1999 An upper bound for the sum \(\sum_{n=a+1}^{a+H}f(n)\) for a certain class of functions \(f\). Zbl 0743.11042Dobrowolski, Edward; Williams, Kenneth S. 5 1992 Calculation of the class numbers of imaginary cyclic quartic fields. Zbl 0615.12009Hardy, K.; Hudson, R. H.; Richman, D.; Williams, K. S.; Holtz, N. M. 5 1986 On the evaluation of the Legendre symbol \(((A+B\sqrt{m})/p)\). Zbl 0524.10002Williams, Kenneth S.; Hardy, Kenneth; Friesen, Christian 5 1985 On the class numbers of \(Q(\sqrt{\pm 2p})\) modulo 16, for \(p=1 (\mod 8)\) a prime. Zbl 0491.12003Kaplan, Pierre; Williams, Kenneth S. 5 1982 Class numbers and biquadratic reciprocity. Zbl 0473.12004Williams, Kenneth S.; Currie, James D. 5 1982 On the class number of \(Q(\sqrt{-p})\) modulo 16, for \(p\equiv 1\) (mod 8) a prime. Zbl 0393.12008Williams, Kenneth S. 5 1981 A rational sixteenth power reciprocity law. Zbl 0363.10003Leonard, Philip A.; Williams, Kenneth S. 5 1977 The cyclotomic numbers of order eleven. Zbl 0267.12009Leonard, Philip A.; Williams, Kenneth S. 5 1975 On the number of distinguished representations of a group element. Zbl 0248.20018Jacobson, David; Williams, Kenneth S. 5 1972 Note on Dickson’s permutation polynomials. Zbl 0235.12011Williams, Kenneth S. 5 1971 On exceptional polynomials. Zbl 0159.05304Williams, K. S. 5 1968 The distribution of the residues of a quartic polynomial. Zbl 0171.00702McCann, K.; Williams, K. S. 5 1967 The distribution of solutions of congruences. Zbl 0158.29603Chalk, John H. H.; Williams, K. S. 5 1965 Some new theta function identities with applications to sextenary quadratic forms. Zbl 1235.11039Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 4 2009 On Liouville’s twelve squares theorem. Zbl 1242.11029Williams, Kenneth S. 4 2008 On the representations of a positive integer by the forms \(x^2+y^2+z^2+2t^2\) and \(x^2+2y^2+2z^2+2t^2\). Zbl 1213.11089Williams, Kenneth S. 4 2008 The simplest \(D_4\)-octics. Zbl 1144.11074Spearman, Blair K.; Williams, Kenneth S. 4 2008 \(A_4\)-sextic fields with a power basis. Zbl 1229.11136Eloff, Daniel; Spearman, Blair K.; Williams, Kenneth S. 4 2007 The index of a cyclic quartic field. Zbl 1049.11111Spearman, Blair K.; Williams, Kenneth S. 4 2003 An arithmetic proof of Jacobi’s eight squares theorem. Zbl 1004.11017Williams, Kenneth S. 4 2001 Pascal’s triangle mod 9. Zbl 0874.11024Huard, James G.; Spearman, Blair K.; Williams, Kenneth S. 4 1997 On solvable quintics \(X^ 5 + aX +b\) and \(X^ 5 + aX^ 2 +b\). Zbl 0862.12002Spearman, Blair K.; Williams, Kenneth S. 4 1996 A refinement of H. C. Williams’ \(q\)th root algorithm. Zbl 0782.11040Williams, Kenneth S.; Hardy, Kenneth 4 1993 A congruence relating the class numbers of complex quadratic fields. Zbl 0557.12003Hardy, Kenneth; Williams, Kenneth S. 4 1986 Extensions of theorems of Cunningham-Aigner and Hasse-Evans. Zbl 0505.10001Hudson, Richard H.; Williams, Kenneth S. 4 1983 On Scholz’s reciprocity law. Zbl 0372.12004Williams, Kenneth S. 4 1977 Note on a cubic character sum. Zbl 0307.10038Williams, Kenneth S. 4 1975 Jacobi sums and a theorem of Brewer. Zbl 0297.10023Leonard, Philip A.; Williams, Kenneth S. 4 1975 A Diophantine system of Dickson. Zbl 0299.12017Leonard, Philip A.; Williams, Kenneth S. 4 1974 Finite transformation formulae involving the Legendre symbol. Zbl 0207.05501Williams, K. S. 4 1970 Quadratic forms and a product-to-sum formula. Zbl 1278.11043Williams, Kenneth S. 3 2013 On the quaternary forms \(x^2+y^2+2z^2+3t^2\), \(x^2+2y^2+2z^2+6t^2\), \(x^2+3y^2+3z^2+6t^2\) and \(2x^2+3y^2+6z^2+6t^2\). Zbl 1255.11020Alaca, Ayşe; Williams, Kenneth S. 3 2012 Everything you wanted to know about \(ax^2+by^2+cz^2+dt^2\) but were afraid to ask. Zbl 1440.11047Williams, Kenneth S. 1 2018 Evaluation of some \(q\)-integrals in terms of the Dedekind eta function. Zbl 1429.11079Doyle, Greg; Williams, Kenneth S. 1 2018 A positive-definite ternary quadratic form does not represent all positive integers. Zbl 1412.11065Doyle, Greg; Williams, Kenneth S. 1 2017 Infinite products with coefficients which vanish on certain arithmetic progressions. Zbl 1429.11078Alaca, Ayşe; Alaca, Şaban; Aygin, Zafer Selcuk; Williams, Kenneth S. 1 2017 Representation numbers of certain quaternary quadratic forms in a genus consisting of a single class. Zbl 1350.11050Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 1 2016 A “four integers” theorem and a “five integers” theorem. Zbl 1327.11024Williams, Kenneth S. 5 2015 Ternary quadratic forms and eta quotients. Zbl 1336.11033Williams, Kenneth S. 2 2015 Double Gauss sums. Zbl 1366.11092Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 1 2014 Ternary quadratic forms \(ax^2+by^2+cz^2\) representing all positive integers \(8k+4\). Zbl 1327.11023Williams, Kenneth S. 1 2014 On the number of representations of a positive integer as a sum of two binary quadratic forms. Zbl 1302.11013Alaca, Şaban; Pehlivan, Lerna; Williams, Kenneth S. 1 2014 Quadratic forms and a product-to-sum formula. Zbl 1278.11043Williams, Kenneth S. 3 2013 The parents of Jacobi’s four squares theorem are unique. Zbl 1305.11026Williams, Kenneth S. 2 2013 Fourier series of a class of eta quotients. Zbl 1252.11043Williams, Kenneth S. 13 2012 Some arithmetic identities involving divisor functions. Zbl 1318.11001Alaca, Şaban; Uygul, Faruk; Williams, Kenneth S. 6 2012 On the quaternary forms \(x^2+y^2+2z^2+3t^2\), \(x^2+2y^2+2z^2+6t^2\), \(x^2+3y^2+3z^2+6t^2\) and \(2x^2+3y^2+6z^2+6t^2\). Zbl 1255.11020Alaca, Ayşe; Williams, Kenneth S. 3 2012 Some product-to-sum identities. Zbl 1282.11039Alaca, Şaban; Pehlivan, Lerna; Williams, Kenneth S. 1 2012 Number theory in the spirit of Liouville. Zbl 1227.11002Williams, Kenneth S. 37 2011 Restricted Eisenstein series and certain convolution sums. Zbl 1245.11095Alaca, Ayşe; Alaca, Şaban; Uygul, Faruk; Williams, Kenneth S. 2 2011 The number of representations of a positive integer by certain octonary quadratic forms. Zbl 1213.11087Alaca, Şaban; Williams, Kenneth S. 23 2010 Fourteen octonary quadratic forms. Zbl 1228.11049Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 9 2010 Sextenary quadratic forms and an identity of Klein and Fricke. Zbl 1228.11050Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 3 2010 Representations by sextenary quadratic forms whose coefficients are 1, 2 and 4. Zbl 1213.11085Alaca, Ayşe; Alaca, Şaban; Uygul, Faruk; Williams, Kenneth S. 2 2010 The number of representations of a positive integer by certain quaternary quadratic forms. Zbl 1221.11097Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 12 2009 Some new theta function identities with applications to sextenary quadratic forms. Zbl 1235.11039Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 4 2009 Sums of \(4k\) squares: a polynomial approach. Zbl 1239.11044Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 3 2009 Some infinite products of Ramanujan type. Zbl 1234.11141Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 3 2009 The convolution sum \(\sum_{m<n/16}\sigma(m)\sigma(n-16m)\). Zbl 1132.11305Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 24 2008 Theta functions identities and representations by certain quaternary quadratic forms. Zbl 1163.11029Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 17 2008 Theta function identities and representations by certain quaternary quadratic forms. II. Zbl 1209.11041Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 7 2008 Seven octonary quadratic forms. Zbl 1171.11021Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 6 2008 On Liouville’s twelve squares theorem. Zbl 1242.11029Williams, Kenneth S. 4 2008 On the representations of a positive integer by the forms \(x^2+y^2+z^2+2t^2\) and \(x^2+2y^2+2z^2+2t^2\). Zbl 1213.11089Williams, Kenneth S. 4 2008 The simplest \(D_4\)-octics. Zbl 1144.11074Spearman, Blair K.; Williams, Kenneth S. 4 2008 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 \(x^2 +2y^2+2z^2+2t^2+2u^2+4v^2\). Zbl 1213.11086Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 3 2008 Arithmetic progressions and binary quadratic forms. Zbl 1204.11073Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 1 2008 Evaluation of the convolution sums \(\sum _{l+6m=n}\sigma (l)\sigma (m)\) and \(\sum _{2l+3m=n}\sigma (l)\sigma (m)\). Zbl 1169.11002Alaca, Şaban; Williams, Kenneth S. 32 2007 Nineteen quaternary quadratic forms. Zbl 1131.11025Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 30 2007 Evaluation of the convolution sums \(\sum_{l+18m=n}\sigma(l)\sigma(m)\) and \(\sum_{2l+9m=n}\sigma(l)\sigma(m)\). Zbl 1163.11003Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 24 2007 Evaluation of the convolution sums \(\sum_{l+24m=n}\sigma(l)\sigma(m)\) and \(\sum_{3l+8m=n}\sigma(l)\sigma(m)\). Zbl 1132.11304Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 17 2007 Jacobi’s identity and representations of integers by certain quaternary quadratic forms. Zbl 1209.11040Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 10 2007 On the quaternary forms \(x^2+y^2+z^2+5t^2\), \(x^2+y^2+5z^2+5t^2\) and \(x^2+5y^2+5z^2+5t^2\). Zbl 1196.11058Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 5 2007 \(A_4\)-sextic fields with a power basis. Zbl 1229.11136Eloff, Daniel; Spearman, Blair K.; Williams, Kenneth S. 4 2007 Lambert series and Liouville’s identities. Zbl 1113.11004Alaca, A.; Alaca, Ş.; McAfee, E.; Williams, K. S. 3 2007 The simplest proof of Jacobi’s six squares theorem. Zbl 1204.11072Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 1 2007 An infinite class of identies. Zbl 1115.33017Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 1 2007 The convolution sum \(\sum_{m<n/8} \sigma(m) \sigma(n-8m)\). Zbl 1130.11006Williams, Kenneth S. 29 2006 Evaluation of the convolution sums \(\sum_{l+12m=n}\sigma(l)\sigma(m)\) and \(\sum_{3l+4m=n}\sigma(l)\sigma(m)\). Zbl 1154.11003Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 23 2006 On the two-dimensional theta functions of the Borweins. Zbl 1127.11035Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 22 2006 Evaluation of two convolution sums involving the sum of divisors function. Zbl 1163.11301Lemire, Mathieu; Williams, Kenneth S. 18 2006 On the number of representations of \(n\) by \(ax^2+bxy+cy^2\). Zbl 1095.11024Sun, Zhihong; Williams, Kenneth S. 11 2006 On a double series of Chan and Ong. Zbl 1163.11318Williams, Kenneth S. 6 2006 Values of the Euler phi function not divisible by a given odd prime. Zbl 1216.11089Spearman, Blair K.; Williams, Kenneth S. 3 2006 Ramanujan identities and Euler products for a type of Dirichlet series. Zbl 1155.11026Sun, Zhihong; Williams, Kenneth S. 3 2006 Evaluation of complete elliptic integrals of the first kind at singular moduli. Zbl 1139.11025Muzaffar, Habib; Williams, Kenneth S. 2 2006 Cyclic cubic fields of given conductor and given index. Zbl 1104.11047Silvester, Alan K.; Spearman, Blair K.; Williams, Kenneth S. 2 2006 Cyclic quartic fields with a unique normal integral basis. Zbl 1172.11038Spearman, Blair K.; Williams, Kenneth S. 1 2006 \(\text{PSL}(2,5)\) sextic fields with a power basis. Zbl 1096.11038Spearman, Blair K.; Watanabe, Aya; Williams, Kenneth S. 1 2006 The convolution sum \(\sum_{m<n/9}\sigma(m)\sigma(n-9m)\). Zbl 1082.11003Williams, Kenneth S. 24 2005 Evaluation of some convolutions sums involving the sum of divisors functions. Zbl 1094.11004Cheng, Nathalie; Williams, Kenneth S. 9 2005 Dihedral quintic fields with a power basis. Zbl 1161.11393Lavallee, Melissa J.; Spearman, Blair K.; Williams, Kenneth S.; Yang, Qiduan 5 2005 Irreducible quartic polynomials with factorization modulo \(p\). Zbl 1211.11037Driver, Eric; Leonard, Philip A.; Williams, Kenneth S. 2 2005 Watson’s method of solving a quintic equation. Zbl 1102.12001Lavallee, Melisa J.; Spearman, Blair K.; Williams, Kenneth S. 1 2005 Introductory algebraic number theory. Zbl 1035.11001Alaca, Şaban; Williams, Kenneth S. 32 2004 Convolution sums involving the divisor function. Zbl 1156.11301Cheng, Nathalie; Williams, Kenneth S. 7 2004 On the number of representations of a positive integer by a binary quadratic form. Zbl 1102.11024Kaplan, Pierre; Williams, Kenneth S. 3 2004 Normal integral bases for Emma Lehmer’s parametric family of cyclic quintics. Zbl 1158.11343Spearman, Blair K.; Williams, Kenneth S. 2 2004 Evaluation of Weber’s functions at quadratic irrationalities. Zbl 1077.11031Muzaffar, Habib; Williams, Kenneth S. 2 2004 The factorization of 2 in cubic fields with index 2. Zbl 1080.11079Alaca, Saban; Spearman, Blair K.; Williams, Kenneth S. 1 2004 A cubic transformation formula for \(_{2}F_{1}(\frac{1}{3},\frac{2}{3};1;z)\) and some arithmetic convolution formulae. Zbl 1060.11003Williams, Kenneth S. 1 2004 On Voronoi’s method for finding an integral basis of a cubic field. Zbl 1061.11056Alaca, Şaban; Williams, Kenneth S. 1 2004 \(p\)-integral bases of a quartic field defined by a trinomial \(x^4+ax+b\). Zbl 1050.11091Alaca, Şaban; Williams, Kenneth S. 1 2004 The index of a cyclic quartic field. Zbl 1049.11111Spearman, Blair K.; Williams, Kenneth S. 4 2003 The discriminant of a cyclic field of odd prime degree. Zbl 1074.11059Spearman, Blair K.; Williams, Kenneth S. 2 2003 \(n=\Delta +\Delta +2(\Delta + \Delta)\). Zbl 1041.11022Williams, Kenneth S. 1 2003 The index of a dihedral quartic field. Zbl 1049.11115Silvester, Alan K.; Spearman, Blair K.; Williams, Kenneth S. 1 2003 Sums of twelve squares. Zbl 1023.11018Huard, James G.; Williams, Kenneth S. 1 2003 Elementary evaluation of certain convolution sums involving divisor functions. Zbl 1062.11005Huard, James G.; Ou, Zhiming M.; Spearman, Blair K.; Williams, Kenneth S. 43 2002 Cubic fields with index 2. Zbl 0992.11058Spearman, Blair K.; Williams, Kenneth S. 3 2002 A restricted Epstein zeta function and the evaluation of some definite integrals. Zbl 1037.11023Muzaffar, Habib; Williams, Kenneth S. 2 2002 The discriminant of a dihedral quintic field defined by a trinomial \(X^5+aX+b\). Zbl 1040.11076Spearman, Blair K.; Williams, Kenneth S. 1 2002 The genera representing a positive integer. Zbl 0998.11018Kaplan, Pierre; Williams, Kenneth S. 1 2002 An arithmetic proof of Jacobi’s eight squares theorem. Zbl 1004.11017Williams, Kenneth S. 4 2001 The cubic congruence \(x^3+Ax^2+Bx+C\equiv 0\pmod p\) and binary quadratic forms. II. Zbl 1017.11017Spearman, Blair K.; Williams, Kenneth S. 2 2001 Integers which are discriminants of bicyclic or cyclic quartic fields. Zbl 1038.11068Spearman, Blair K.; Williams, Kenneth S. 1 2001 On the density of cyclic quartic fields. Zbl 0996.11068Ou, Zhiming M.; Williams, Kenneth S. 1 2001 Congruences for \(L\)-functions. Zbl 0972.11001Urbanowicz, Jerzy; Williams, Kenneth S. 7 2000 The simplest arithmetic proof of Jacobi’s four squares theorem. Zbl 0958.11029Spearman, Blair K.; Williams, Kenneth S. 6 2000 Pure quintic fields defined by trinomials. Zbl 0970.11040Spearman, Blair K.; Williams, Kenneth S. 2 2000 Quartic trinomials with Galois groups \(A_4\) and \(V_4\). Zbl 0987.11063Spearman, Blair K.; Williams, Kenneth S. 2 2000 Some Lambert series expansions of products of theta functions. Zbl 0938.11017Williams, Kenneth S. 7 1999 Values of the Dedekind eta function at quadratic irrationalities. Zbl 0936.11026van der Poorten, Alfred; Williams, Kenneth S. 5 1999 Dihedral quintic polynomials and a theorem of Galois. Zbl 1056.12004Spearman, Blair K.; Williams, Kenneth S. 1 1999 DeMoivre’s quintic and a theorem of Galois. Zbl 0964.11047Spearman, Blair K.; Williams, Kenneth S. 1 1999 Gauss and Jacobi sums. Zbl 0906.11001Berndt, Bruce C.; Evans, Ronald J.; Williams, Kenneth S. 266 1998 Pascal’s triangle (mod 8). Zbl 0889.11007Huard, James G.; Spearman, Blair K.; Williams, Kenneth S. 8 1998 An explicit integral basis for a pure cubic field. Zbl 1069.11512Spearman, Blair K.; Williams, Kenneth S. 2 1998 Pascal’s triangle mod 9. Zbl 0874.11024Huard, James G.; Spearman, Blair K.; Williams, Kenneth S. 4 1997 On a formula of Dirichlet. Zbl 0934.11017Kaplan, Pierre; Williams, Kenneth S. 2 1997 The conductor of a cyclic quartic field using Gauss sums. Zbl 0898.11041Spearman, B. K.; Williams, K. S. 2 1997 Quadratic subfields of the splitting field of a dihedral quintic trinomial \(x^5+ax+b\). Zbl 0922.11092Spearman, Blair K.; Spearman, Laura Y.; Williams, Kenneth S. 2 1997 ...and 140 more Documents all cited Publications top 5 cited Publications all top 5 Cited by 986 Authors 67 Williams, Kenneth S. 23 Sun, Zhihong 22 Alaca, Şaban 20 Spearman, Blair Kenneth 17 Alaca, Ayşe 17 Evans, Ronald J. 13 Xia, Ernest X. W. 12 Feng, Tao 11 Kim, Daeyeoul 11 Momihara, Koji 11 Xiang, Qing 11 Yue, Qin 10 Li, Chengju 9 Aygin, Zafer Selçuk 9 Srivastava, Hari Mohan 9 Yao, Olivia Xiang Mei 9 Ye, Dongxi 8 Dilcher, Karl 8 Halter-Koch, Franz 8 Sun, Zhi-Wei 8 Zeng, Xiangyong 7 Baoulina, Ioulia N. 7 Coffey, Mark William 7 Tsumura, Hirofumi 6 Barman, Rupam 6 Berndt, Bruce Carl 6 Kokluce, Bulent 6 Louboutin, Stéphane R. 6 Ramakrishnan, Balakrishnan 6 Sahu, Brundaban 6 Tu, Ziran 5 Bibak, Khodakhast 5 Cao, Wei 5 Cho, Bumkyu 5 Cooper, Shaun 5 Helleseth, Tor 5 Heng, Ziling 5 Hudson, Richard H. 5 Jones, Lenny K. 5 Kaplan, Pierre 5 Kim, Aeran 5 Kim, Minsoo 5 Leonard, Philip A. 5 McCarthy, Dermot 5 Moslehian, Mohammad Sal 5 Park, Yoon Kyung 5 Saikia, Neelam 5 Saito, Kichisuke 4 Cosgrave, John B. 4 Feng, Keqin 4 Garcia, Stephan Ramon 4 Gurak, Stanley J. 4 Hou, Xiang-Dong 4 Hu, Su 4 Kapron, Bruce M. 4 Kim, Henry H. 4 Kwon, Soonhak 4 Luca, Florian 4 Moree, Pieter 4 Nakahara, Toru 4 Ntienjem, Ebénézer 4 Oh, Byeong-Kweon 4 Park, Ho-Jin 4 Pehlivan, Lerna 4 Seeger, Alberto 4 Singh, Anup Kumar 4 Srinivasan, Venkatesh 4 Toh, Pee Choon 4 van Wamelen, Paul B. 4 Yang, Jing 4 Yang, Qiduan 4 Zaharescu, Alexandru 3 Ahn, Jaehyun 3 Alahmadi, Adel N. 3 Ayad, Mohamed 3 Blagouchine, Iaroslav V. 3 Bradley, David M. 3 Cai, Tianxin 3 Carlet, Claude 3 Chamberland, Marc 3 Chan, Heng Huat 3 Chern, Shane 3 Chou, Wunseng 3 Cochrane, Todd 3 Cvijović, Djurdje 3 Dadipour, Farzad 3 Ding, Cunsheng 3 Doyle, Greg 3 El Bachraoui, Mohamed 3 Friedlander, John Benjamin 3 Gaál, István 3 Greene, John 3 Hijazi, Rola Asaad 3 Huard, James G. 3 Kendirli, Bariṣ 3 Kesicioğlu, Yavuz 3 Kihel, Omar 3 Kim, Chang Heon 3 Llorens-Fuster, Enrique 3 Merca, Mircea ...and 886 more Authors all top 5 Cited in 178 Serials 145 Journal of Number Theory 84 International Journal of Number Theory 57 Finite Fields and their Applications 36 Proceedings of the American Mathematical Society 31 The Ramanujan Journal 27 Mathematics of Computation 25 Journal of Mathematical Analysis and Applications 25 Designs, Codes and Cryptography 20 Journal de Théorie des Nombres de Bordeaux 19 Rocky Mountain Journal of Mathematics 14 Cryptography and Communications 13 Functiones et Approximatio. Commentarii Mathematici 12 Bulletin of the Australian Mathematical Society 12 Discrete Mathematics 11 Journal of Combinatorial Theory. Series A 10 Acta Arithmetica 10 Manuscripta Mathematica 10 Integers 9 Archiv der Mathematik 9 Transactions of the American Mathematical Society 9 Linear Algebra and its Applications 8 Journal of Algebra 8 Journal of Inequalities and Applications 7 Advances in Mathematics 7 Proceedings of the Japan Academy. Series A 7 Advances in Applied Mathematics 6 Communications in Algebra 6 European Journal of Combinatorics 5 Annales de l’Institut Fourier 5 Acta Mathematica Hungarica 5 Applicable Algebra in Engineering, Communication and Computing 5 Journal of Algebraic Combinatorics 4 Israel Journal of Mathematics 4 Applied Mathematics and Computation 4 Glasgow Mathematical Journal 4 International Journal of Mathematics and Mathematical Sciences 4 Journal of Computational and Applied Mathematics 4 Mathematische Annalen 4 Mathematische Zeitschrift 4 Tokyo Journal of Mathematics 4 Abstract and Applied Analysis 4 Acta Mathematica Sinica. English Series 4 Journal of Algebra and its Applications 4 Advances in Difference Equations 4 European Journal of Pure and Applied Mathematics 4 Science China. Mathematics 4 Afrika Matematika 4 Research in Number Theory 3 Discrete Applied Mathematics 3 Journal of Mathematical Physics 3 Mathematical Notes 3 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 3 Compositio Mathematica 3 Inventiones Mathematicae 3 Journal of Combinatorial Theory. Series B 3 Kyungpook Mathematical Journal 3 Journal of Symbolic Computation 3 Journal of the Ramanujan Mathematical Society 3 Journal of Mathematical Sciences (New York) 3 Annals of Combinatorics 3 Nihonkai Mathematical Journal 3 Journal of Classical Analysis 2 American Mathematical Monthly 2 Communications in Mathematical Physics 2 Indian Journal of Pure & Applied Mathematics 2 Periodica Mathematica Hungarica 2 Ukrainian Mathematical Journal 2 Czechoslovak Mathematical Journal 2 Illinois Journal of Mathematics 2 Journal of the Korean Mathematical Society 2 Journal of the Mathematical Society of Japan 2 Journal of Pure and Applied Algebra 2 Nagoya Mathematical Journal 2 Osaka Journal of Mathematics 2 Proceedings of the Edinburgh Mathematical Society. Series II 2 Results in Mathematics 2 Theoretical Computer Science 2 Forum Mathematicum 2 International Journal of Mathematics 2 New Zealand Journal of Mathematics 2 Turkish Journal of Mathematics 2 The Electronic Journal of Combinatorics 2 Acta Mathematica et Informatica Universitatis Ostraviensis 2 Doklady Mathematics 2 Honam Mathematical Journal 2 Journal of Applied Mathematics 2 Comptes Rendus. Mathématique. Académie des Sciences, Paris 2 Journal of Applied Mathematics and Computing 2 Mediterranean Journal of Mathematics 2 Proceedings of the Japan Academy 2 Annales Mathématiques du Québec 2 Open Mathematics 2 Algebraic Combinatorics 1 Computers & Mathematics with Applications 1 Journal d’Analyse Mathématique 1 Lithuanian Mathematical Journal 1 The Mathematical Gazette 1 Nuclear Physics. B 1 Problems of Information Transmission 1 Reviews of Modern Physics ...and 78 more Serials all top 5 Cited in 41 Fields 772 Number theory (11-XX) 113 Combinatorics (05-XX) 79 Special functions (33-XX) 56 Information and communication theory, circuits (94-XX) 40 Field theory and polynomials (12-XX) 38 Algebraic geometry (14-XX) 31 Functional analysis (46-XX) 20 Sequences, series, summability (40-XX) 18 Real functions (26-XX) 16 Linear and multilinear algebra; matrix theory (15-XX) 14 Group theory and generalizations (20-XX) 14 Computer science (68-XX) 13 Functions of a complex variable (30-XX) 12 Commutative algebra (13-XX) 12 Harmonic analysis on Euclidean spaces (42-XX) 11 Operator theory (47-XX) 11 Quantum theory (81-XX) 7 Approximations and expansions (41-XX) 7 Geometry (51-XX) 7 Convex and discrete geometry (52-XX) 7 Probability theory and stochastic processes (60-XX) 7 Numerical analysis (65-XX) 6 Associative rings and algebras (16-XX) 6 Dynamical systems and ergodic theory (37-XX) 5 History and biography (01-XX) 4 Order, lattices, ordered algebraic structures (06-XX) 4 Nonassociative rings and algebras (17-XX) 4 Manifolds and cell complexes (57-XX) 3 \(K\)-theory (19-XX) 3 Partial differential equations (35-XX) 3 Statistical mechanics, structure of matter (82-XX) 3 Operations research, mathematical programming (90-XX) 2 Measure and integration (28-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Ordinary differential equations (34-XX) 1 General and overarching topics; collections (00-XX) 1 Category theory; homological algebra (18-XX) 1 Integral equations (45-XX) 1 Differential geometry (53-XX) 1 General topology (54-XX) 1 Global analysis, analysis on manifolds (58-XX) Citations by Year