Download P. 538,-1. 11. For -£

562
CORRIGENDA.
ON HECKE'S MODULAR FUNCTIONS, ZETA FUNCTIONS, AND SOME
OTHER
ANALYTIC FUNCTIONS IN THE THEORY OF NUMBERS
[Proc. London Math. Soc. (2), 32 (1931), 501-556.]
L . J. MORDELL.
Prof. Siegel has been kind enough to point out to me the reason for the difference between
my formula (6.23), p. 544, and the corresponding formula on p. 166 of Hecke's paper*, my
formula containing a constant Cn which is absent from*Hecke's formula. Hecka, in deducing
his formula on p. 163, line 6 up, from the first formula on the same page (when k = 1) by
deforming the path of integration, has overlooked the residue arising from the pole s = 0.
The investigation of the simplest form and properties of Cn would be interesting.
I am also indebted to Prof. Siegel for the majority of the following corrections:
P.
P.
P.
P.
P.
505, formula 2 . 3 . For n c (mod a) read /x~c (mod a).
534, 1. 7 up. For 1, f, e8, *», ... read ± 1 , ±e, ±e 2 , ±*3, ....
536,1. 11. In the exponent, for h'e-'c read h'e-'c'.
537, 1. 5 For a read a.
537, 1. 10. For a = B = l read a = 0 = 0.
P. 538,-1.
11. For -£read - 2TTA
£-.
'
2ir
P. 538, 1. 2 up. For 77 +- T sgn t) read V + T sgn A.
P. 540, formula (6 .10). For -2nir logf read - 2 I T log f.
P. 541, foot.
For | A | read - L ^ .
P.
P.
P.
P.
P.
P.
P.
For S(Ap)~S(<r'P) (modaQD) rend S{kp)-S(o'p) (mod^QD).
For c.r read IT.
For D(iv; p , a') read D(iv', <r', a').
In the exponent, for —r(ft-t- + //2/£2) read —f2r \ ^ |.
For P read v
For £, v read {, P.
For > read ^ .
542, 1. 8.
544, 1. 4.
544, 1. 5.
548, 1. 4.
551, 1. 1.
551, 1. 2.
553, 1. 8.
ON SOME CLASSES OF FOURIEE SERIES
[Proc. London Math. Soc. (2), 33 (1932), 287-327.]
S. VEBBLUNSKY.
Equation (36) on p. 320 of this paper is false. A consequence of this is that the conditions
obtained in §§11, 12, and which are stated to be necessary and sufficient, have only been
shown to be sufficient. To obtain conditions which are both necessary and sufficient, we
observe that, if {<7,»(£)} denote a sequence of integrable functions, then, in order that fov
every bounded f(t) we have
lim \i'f(t)gn(t)dt = O,
(1)
n—>» Jo
it is necessary and sufficient that (i) there is a constant K such that
t\gn(t)\dt<K,
Jo
• Journal fUr Math., 157 (1927), 159-170.
(2)
CORRIGENDA.
563
and (ii), given a measurable set e in (0, 2ir),
lim ( gn(t)dt = O.
(3)
This is an easy congruence of Lemma 2, p . 293.
To obtain condition (2, 6), we must characterize the functions g(t) for which equation (33),
p. 320, is true. We amend the definition of class (b) as follows.
Amended definition of class (b). Let g(t) be an integrable periodic function.
let there correspond a number vn of non-overlapping intervals in (0, 2ir),
{(<,<.''>, |Bj«))}
such that
Let
(i = 1,2, . . . , , „ ) ,
To each n,
(4)
lim 2* (flC'O-<,(»)) = 0.
(/„(/)= s
{g(t-pW)-g(t-a\">)}.
If there is a constant K such that (2) holds, and if, given a measurable set e in (0, 2ir), (3) holds,
then g(t) is absolutely continuous in mean. The sequence {<7(lJ(0} ot functions is uniformly
absolutely continuous in mean, if, with the notation
I"" | g[r> (t) | dt < K ;
lim [ g[r>(t)dt-O,
(5)
Jo
n—>» J« '
where K is an absolute constant, and, for a given e, the limit is uniform in r.
The propositions enunciated in §11 become true with this amended definition. For §12,
we must introduce an amended definition of i?-integrability in mean.
Amended definition of B-integrability in mean.
Let g(t) be an integrable periodic
function. To each n let there correspond a number v,, of non-overlapping intervals (4) in
(0, 2ir) such that
lim Max (/B(">-al">) = 0 .
we have
Let y\"\ 5(Hl be arbitrarily chosen to satisfy o(") ^ yW < 8|") ^ &W. Let
gH(t) = ^ {g(t-$(»))-g(t-y^)}
(/S(»)-«(-0).
If there is a constant K such that (2) holds, and if, given a measurable set e in (0, 2ir), (3) holds,
then g(t) is .R-integrable in mean. The sequence i<7-v)(0} ^s uniformly .R-integrable in mean,
if, with the notation
#>(«) = 2 {9{r)(t-hW)-g('>(t-y['»)}
(J8j»»-o}»)),
the relations (5) hold, where K is an absolute constant, and, for a given e, the limit is uniform
in r.
With this amended definition Lemma 8 is true. This lemma gives us a form of the condition (2, 3). I am, however, unable to give an amended definition of a function of 22-integrable
variation in mean, and thus to give the condition (2, 3) by characterizing the function of which
2 \,, sin nxjn is the Fourier series.