# Errata, Remarks, and Reviews for  GAUSS AND JACOBI SUMS

Bruce  C.  Berndt,  Ronald  J.  Evans, and  Kenneth  S.  Williams
Wiley-Interscience, N.Y., 1998

Errata

(1) We are very grateful to Richard A. Mollin for pointing
out that our proof of the cubic reciprocity law (Theorem
8.1.7) is only valid when the primes
and   on  p. 238
are distinct. Thus our proof only verifies the theorem under
and   have coprime norms.
In order to complete the proof, we must verify that the cubic
residue symbols    and
are equal, where  is a
primary Eisenstein prime whose norm p is congruent to 1
modulo 3,  and   is the complex conjugate of
.   We can
give a short proof of this as follows, using what was already
proved so far.  First, writing ,  we have

Next, reversing the roles of
and  ,  we obtain

The symbols on the right of these equalities are equal,
since   is congruent to
modulo . Hence the symbols
on the left are equal, and the proof is complete.
Remark:   Since the symbols on the left of these equalities
are complex conjugates of each other by Proposition 8.1.3,
it follows that they each equal 1. Thus    is a cubic
residue modulo . But as was pointed out on p. 239,

where  is defined (up to sign) by the quadratic form
Thus we obtain the interesting application that
r  is always a cubic residue of p.  For example,
r =  7
is a cubic residue of the prime  p = 73.

(2) We are very grateful to Charles Helou for pointing out
the following list of mistakes.  On p. 271, in Theorem 9.2.6,
replace each   t  by  .
On p. 362, before formula (11.6.1),
the reference should be to Theorem 2.1.3(a)  instead of
Theorem 1.1.4(a).  On p. 365, after (11.6.4), replace
by  .
On p. 444,  in formula (13.2.6), replace
by  .
On p. 468,  three and four lines after
(14.1.1), the nonzero ideals of    and the
nonzero elements
in   must be taken
relatively prime to  k.
On p. 470, in each of the two lines above (14.2.2),  replace
M  by   (k-1)M.

(3) On p. 388, line 6, replace   ab
by  2ab.

(4) On p. 41, line 13, replace "The inner sum on
vanishes"
by  "The inner sum on   vanishes".

(Note that the sum on   y  vanishes
unless   =  1 + pt
with  ,
while for each such nonzero   t ,  the sum
on  x  vanishes  because the subsum over
those  x  congruent
to   c  (mod p)   vanishes for
each fixed  c.)

(5) In Problem 22 on p. 336, we take k to be q-1.

(6) Grammatical mistakes:
On p. 531, line 2, capitalize
the word sums Throughout the text, the possessive form
Gauss' should perhaps be replaced by Gauss's.

Notes

Research Problem 6 on p. 496 has been solved for g(12) by
R. J. Evans [Gauss sums of orders six and twelve, Can. Math.
Bull. 44 (2001), 22-26 ]. (The problem for  g(8)  is still open.)
Also in this paper is a new evaluation of  g(6) which is
more elegant than that given on p. 156 in Theorem 4.1.4 of the book.

Research Problems 9, 10 on pp. 496--497 have been solved by
R. J. Evans, M. Minei, and B. Yee [Incomplete higher order Gauss sums,
J. Math. Anal. Appl. 281 (2003), 454-476].

Research Problem 11 on p. 497 has been solved by R. J. Evans,
[Nonexistence of twentieth power residue difference sets,
Acta Arith. 84 (1999), 397-402 ].

Research Problem 25 on p. 498 has been solved by R. J. Evans
[Classical congruences for parameters in binary quadratic forms,
Finite Fields and their Applications 7 (2001), 110-124.]

Research Problem 26 on p. 498 has been solved by R. J. Evans
[Extensions of classical congruences for parameters in
binary quadratic forms,  Acta Arith. 100 (2001), 349-364 ].

Reviews

Last updated  July, 2011