William D. Banks Papers

1995

• The Casselman-Shalika formula for a distinguished model, W. Banks, Proc. Amer. Math. Soc. 123 (1995), no. 3, 681-692.

1997

• Twisted symmetric-square L-functions and the nonexistence of Siegel zeros on GL(3), W. Banks, Duke Math. J. 87 (1997), no. 2, 343-353.

1998

• Heredity of Whittaker models on the metaplectic group, W. Banks, Pacific J. Math. 185 (1998), no. 1, 89-96.

• A corollary to Bernstein's theorem and Whittaker functionals on the metaplectic group, W. Banks, Math. Res. Lett. 5 (1998), 781-790.

1999

• Block-compatible metaplectic cocycles, W. Banks, J. Levy and M. Sepanski, J. Reine Angew. Math. 507 (1999), 131-163.

2000

• An identification scheme based on sparse polynomials, W. Banks, D. Lieman and I. Shparlinski, in Public Key Cryptography, Third International Workshop on Practice and Theory in Public Key Cryptography, PKC2000 (Melbourne), Lecture Notes in Comput. Sci. 1751, Springer-Verlag, Berlin (2000).

• Nonlinear complexity of the Naor-Reingold pseudo-random function, W. Banks, F. Griffin, D. Lieman and I. Shparlinski, in Proceedings of ICISC`99 (Seoul), Lecture Notes in Comput. Sci. 1787, Springer-Verlag, Berlin (2000).

• An extremely small and efficient identification scheme, W. Banks, D. Lieman and I. Shparlinski, in Proceedings of ACISP2000 (Brisbane), Lecture Notes in Comput. Sci. 1841, Springer-Verlag, Berlin (2000).

2001

• Cryptographic applications of sparse polynomials over finite rings, W. Banks, D. Lieman, I. Shparlinski and V. To, in Proceedings of ICISC2000 (Seoul), Lecture Notes in Comput. Sci. 2015, Springer-Verlag, Berlin (2001).

• On the dimension of the Jacquet module of a certain induced representation, W. Banks, D. Bump and D. Lieman, in Ideal Theoretic Methods in Commutative Algebra, Lecture Notes in Pure and Applied Math. 220, Marcel Dekker, Inc., New York (2001).

• Some unusual identities for special values of the Riemann zeta function, W. Banks, Ramanujan J. 5 (2001), no. 2, 153-157.

• Distribution of inverses in polynomial rings, W. Banks and I. Shparlinski, Indag. Math. (N.S.) 12 (2001), no. 3, 303-315.

2002

• On the number of sparse RSA exponents, W. Banks and I. Shparlinski, J. Number Theory 95 (2002), no. 2, 340-350.

• Average normalisations of elliptic curves, W. Banks and I. Shparlinski, Bull. Austral. Math. Soc. 66 (2002), 353-358.

• Character sums over integers with restricted $g$-ary digits, W. Banks, A. Conflitti and I. Shparlinski, Illinois J. Math. 46 (2002), no. 3, 819-836.

2003

• Whittaker-Fourier coefficients of metaplectic Eisenstein series, W. Banks, D. Bump and D. Lieman, Compositio Math. 135 (2003), no. 2, 153-178.

• Matrix inequalities with applications to the theory of iterated kernels, W. Banks, A. Harcharras, S. Neuwirth and E. Ricard, Linear Algebra Appl. 362 (2003), 275-286.

• A variant of NTRU with non-invertible polynomials, W. Banks and I. Shparlinski, in Progress in Cryptology - Indocrypt 2002, Lecture Notes in Comput. Sci. 2551, Springer-Verlag, Berlin (2003).

• Short Kloosterman sums for polynomials over finite fields, W. Banks, A. Harcharras and I. Shparlinski, Canad. J. Math. 55 (2003), 225-246.

• New examples of noncommutative $\Lambda(p)$ sets, W. Banks and A. Harcharras, Illinois J. Math. 47 (2003), 1063-1078.

2004

• Exponential sums over Mersenne numbers, W. Banks, A. Conflitti, J. Friedlander and I. Shparlinski, Compositio Math. 140 (2004), 15-30.

• Number theoretic designs for directed regular graphs of small diameter, W. Banks, A. Conflitti and I. Shparlinski, SIAM J. Discrete Math. 17 (2004), no. 3, 377-383.

• Squares from products of integers, W. Banks and A. van der Poorten, Austral. Math. Soc. Gaz. 31 (2004), no. 1, 40-42.

• Congruences and exponential sums with the Euler function, W. Banks and I. Shparlinski, in High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, 49-59, Fields Inst. Commun. 41, Amer. Math. Soc., Providence, RI, 2004.

• Multiplicative structure of values of the Euler function, W. Banks, J. Friedlander, C. Pomerance and I. Shparlinski, in High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, 29-47, Fields Inst. Commun. 41, Amer. Math. Soc., Providence, RI, 2004.

• Arithmetic properties of numbers with restricted digits, W. Banks and I. Shparlinski, Acta Arith. 112 (2004), no. 4, 313-332.

• On the norm of an idempotent Schur multiplier on the Schatten class, W. Banks and A. Harcharras, Proc. Amer. Math. Soc. 132 (2004), no. 7, 2121-2125.

• Smooth values of shifted primes in arithmetic progressions, W. Banks, A. Harcharras and I. Shparlinski, Michigan Math. J. 52 (2004), no. 3, 603-618.

• Almost all palindromes are composite, W. Banks, D. Hart and M. Sakata, Math. Res. Lett. 11 (2004) nos. 5-6, 853-868.

2005

• On the average value of divisor sums in arithmetic progressions, W. Banks, R. Heath-Brown and I. Shparlinski, Int. Math. Res. Not. (2005), no. 1, 1-25.

• Concatenations with binary recurrent sequences, W. Banks and F. Luca, J. Integer Seq. 8 (2005), no. 1, Article 05.1.3, 18 pp. (electronic)

• On the value set of $n!$ modulo a prime, W. Banks, F. Luca, I. Shparlinski and H. Stichtenoth, Turkish J. Math. 29 (2005), 169-174.

• Irrationality of power series for various number theoretic functions, W. Banks, F. Luca and I. Shparlinski, Manuscripta Math. 117 (2005), no. 2, 183-197.

• Values of arithmetical functions equal to a sum of two squares, W. Banks, F. Luca, F. Saidak and I. Shparlinski, Quart. J. Math. Oxford Ser. 56 (2005), 123-139.

• Nonaliquots and Robbins numbers, W. Banks and F. Luca, Colloq. Math. 103 (2005), 27-32.

• Values of the Euler function in various sequences, W. Banks, K. Ford, F. Luca, F. Pappalardi and I. Shparlinski, Monatsh. Math. 146 (2005), 1-19.

• Compositions with the Euler and Carmichael functions, W. Banks, F. Luca, F. Saidak and P. Stanica, Abh. Math. Sem. Hamburg 75 (2005), 215-243.

• Uniform distribution of the fractional part of the average prime divisor, W. Banks, M. Garaev, F. Luca and I. Shparlinski, Forum Math. 17 (2005), no. 6, 885-901.

• Towards faster cryptosystems, II, W. Banks, in Public Key Cryptography, Proceedings of Symposia in Pure Mathematics, 62, American Mathematical Society, 2005.

• Roughly squarefree values of the Euler and Carmichael functions, W. Banks and F. Luca, Acta Arith. 120 (2005), 211-230.

• Prime divisors of palindromes, W. Banks and I. Shparlinski, Period. Math. Hungar. 51 (2005), 1-10.

• Some divisibility properties of the Euler function, W. Banks, F. Luca and I. Shparlinski, Glasgow Math. J. 47 (2005), no. 3, 517-528.

• Distributional properties of the largest prime factor, W. Banks, G. Harman and I. Shparlinski, Michigan Math. J. 53 (2005), 665-681.

• On certain sums related to multiple divisibility by the largest prime factor, W. Banks, F. Luca and I. Shparlinski, Ann. Sci. Math. Quebec 29 (2005), no. 2, 131-145.

2006

• Arithmetic properties of $\varphi(n)/\lambda(n)$ and the structure of the multiplicative group modulo $n$, W. Banks, F. Luca and I. Shparlinski, Comment. Math. Helv. 81 (2006), no. 1, 1-22.

• Incomplete character sums and Diffie-Hellman triples, W. Banks, J. Friedlander, S. Konyagin and I. Shparlinski, Math. Proc. Cambridge Philos. Soc. 140 (2006), 193-206.

• Coincidences in the values of the Euler and Carmichael functions, W. Banks, J. Friedlander, F. Luca, F. Pappalardi and I. Shparlinski, Acta Arith. 122 (2006), no. 3, 207-234.

• Non-residues and primitive roots in Beatty sequences, W. Banks and I. Shparlinski, Bull. Austral. Math. Soc. 73 (2006), 433-443.

• On integers with a special divisibility property, W. Banks and F. Luca, Arch. Math. (Brno) 42 (2006), 31-42.

• Multiplicative character sums with the $g$-ary sum of digits function, W. Banks and I. Shparlinski, Ramanujan J. 11 (2006), no. 2, 215-219.

• Double character sums over elliptic curves and finite fields, W. Banks, J. Friedlander, M. Garaev and I. Shparlinski, Pure Appl. Math. Q. 2 (2006), no. 1, 179-197.

• Short character sums with Beatty sequences, W. Banks and I. Shparlinski, Math. Res. Lett. 13 (2006), 539-547.

• Character sums with exponential functions over smooth numbers, W. Banks, J. Friedlander, M. Garaev and I. Shparlinski, Indag. Math. (N.S.) 17 (2006), no. 2, 157-168.

• Values of the Euler function free of $k$-th powers, W. Banks and F. Pappalardi, J. Number Theory 120 (2006), no. 2, 326-348.

• Congruences and rational exponential sums with the Euler function, W. Banks and I. Shparlinski, Rocky Mountain J. Math. 36 (2006), 1415-1426.

• Common divisors of the Euler function at related arguments, W. Banks, F. Luca and I. Shparlinski, Acta Sci. Math. (Szeged) 72 (2006), no. 3-4, 525-536.

• Average value of the Euler function on binary palindromes, W. Banks and I. Shparlinski, Bull. Pol. Acad. Sci. Math. 54 (2006), no. 2, 95-101.

2007

• Prime divisors of Beatty sequences, W. Banks and I. Shparlinski, J. Number Theory 123 (2007), no. 2, 413-425.

• On rough and smooth neighbours, W. Banks, F. Luca and I. Shparlinski, Rev. Mat. Complut. 20 (2007), no. 1, 109-118.

• On values taken by the largest prime factor of shifted primes, W. Banks and I. Shparlinski, J. Aust. Math. Soc. 82 (2007), 133-147.

• Integers with a large smooth divisor, W. Banks and I. Shparlinski, Integers 7 (2007), A17, 11 pp. (electronic)

• Sums of prime divisors and Mersenne numbers, W. Banks and F. Luca, Houston J. Math. 33 (2007), no. 2, 403-413 (electronic).

• When the sum of aliquots divides the totient, W. Banks and F. Luca, Proc. Edinb. Math. Soc. 50 (2007), 563-569.

• Composite integers $n$ for which $\varphi(n)\mid n-1$, W. Banks and F. Luca, Acta Math. Sinica, English Series 23 (2007), no. 10, 1915-1918.

• Kloosterman sums for modified van der Corput sequences, W. Banks, M. Sakata and F. Saidak, Uniform Distribution Theory 2 (2007), no. 1, 39-52.

• Estimates for Wieferich numbers, W. Banks, F. Luca and I. Shparlinski, Ramanujan J. 14 (2007), no. 3, 361-378.

• Representations of integers as sums of primes from a Beatty sequence, W. Banks, A. Guloglu and W. Nevans, Acta Arith. 130 (2007), no. 3, 255-275.

• Exponential sums with polynomial values of the discrete logarithm, W. Banks and I. Shparlinski, Uniform Distribution Theory 2 (2007), no. 2, 67-72.

2008

• Density of non-residues in Burgess-type intervals and applications, W. Banks, M. Garaev, R. Heath-Brown and I. Shparlinski, Bull. London Math. Soc. 40 (2008), 88-96.

• On the congruence $n\equiv a\pmod{\varphi(n)}$, W. Banks, A. Guloglu and W. Nevans, Integers 8(1) (2008), A59, 8 pp. (electronic)

• Descartes numbers, W. Banks, A. Guloglu, W. Nevans and F. Saidak, in Anatomy of Integers, 167-174, American Mathematical Society, Providence R.I., 2008.

• Congruences and exponential sums with the sum of aliquot divisors function, S. Balasuriya, W. Banks and I. Shparlinski, Int. J. Number Theory 4 (2008), no. 6, 903-909.

• Integers without divisors from a fixed arithmetic progression, W. Banks, F. Luca and J. Friedlander, Forum Math. 20 (2008), no. 6, 1005-1037.

2009

• Arithmetic functions on Beatty sequences, A. Abercrombie, W. Banks and I. Shparlinski, Acta Arith. 136 (2009), 81-89.

• Character sums with Beatty sequences on Burgess-type intervals, W. Banks and I. Shparlinski, in Analytic Number Theory: Essays in Honour of Klaus Roth, Cambridge University Press, 2009.

• Carmichael numbers with a square totient, W. Banks, Canad. Math. Bull. 52(1) (2009), 3-8.

• Uniform distribution of fractional parts related to pseudoprimes, W. Banks, M. Garaev, F. Luca and I. Shparlinski, Canad. J. Math. 61 (2009), no. 3, 481-502.

• The Nicolas and Robin inequalities with sums of two squares, W. Banks, D. Hart, P. Moree and W. Nevans, Monatsh. Math. 157 (2009), 303-322.

• Values of the Carmichael function equal to a sum of two squares, W. Banks and A. Guloglu, Turkish J. Math. 33 (2009), no. 1, 9-16.

• Prime numbers with Beatty sequences, W. Banks and I. Shparlinski, Colloq. Math. 115 (2009), no. 2, 147-157.

• A variant on the notion of Diophantine $s$-tuples, W. Banks, F. Luca and L. Szalay, Glasg. Math. J. 51 (2009), no. 1, 83-89.

• Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height, W. Banks and I. Shparlinski, Israel J. Math. 173 (2009), 253-277.

• Multiplicative character sums with twice-differentiable functions, W. Banks and I. Shparlinski, Q. J. Math. 60 (2009), no. 4, 401-411.

• A remark on Giuga's conjecture and Lehmer's totient problem, W. Banks, W. Nevans and C. Pomerance, Albanian J. Math. 3 (2009), no. 2, 81-85.

2010

• Sums with convolutions of Dirichlet characters, W. Banks and I. Shparlinski, Manuscripta Math. 133 (2010) no. 1, 105-114.

• On Carmichael numbers in arithmetic progressions, W. Banks and C. Pomerance, J. Aust. Math. Soc. 88 (2010), 313-321.

• On primitive Dirichlet characters and the Riemann hypothesis, W. Banks, A. Guloglu and W. Nevans, J. Number Theory 130 (2010), no. 3, 574-579.

2011

• Sums and products with smooth numbers, W. Banks and D. Covert, J. Number Theory 131 (2011), no. 6, 985-993.

• Power totients with almost primes, W. Banks and F. Luca, Integers 11 (2011), 307-313.

• Carmichael numbers composed of primes from a Beatty sequence, W. Banks and A. Yeager, Colloq. Math. 125 (2011), no. 1, 129-137.

2012

• On group structures realized by elliptic curves over arbitrary finite fields, W. Banks, F. Pappalardi and I. Shparlinski, Experiment. Math. 21 (2012), no. 1, 11-25.

• Exponential and character sums with Mersenne numbers, W. Banks, J. Friedlander, M. Garaev and I. Shparlinski, J. Aust. Math. Soc. 92 (2012), 1-13.

• Carmichael numbers with a totient of the form $a^2+nb^2$, W. Banks, Monatsh. Math. 167 (2012), no. 2, 157-163.

• On repeated values of the Riemann zeta function on the critical line, W. Banks and S. Kang, Experiment. Math. 12 (2012), no. 2, 132-140.

2013

• Piatetski-Shapiro sequences, R. Baker, W. Banks, J. Brudern, I. Shparlinski and A. Weingartner, Acta Arith. 157 (2013), no. 1, 37-68.

• Self-intersections of the Riemann zeta function on the critical line, W. Banks, V. Castillo-Garate, L. Fontana and C. Morpurgo, J. Math. Anal. Appl. 406 (2013), no. 2, 475-481.

• Optimal primitive sets with restricted primes, W. Banks and G. Martin, Integers 13 (2013), A69, 10pp. (electronic)

• Carmichael meets Chebotarev, W. Banks, A. Guloglu and A. Yeager, Canad. Math. Bull. 56 (2013), no. 4, 695-708.

2014

• Waring's problem for Beatty sequences and a local to global principle, W. Banks, A. Guloglu and R. Vaughan, J. Theor. Nombres Bordeaux 26 (2014), no. 1, 1-16.

• Piatetski-Shapiro primes from almost primes, R. Baker, W. Banks, Z. Guo and A. Yeager Monat. Math. 174 (2014), no. 3, 357-370.

2015

• Sierpinski and Carmichael numbers, W. Banks, C. Finch, F. Luca, C. Pomerance and P. Stanica, Trans. AMS 367 (2015), no. 1, 355-376.

• Consecutive primes in tuples, T. Freiberg, W. Banks and C. Turnage-Butterbaugh, Acta Arith. 167 (2015), no. 3, 261-266.

TO APPEAR

• Character sums with Piatetski-Shapiro sequences, W. Banks and R. Baker, to appear in Quart. J. Math.

• On Gauss sums and the evaluation of Stechkin's constant, W. Banks and I. Shparlinski, to appear in Math. Comp.

• Fractional parts of Dedekind sums, W. Banks and I. Shparlinski, to appear in Int. J. Number Theory