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*