R ENDICONTI
del
S EMINARIO M ATEMATICO
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U NIVERSITÀ DI
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L. C ARLITZ
Some formulas related to Gauss’s sum
Rendiconti del Seminario Matematico della Università di Padova,
tome 41 (1968), p. 222-226
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SOME FORMULAS RELATED TO GAUSS’S SUM
L. CARLITZ
1. Chowla
where m, n
are
[1]]
has
proved
arbitrary
odd
*)
the formula
positive integers. By contour integration
it is shown that
where m, n
Comparison
*)
are
odd and
of (2) and
Indirizzo dell’A.:
positive. Ramanujan [2] proved
that
(3) gives (1).
Dept.
of
Math.,
Duke
Univ., Durham, N. C., USA.
223
In the
of
note we shall give a
shall prove the slightly
present
(1). Indeed
we
where m, n are odd, (a, 2mn~
the Legendre-Jacobi symbol.
In addition
we
=
1, aa’
«
For a, = a’
simple elementary proof
general formula
more
(mod
=
-
and
prove
-
where
m
2. If
is odd and (a, mn)
m
is odd and ~
Let a be
Let aa’
an
=
1
=
1.
=
I
integer prime
(mod 2~nn) ;
to
then
it is clear that
Then
(a/mn)
1, (4) reduces
to
is
(1).
224
so
that
Since
and
(6) become8
We shall
now
where a and n
are
show that
any odd
and
which
evidently yields (6).
integers. Indeed
225
Substituting
But, by
where
a little
a
from
so
in
is the
manipulation
now m
=
we
get
Legendre-Jacobi symbol.
to
now
the
1
sum
is odd and (a, mn)
that
Let aa’
(7)
familiar formula for Gaussian sums,
3. We consider
where
(3)
(mod mn) ;
then
=
1.
Thus
(9) reduces
after
226
It follows that
by (10). This evidently completes
the
proof of (5).
REFERENCES
[1]
S.
[2]
S.
CHOWLA, Some formalae of the Gauss sum type (II), Tohokn Mathematical
Journal, vol. 32 (1929-30), pp. 352-353.
RAMANUJAN, Some definite integrals connected with Gauss’s sums, Messenger of
Mathematics, vol. 44 (1915). pp. 75-85.
Manoscritto
pervenuto
in redazione il 8-6-68.