Edit Profile (opens in new tab) Williams, Kenneth S. Co-Author Distance Author ID: williams.kenneth-s Published as: Williams, Kenneth S.; Williams, K. S.; Williams, kenneth S.; Wiliams, Kenneth S.; Williams, Kenneth more...less Homepage: http://people.math.carleton.ca/~williams/index.html External Links: MGP · dblp · GND Documents Indexed: 384 Publications since 1964, including 11 Books 2 Contributions as Editor Reviewing Activity: 95 Reviews Co-Authors: 59 Co-Authors with 272 Joint Publications 586 Co-Co-Authors all top 5 Co-Authors 114 single-authored 80 Spearman, Blair Kenneth 44 Alaca, Şaban 33 Alaca, Ayşe 22 Hudson, Richard H. 21 Kaplan, Pierre 21 Leonard, Philip A. 19 Hardy, Kenneth 16 Huard, James G. 10 Zhang, Nanyue 9 Pehlivan, Lerna 6 Buell, Duncan A. 6 Doyle, Greg 6 Friesen, Christian 6 Lemire, Mathieu F. 5 Aygin, Zafer Selçuk 5 Muskat, Joseph B. 4 Buck, Nicholas 4 Lavallee, Melisa J. 4 McAfee, Erin 4 Richman, David Ross 4 Uygul, Faruk 3 Eloff, Daniel 3 Evans, Ronald J. 3 McCann, Kevin S. 3 Muzaffar, Habib Bin 3 Ou, Zhiming M. 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 Münkel, Florian 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 deMille, Iain 1 Dlab, Vlastimil 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, Dunxue 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 American Mathematical Monthly 16 Journal of Number Theory 15 Pacific Journal of Mathematics 15 International Journal of Number Theory 14 Rocky Mountain Journal of Mathematics 13 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 Integers 6 Journal of Combinatorics and Number Theory 5 Archiv der Mathematik 4 Bulletin of the Australian Mathematical Society 4 Indian Journal of Pure & Applied Mathematics 4 Journal für die Reine und Angewandte Mathematik 4 Utilitas Mathematica 4 The Ramanujan Journal 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 Carleton-Ottawa Mathematical Lecture Note Series 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 Debrecen 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 Turkish 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 Mathematics and its Applications (Dordrecht) 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 360 Number theory (11-XX) 14 Field theory and polynomials (12-XX) 9 Special functions (33-XX) 5 Combinatorics (05-XX) 5 Sequences, series, summability (40-XX) 3 General and overarching topics; collections (00-XX) 3 Commutative algebra (13-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 Open 261 Publications have been cited 1,920 times in 1,158 Documents Cited by ▼ Year ▼ Gauss and Jacobi sums. Zbl 0906.11001Berndt, Bruce C.; Evans, Ronald J.; Williams, Kenneth S. 327 1998 Elementary evaluation of certain convolution sums involving divisor functions. Zbl 1062.11005Huard, James G.; Ou, Zhiming M.; Spearman, Blair K.; Williams, Kenneth S. 56 2002 Introductory algebraic number theory. Zbl 1035.11001Alaca, Şaban; Williams, Kenneth S. 53 2004 A simple norm inequality. Zbl 0178.16202Dunkl, Charles F.; Williams, K. S. 52 1964 Number theory in the spirit of Liouville. Zbl 1227.11002Williams, Kenneth S. 47 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. 38 2007 Note on cubics over \(GF(2^n)\) and \(GF(3^n)\). Zbl 0321.12029Williams, Kenneth S. 35 1975 The convolution sum \(\sum_{m<n/8} \sigma(m) \sigma(n-8m)\). Zbl 1130.11006Williams, Kenneth S. 33 2006 Nineteen quaternary quadratic forms. Zbl 1131.11025Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 33 2007 Integers of biquadratic fields. Zbl 0205.35401Williams, Kenneth S. 32 1970 The convolution sum \(\sum_{m<n/16}\sigma(m)\sigma(n-16m)\). Zbl 1132.11305Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 30 2008 The number of representations of a positive integer by certain octonary quadratic forms. Zbl 1213.11087Alaca, Şaban; Williams, Kenneth S. 29 2010 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. 29 2007 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. 28 2006 The convolution sum \(\sum_{m<n/9}\sigma(m)\sigma(n-9m)\). Zbl 1082.11003Williams, Kenneth S. 28 2005 On the two-dimensional theta functions of the Borweins. Zbl 1127.11035Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 25 2006 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. 23 2007 Some results on the generalized Stieltjes constants. Zbl 0808.11054Zhang, Nanyue; Williams, Kenneth S. 22 1994 Evaluation of two convolution sums involving the sum of divisors function. Zbl 1163.11301Lemire, Mathieu; Williams, Kenneth S. 22 2006 Integral bases for quartic fields with quadratic subfields. Zbl 0826.11048Huard, James G.; Spearman, Blair K.; Williams, Kenneth S. 22 1995 On Tornheim’s double series. Zbl 0858.40008Huard, James G.; Williams, Kenneth S.; Zhang, Nanyue 21 1996 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. 19 1987 Mertens’ theorem for arithmetic progressions. Zbl 0286.10022Williams, Kenneth S. 19 1974 Theta functions identities and representations by certain quaternary quadratic forms. Zbl 1163.11029Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 18 2008 Fourier series of a class of eta quotients. Zbl 1252.11043Williams, Kenneth S. 15 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. 13 2009 Jacobi’s identity and representations of integers by certain quaternary quadratic forms. Zbl 1209.11040Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 13 2007 Evaluation of some convolutions sums involving the sum of divisors functions. Zbl 1094.11004Cheng, Nathalie; Williams, Kenneth S. 12 2005 On the number of representations of \(n\) by \(ax^2+bxy+cy^2\). Zbl 1095.11024Sun, Zhihong; Williams, Kenneth S. 12 2006 Quartics over GF(2\(^n\)). Zbl 0274.12013Leonard, Philip A.; Williams, Kenneth S. 12 1973 \(A_4\)-sextic fields with a power basis. Zbl 1229.11136Eloff, Daniel; Spearman, Blair K.; Williams, Kenneth S. 11 2007 On the divisibility of the class numbers of \(\mathbb{Q}(\sqrt{-p})\) and \(\mathbb{Q}(\sqrt{-2p})\) by 16. Zbl 0431.12007Leonard, Philip A.; Williams, Kenneth S. 11 1982 Fourteen octonary quadratic forms. Zbl 1228.11049Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 11 2010 Binomial coefficients and Jacobi sums. Zbl 0551.12015Hudson, Richard H.; Williams, Kenneth S. 10 1984 Special values of the Lerch zeta function and the evaluation of certain integrals. Zbl 0785.11046Williams, Kenneth S.; Zhang, Nanyue 10 1993 Some series representations of \(\zeta(2n+1)\). Zbl 0808.11053Zhang, Nanyue; Williams, Kenneth S. 10 1993 Theta function identities and representations by certain quaternary quadratic forms. II. Zbl 1209.11041Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 10 2008 Evaluation of two trigonometric sums. Zbl 0820.11010Williams, Kenneth S.; Zhang, Nan-Yue 10 1994 Congruences for \(L\)-functions. Zbl 0972.11001Urbanowicz, Jerzy; Williams, Kenneth S. 10 2000 The integers of a cyclic quartic field. Zbl 0707.11078Hudson, R. H.; Williams, K. S. 9 1990 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 The Chowla-Selberg formula for genera. Zbl 0855.11018Huard, James G.; Kaplan, Pierre; Williams, Kenneth S. 9 1995 Pascal’s triangle (mod 8). Zbl 0889.11007Huard, James G.; Spearman, Blair K.; Williams, Kenneth S. 9 1998 The cyclotomic numbers of order seven. Zbl 0311.12008Leonard, Philip A.; Williams, Kenneth S. 9 1975 Evaluation of character sums connected with elliptic curves. Zbl 0369.10025Williams, Kenneth S. 9 1979 On the class number of \(Q(\sqrt{-p})\) modulo 16, for \(p\equiv 1\) (mod 8) a prime. Zbl 0393.12008Williams, Kenneth S. 9 1981 Pell’s equations \(X^ 2-mY^ 2=-1,-4\) and continued fractions. Zbl 0596.10013Kaplan, Pierre; Williams, Kenneth S. 8 1986 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. 8 2007 An Artin character and representations of primes by binary quadratic forms. II. Zbl 0485.12003Halter-Koch, Franz; Kaplan, Pierre; Williams, Kenneth S. 8 1982 Dihedral quintic fields with a power basis. Zbl 1161.11393Lavallee, Melissa J.; Spearman, Blair K.; Williams, Kenneth S.; Yang, Qiduan 8 2005 Relative integral bases for quartic fields over quadratic subfields. Zbl 0853.11090Spearman, B. K.; Williams, K. S. 8 1996 Some Lambert series expansions of products of theta functions. Zbl 0938.11017Williams, Kenneth S. 8 1999 Values of the Dedekind eta function at quadratic irrationalities. Zbl 0936.11026van der Poorten, Alfred; Williams, Kenneth S. 8 1999 Note on Salie’s sum. Zbl 0224.10039Williams, Kenneth S. 8 1971 On \[ \sum_{n=1}^ \infty (1/n^{2k}) \]. Zbl 0224.40008Williams, Kenneth S. 8 1971 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. 7 1990 Convolution sums involving the divisor function. Zbl 1156.11301Cheng, Nathalie; Williams, Kenneth S. 7 2004 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 Application of the Hurwitz zeta function to the evaluation of certain integrals. Zbl 0791.11045Zhang, Nanyue; Williams, Kenneth S. 7 1993 Seven octonary quadratic forms. Zbl 1171.11021Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 7 2008 Some arithmetic identities involving divisor functions. Zbl 1318.11001Alaca, Şaban; Uygul, Faruk; Williams, Kenneth S. 7 2012 Class numbers and biquadratic reciprocity. Zbl 0473.12004Williams, Kenneth S.; Currie, James D. 7 1982 On a double series of Chan and Ong. Zbl 1163.11318Williams, Kenneth S. 7 2006 A restricted Epstein zeta function and the evaluation of some definite integrals. Zbl 1037.11023Muzaffar, Habib; Williams, Kenneth S. 7 2002 On Euler’s criterion for cubic nonresidues. Zbl 0273.10002Williams, Kenneth S. 7 1975 The simplest arithmetic proof of Jacobi’s four squares theorem. Zbl 0958.11029Spearman, Blair K.; Williams, Kenneth S. 7 2000 On the imaginary bicyclic biquadratic fields with class-number 2. Zbl 0379.12002Buell, D. A.; Williams, H. C.; Williams, K. S. 7 1977 The \(n\)th power of a \(2\times 2\) matrix. Zbl 0789.15015Williams, Kenneth S. 6 1992 Some new residuacity criteria. Zbl 0456.10001Hudson, Richard H.; Williams, Kenneth S. 6 1980 Ramanujan identities and Euler products for a type of Dirichlet series. Zbl 1155.11026Sun, Zhihong; Williams, Kenneth S. 6 2006 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. 6 1992 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 Note on Dickson’s permutation polynomials. Zbl 0235.12011Williams, Kenneth S. 6 1971 The index of a cyclic quartic field. Zbl 1049.11111Spearman, Blair K.; Williams, Kenneth S. 6 2003 A rational octic reciprocity law. Zbl 0311.10004Williams, Kenneth S. 6 1976 The quadratic and quartic character of certain quadratic units. I. Zbl 0335.12006Leonard, Philip A.; Williams, Kenneth S. 6 1977 The quadratic and quartic character of certain quadratic units. II. Zbl 0386.12002Leonard, Philip A.; Williams, Kenneth S. 6 1979 On the evaluation of the Legendre symbol \(((A+B\sqrt{m})/p)\). Zbl 0524.10002Williams, Kenneth S.; Hardy, Kenneth; Friesen, Christian 5 1985 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 A “four integers” theorem and a “five integers” theorem. Zbl 1327.11024Williams, Kenneth S. 5 2015 The simplest \(D_4\)-octics. Zbl 1144.11074Spearman, Blair K.; Williams, Kenneth S. 5 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. 5 2008 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 \(\text{PSL}(2,5)\) sextic fields with a power basis. Zbl 1096.11038Spearman, Blair K.; Watanabe, Aya; Williams, Kenneth S. 5 2006 Characterization of solvable quintics \(x^ 5+ ax+ b\). Zbl 0844.12002Spearman, Blair K.; Williams, Kenneth S. 5 1994 The cyclotomic numbers of order eleven. Zbl 0267.12009Leonard, Philip A.; Williams, Kenneth S. 5 1975 An arithmetic proof of Jacobi’s eight squares theorem. Zbl 1004.11017Williams, Kenneth S. 5 2001 The distribution of solutions of congruences. Zbl 0158.29603Chalk, John H. H.; Williams, K. S. 5 1965 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 On the number of distinguished representations of a group element. Zbl 0248.20018Jacobson, David; Williams, Kenneth S. 5 1972 Some new theta function identities with applications to sextenary quadratic forms. Zbl 1235.11039Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 5 2009 Note on a result of Barrucand and Cohn. Zbl 0325.10002Williams, Kenneth S. 5 1976 A Diophantine system of Dickson. Zbl 0299.12017Leonard, Philip A.; Williams, Kenneth S. 5 1974 Cubic fields with index 2. Zbl 0992.11058Spearman, Blair K.; Williams, Kenneth S. 5 2002 A rational sixteenth power reciprocity law. Zbl 0363.10003Leonard, Philip A.; Williams, Kenneth S. 5 1977 On Scholz’s reciprocity law. Zbl 0372.12004Williams, Kenneth S. 5 1977 A congruence relating the class numbers of complex quadratic fields. Zbl 0557.12003Hardy, Kenneth; Williams, Kenneth S. 4 1986 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. 4 2014 Beyond the Basel problem. I. Zbl 07687521Williams, Kenneth S. 1 2023 An application of binary quadratic forms of discriminant \(-31\) to modular forms. Zbl 1501.11056Aygin, Zafer Selcuk; Williams, Kenneth S. 1 2022 Derivable quadratic forms and representation numbers. Zbl 1478.11056Aygin, Zafer Selcuk; Williams, Kenneth S. 1 2021 Representation numbers of spinor regular ternary quadratic forms. Zbl 1506.11056Aygin, Zafer Selcuk; Doyle, Greg; Münkel, Florian; Pehlivan, Lerna; Williams, Kenneth S. 1 2021 Prime-universal quadratic forms \(ax^2+by^2+cz^2\) and \(ax^2+by^2+cz^2+dw^2\). Zbl 1445.11021Doyle, Greg; Williams, Kenneth S. 1 2020 Representation numbers of seven quaternary quadratic forms each in a genus consisting of only two classes. Zbl 1493.11080Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth 1 2020 Positive integers represented by regular primitive positive-definite integral ternary quadratic forms. Zbl 1467.11040Doyle, Greg; Muskat, Joseph B.; Pehlivan, Lerna; Williams, Kenneth S. 2 2019 \((k,l)\)-universality of ternary quadratic forms \(ax^2 + by^2 + cz^2\). Zbl 1450.11032Pehlivan, Lerna; Williams, Kenneth S. 2 2018 Evaluation of some \(q\)-integrals in terms of the Dedekind eta function. Zbl 1429.11079Doyle, Greg; Williams, Kenneth S. 1 2018 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 Some arithmetic convolution identities. Zbl 1422.11007Williams, Kenneth S. 3 2017 A positive-definite ternary quadratic form does not represent all positive integers. Zbl 1412.11065Doyle, Greg; Williams, Kenneth S. 2 2017 Infinite products with coefficients which vanish on certain arithmetic progressions. Zbl 1429.11078Alaca, Ayşe; Alaca, Şaban; Aygin, Zafer Selcuk; Williams, Kenneth S. 2 2017 A class of subsums of Euler’s sum. Zbl 1364.11009Williams, Kenneth S. 1 2016 Bounds for the representations of integers by positive quadratic forms. Zbl 1398.11068Williams, Kenneth S. 1 2016 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 Historical remark on Ramanujan’s tau function. Zbl 1336.11037Williams, Kenneth S. 1 2015 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. 4 2014 Double Gauss sums. Zbl 1366.11092Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 3 2014 The power series expansion of certain infinite products \(q^{r}\prod_{n=1}^{\infty}(1-q^{n})^{a_{1}}(1-q^{2n})^{a_{2}}\cdots(1-q^{mn})^{a_{m}}\). Zbl 1285.11064Pehlivan, Lerna; Williams, Kenneth S. 2 2014 Ternary quadratic forms \(ax^2+by^2+cz^2\) representing all positive integers \(8k+4\). Zbl 1327.11023Williams, 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 Euler products in Ramanujan’s lost notebook. Zbl 1281.11086Berndt, Bruce C.; Kim, Byungchan; Williams, Kenneth S. 1 2013 Fourier series of a class of eta quotients. Zbl 1252.11043Williams, Kenneth S. 15 2012 Some arithmetic identities involving divisor functions. Zbl 1318.11001Alaca, Şaban; Uygul, Faruk; Williams, Kenneth S. 7 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. 4 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. 47 2011 Lifting monogenic cubic fields to monogenic sextic fields. Zbl 1237.11045Lavallee, Melisa J.; Spearman, Blair K.; Williams, Kenneth S. 4 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. 29 2010 Fourteen octonary quadratic forms. Zbl 1228.11049Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 11 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. 13 2009 Some new theta function identities with applications to sextenary quadratic forms. Zbl 1235.11039Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 5 2009 Some infinite products of Ramanujan type. Zbl 1234.11141Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 3 2009 Sums of \(4k\) squares: a polynomial approach. Zbl 1239.11044Alaca, 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. 30 2008 Theta functions identities and representations by certain quaternary quadratic forms. Zbl 1163.11029Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 18 2008 Theta function identities and representations by certain quaternary quadratic forms. II. Zbl 1209.11041Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 10 2008 Seven octonary quadratic forms. Zbl 1171.11021Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 7 2008 The simplest \(D_4\)-octics. Zbl 1144.11074Spearman, Blair K.; Williams, Kenneth S. 5 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. 5 2008 On Liouville’s twelve squares theorem. Zbl 1242.11029Williams, 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 Indices of integers in cyclic cubic fields. Zbl 1247.11135Spearman, B. K.; Williams, K. S. 2 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. 38 2007 Nineteen quaternary quadratic forms. Zbl 1131.11025Alaca, Ayşe; Alaca, Şaban; Lemire, Mathieu F.; Williams, Kenneth S. 33 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. 29 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. 23 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. 13 2007 \(A_4\)-sextic fields with a power basis. Zbl 1229.11136Eloff, Daniel; Spearman, Blair K.; Williams, Kenneth S. 11 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. 8 2007 Lambert series and Liouville’s identities. Zbl 1113.11004Alaca, A.; Alaca, Ş.; McAfee, E.; Williams, K. S. 3 2007 An infinite class of identies. Zbl 1115.33017Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 1 2007 The simplest proof of Jacobi’s six squares theorem. Zbl 1204.11072Alaca, 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. 33 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. 28 2006 On the two-dimensional theta functions of the Borweins. Zbl 1127.11035Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. 25 2006 Evaluation of two convolution sums involving the sum of divisors function. Zbl 1163.11301Lemire, Mathieu; Williams, Kenneth S. 22 2006 On the number of representations of \(n\) by \(ax^2+bxy+cy^2\). Zbl 1095.11024Sun, Zhihong; Williams, Kenneth S. 12 2006 On a double series of Chan and Ong. Zbl 1163.11318Williams, Kenneth S. 7 2006 Ramanujan identities and Euler products for a type of Dirichlet series. Zbl 1155.11026Sun, Zhihong; Williams, Kenneth S. 6 2006 \(\text{PSL}(2,5)\) sextic fields with a power basis. Zbl 1096.11038Spearman, Blair K.; Watanabe, Aya; Williams, Kenneth S. 5 2006 Values of the Euler phi function not divisible by a given odd prime. Zbl 1216.11089Spearman, Blair K.; Williams, Kenneth S. 4 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 The prime ideal factorization of 2 in pure quartic fields with index 2. Zbl 1175.11061Spearman, Blair K.; Williams, Kenneth S. 1 2006 The convolution sum \(\sum_{m<n/9}\sigma(m)\sigma(n-9m)\). Zbl 1082.11003Williams, Kenneth S. 28 2005 Evaluation of some convolutions sums involving the sum of divisors functions. Zbl 1094.11004Cheng, Nathalie; Williams, Kenneth S. 12 2005 Dihedral quintic fields with a power basis. Zbl 1161.11393Lavallee, Melissa J.; Spearman, Blair K.; Williams, Kenneth S.; Yang, Qiduan 8 2005 Irreducible quartic polynomials with factorization modulo \(p\). Zbl 1211.11037Driver, Eric; Leonard, Philip A.; Williams, Kenneth S. 3 2005 Watson’s method of solving a quintic equation. Zbl 1102.12001Lavallee, Melisa J.; Spearman, Blair K.; Williams, Kenneth S. 2 2005 Introductory algebraic number theory. Zbl 1035.11001Alaca, Şaban; Williams, Kenneth S. 53 2004 Convolution sums involving the divisor function. Zbl 1156.11301Cheng, Nathalie; Williams, Kenneth S. 7 2004 \(p\)-integral bases of a quartic field defined by a trinomial \(x^4+ax+b\). Zbl 1050.11091Alaca, Şaban; Williams, Kenneth S. 4 2004 On the number of representations of a positive integer by a binary quadratic form. Zbl 1102.11024Kaplan, Pierre; Williams, Kenneth S. 3 2004 Evaluation of Weber’s functions at quadratic irrationalities. Zbl 1077.11031Muzaffar, Habib; Williams, Kenneth S. 2 2004 Normal integral bases for Emma Lehmer’s parametric family of cyclic quintics. Zbl 1158.11343Spearman, Blair K.; Williams, Kenneth S. 2 2004 On Voronoi’s method for finding an integral basis of a cubic field. Zbl 1061.11056Alaca, Şaban; 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 The factorization of 2 in cubic fields with index 2. Zbl 1080.11079Alaca, Saban; Spearman, Blair K.; Williams, Kenneth S. 1 2004 The index of a cyclic quartic field. Zbl 1049.11111Spearman, Blair K.; Williams, Kenneth S. 6 2003 The discriminant of a cyclic field of odd prime degree. Zbl 1074.11059Spearman, Blair K.; Williams, Kenneth S. 3 2003 \(n=\Delta +\Delta +2(\Delta + \Delta)\). Zbl 1041.11022Williams, Kenneth S. 2 2003 Sums of twelve squares. Zbl 1023.11018Huard, James G.; Williams, Kenneth S. 1 2003 A relative integral basis over \(\mathbb{Q}(\sqrt{-3})\) for the normal closure of a pure cubic field. Zbl 1064.11070Spearman, Blair K.; Williams, Kenneth S. 1 2003 The index of a dihedral quartic field. Zbl 1049.11115Silvester, Alan K.; Spearman, Blair K.; 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. 56 2002 A restricted Epstein zeta function and the evaluation of some definite integrals. Zbl 1037.11023Muzaffar, Habib; Williams, Kenneth S. 7 2002 Cubic fields with index 2. Zbl 0992.11058Spearman, Blair K.; Williams, Kenneth S. 5 2002 The discriminant of a dihedral quintic field defined by a trinomial \(X^5+aX+b\). Zbl 1040.11076Spearman, Blair K.; Williams, Kenneth S. 3 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. 5 2001 ...and 161 more Documents all cited Publications top 5 cited Publications all top 5 Cited by 1,240 Authors 75 Williams, Kenneth S. 26 Sun, Zhihong 23 Alaca, Şaban 20 Spearman, Blair Kenneth 19 Alaca, Ayşe 16 Evans, Ronald J. 16 Kim, Daeyeoul 15 Xia, Ernest X. W. 12 Aygin, Zafer Selçuk 12 Jones, Lenny K. 11 Feng, Tao 10 Yao, Olivia Xiang Mei 10 Ye, Dongxi 9 Srivastava, Hari Mohan 9 Zeng, Xiangyong 8 Barman, Rupam 8 Berndt, Bruce Carl 8 Dilcher, Karl 8 Gao, Peng 8 Halter-Koch, Franz 8 Köklüce, Bülent 8 Ramakrishnan, Balakrishnan 8 Sahu, Brundaban 8 Xiang, Qing 8 Zhao, Liangyi 7 Baoulina, Ioulia N. 7 Coffey, Mark William 7 Helleseth, Tor 7 Li, Chengju 7 Pehlivan, Lerna 7 Sun, Zhi-Wei 7 Tsumura, Hirofumi 7 Wu, Hai-Liang 7 Yue, Qin 6 Doyle, Greg 6 Kim, Minsoo 6 Louboutin, Stéphane R. 6 McCarthy, Dermot 6 Momihara, Koji 6 Moree, Pieter 6 Park, Yoon Kyung 6 Singh, Anup Kumar 6 Tanti, Jagmohan 6 Tu, Ziran 6 Zaharescu, Alexandru 5 Bibak, Khodakhast 5 Cao, Wei 5 Cho, Bumkyu 5 Cooper, Shaun 5 Garcia, Stephan Ramon 5 Hou, Xiang-Dong 5 Hu, Su 5 Hudson, Richard H. 5 Kaplan, Pierre 5 Kim, Aeran 5 Kwon, Soonhak 5 Leonard, Philip A. 5 Minculete, Nicusor 5 Moslehian, Mohammad Sal 5 Ntienjem, Ebénézer 5 Oh, Byeong-Kweon 5 Saikia, Neelam 5 Saito, Kichisuke 4 Ahmed, Mohammad Helal Uddin 4 Azizi, Abdelmalek 4 Bayad, Abdelmejid 4 Chan, Heng Huat 4 Chems-Eddin, Mohamed Mahmoud 4 Cochrane, Todd 4 Cosgrave, John B. 4 Feng, Keqin 4 Gaál, István 4 Gurak, Stanley J. 4 Hong, Shaofang 4 İkikardes, Nazli Yildiz 4 Kapron, Bruce M. 4 Kim, Henry H. 4 Li, Kangquan 4 Luca, Florian 4 Mesnager, Sihem 4 Nakahara, Toru 4 Ono, Ken 4 Qu, Longjiang 4 Sahmoudi, Mohammed 4 Seeger, Alberto 4 Sofo, Anthony 4 Soullami, Abderazak 4 Srinivasan, Venkatesh 4 Toh, Pee Choon 4 Tripathi, Mohit 4 Tu, Fang-Ting 4 van Wamelen, Paul B. 4 Yang, Qiduan 3 Ahn, Jaehyun 3 Alahmadi, Adel N. 3 Ayad, Mohamed 3 Batir, Necdet 3 Blagouchine, Iaroslav V. 3 Bradley, David M. 3 Bringmann, Kathrin ...and 1,140 more Authors all top 5 Cited in 228 Serials 152 Journal of Number Theory 93 International Journal of Number Theory 60 Finite Fields and their Applications 46 The Ramanujan Journal 38 Proceedings of the American Mathematical Society 32 Journal of Mathematical Analysis and Applications 28 Mathematics of Computation 27 Designs, Codes and Cryptography 24 Rocky Mountain Journal of Mathematics 22 Journal de Théorie des Nombres de Bordeaux 21 Cryptography and Communications 19 Functiones et Approximatio. 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Series II ...and 128 more Serials all top 5 Cited in 49 Fields 983 Number theory (11-XX) 127 Combinatorics (05-XX) 98 Special functions (33-XX) 66 Information and communication theory, circuits (94-XX) 54 Field theory and polynomials (12-XX) 47 Algebraic geometry (14-XX) 39 Functional analysis (46-XX) 26 Real functions (26-XX) 22 Linear and multilinear algebra; matrix theory (15-XX) 20 Sequences, series, summability (40-XX) 18 Group theory and generalizations (20-XX) 16 Commutative algebra (13-XX) 16 Operator theory (47-XX) 15 Functions of a complex variable (30-XX) 15 Harmonic analysis on Euclidean spaces (42-XX) 15 Computer science (68-XX) 14 Quantum theory (81-XX) 13 Numerical analysis (65-XX) 10 Geometry (51-XX) 9 Convex and discrete geometry (52-XX) 9 Probability theory and stochastic processes (60-XX) 8 Associative rings and algebras (16-XX) 7 Dynamical systems and ergodic theory (37-XX) 7 Approximations and expansions (41-XX) 6 Order, lattices, ordered algebraic structures (06-XX) 5 History and biography (01-XX) 5 Partial differential equations (35-XX) 5 Manifolds and cell complexes (57-XX) 4 Nonassociative rings and algebras (17-XX) 4 \(K\)-theory (19-XX) 4 Ordinary differential equations (34-XX) 4 General topology (54-XX) 4 Statistical mechanics, structure of matter (82-XX) 3 Category theory; homological algebra (18-XX) 3 Measure and integration (28-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Operations research, mathematical programming (90-XX) 2 General and overarching topics; collections (00-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Abstract harmonic analysis (43-XX) 2 Statistics (62-XX) 2 Fluid mechanics (76-XX) 1 Integral transforms, operational calculus (44-XX) 1 Integral equations (45-XX) 1 Differential geometry (53-XX) 1 Mechanics of deformable solids (74-XX) 1 Optics, electromagnetic theory (78-XX) 1 Geophysics (86-XX) 1 Mathematics education (97-XX) Citations by Year