Mean values of the Riemann zeta-function ~(s) on the "critical" line o = Res = 1/2 ... more general zeta-functions, but to avoid lengthy technicalities we will treat ...
MEAN VALUES OF CERTAIN ZETA-FUNCTIONS
ON THE CRITICAL LINE
A. Ivic and A. Perelli
i.
UDC 519.2+51!.33
Introduction
Mean values of the Riemann zeta-function ~(s) on the "critical" line o = Res = 1/2 are extensively treated in the literature (see [7] and Titchmarsh [13]). Similarly one can investigate the mean values of the other zeta-functions besides ~(s) on the appropriate "critical" lines. In this paper we shall also consider the Dirichlet L-functions L(s, X), the Dedekind zeta-function ~K(S) (see Landau [9]) of an algebraic number field K and ?(s), the zeta-function associated ~ith a cusp form of weight k for the full modular group (see Apostol [i]). It will transpire from the sequel that our methods may be used to deal with some more general zeta-functions, but to avoid lengthy technicalities we will treat only the case of ~(s), L(s, X), ~K(S), and ~(s). The critical line corresponding to ~(s), L(s, X) and ~K(S) is o = I/2, and the critical line for ,~(s) is o = k/2. One is commonly interested in mean values of i~(l'2+it)I< IL(l'2-mX}l:< I'~ ,,~.: for i$-: an integer. .~(l'2~it)] . . . x .and (--=,, It seemed to us that the case when ~ > 0 is "small" is neglected, and that the aforementioned mean values should be all equal to I. More precisely, one should have ,,
T
1
f i~(1/2+it) Zdt=[
lira y
(1.i)
0
or equivalently T
t .(1 2+iOiXdt
T+o(T] (T~-cc')
(1.2)
0
i f )~ = ;~(T) -~ 0 s u f f i c i e n t l y quickly as and 9(k/2+it). We s h a l l i n v e s t i g a t e the In what follows T will be assumed to be positive function such that ~(T) ~ ~ as constants.
T -~ ~o, a n d t h e same f o r L ( 1 / 2 r a n g e o f X f o r w h i c h ( 1 . 2 ) and sufficiently large, ~(T) will T -+ ~ , c l , c 2 , . . , will denote
+ it, • ~K(1./2 + i t ) its analogues hold. denote an arbitrary positive, absolute
Our main result is the following THEORE~I 1.
F o r 0~~T+o(T) (T-+ oo)
(2.1)
o
holds for any k>~0. In Sec. 3 we shall show that the analogous upper bound estimate holds for 2)~ l an i n t e g e r a n d some a b s o l u t e constants 0 < c z < c 2. Jutila refined t h e a r g u m e n t s o f H e a t h - B r o w n [ 6 ] , who p r o v e d ( 3 . 1 ) w i t h c l , c : p o s s i b l y dep e n d i n g on m~ S u p p o s e O ~ X ~ @ ( T ) l o g l o g T ) -~/~ , a n d t a k e m = [ ( l o g l o g T ) l / 2 ] . By H ~ t d e r ' s inequality for integrals
and the upper bound
7
in (3.1) we have
T
m).12 0
(3.2)
o
since
0 ~X]2m ~ (1/2 + o (1)) + (T) -~/~ (log log T) -~, hence,
as T + ~,
(log T) xl~== 1 + o (1), of z/= = 1 + o (1). From ( 2 . 1 )
and (3.2)
we o b t a i n
(1.3)
for
0~~ 2xlx~ 0
Therefore (1.3) cannot hold for X>C(IoglogT) -1/2, C=3c3, where c 3 is as above. sertion follows also from (1.7) for ~(I/2 + it). 4.
The same as-
The Upper Bound Estimate for L-Functions
To derive (1.4) it suffices to prove the appropriate analogue of the upper bound in (3.1), and then to proceed similarly as in the case of ~(s). Our upper bound result on L-functions, which seems to be of indpendent interest, is contained in THEOREM 2. Let X be a primitive character (mod q) and q ~ T z-s for any given ~ > 0. Then uniformly in q, m ~ l an integer, we have T
f IL(l/2+it, Z) i~/m~r(logT) lira'.
(4.1)
0
In view of (3.1) we may suppose q > i. The result generalizes the upper bound in (3.1), and its proof is based on Jutila's uniform version [8] of the method of Heath-Brown [6] for the estimation of fractional power moments of ~(s). In the following lemmas, which generalize the corresponding lemmas of [6] (e.g. our Lemma 2' is the analogue of Lemma 2 of [6]), we make the transition from ~(s) to L(s, X) with q ~ T l-E, and we take into account that our final result (4.1) must be also uniform in m. All the salient points of the proof will be given, while the details identical with the corresponding proof of [6, 8] are either sketched or left out completely. We begin by supposing that k~0,
~k (,) = ~
and write, for o = R e s > i,
dk (~). % L k (*, Z) = ~] dk (., • . - ' .
n=l
n=l
It is well known (see Chap. 14 of [7] or Heath-Brown tion of n such that for any prime p and e = i, 2,...
[6]) that dk(~) is a multiplicative
r(k+~) Similarly,
since •
is completely multiplicative,
L~(s' ~ = H
(I-%(p)p-0-'=H p
p
(Z p
P I)
with
S
[a(n, Z)!=iz(n){1-
dlfm(nz).
. ,dl/m(nm)
} ]~ 0 will be specified a little later.
If (4.20) holds,
(4.21) then
Jz (l/2) ~ (qT) ("-xti~/" ,,,2 (~-~/2)ta Jx (I/2) T 2 (~/z-o)t,, + e- r*/~o,,= =exp{
0 m~ogr l~
} J~(1/2)+exp(-Ti/lOm)"
That is, for some absolute c > 0, we have
Jx (1/2) 2, we use (5.3) to obtain, for 0~XC(+(T) IogT) -~ and T ~ ~, T
T
0
0
Combining this with the corresponding of Theorem i. 6.
lower bound we obtain (1.6).
This completes the proof
Mean Values in the Critical Strip
One may also consider the problem of mean values of l~( 0, disproving (6.3). Finally we note that for 0 < o < I/2 fixed the asymptotic formula (6.3) holds for 0~ 0 it fails to hold. This follows from the functional equation
(s) =% (s) ~ (1 -s), Z (s) = (2r~/t)o+"-~lz e' (,+,~1,) (1 + O ('~ t i-a)) and t h e p r e v i o u s c a s e .
Similar analysis
may be made f o r L ( s , X), XK(S) and ~(s).
LITERATURE CITED i.
2.
T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, GTM n~ Berlin - H e i d e l b e r g - N e w York, Springer-Verlag (1976). K. Chandrasekharan and R. Narasimhan, "The approximate functional equation for a class of zeta-functions," Math. Ann., 152, 30-64 (1963). 359
3. 4. 5. 6. 7. 8~ 9. I0. ii. 12. 13. 14.
H. Davenport~ Multiplicative Number theory, GMT n~ B e r l i n - H e i d e l b e r g - N e w York, Springer-Verlag (1980). A. Good, "Beitrgge zur Theorie der Dirichletreihen, die Spitzenformen zugeordnet sind," J. Number Theory, 13, 18-65 (1981). A. Good, "The square mean of Dirichlet series associated with cusp forms," Mathematika, 29, 278-295 (1983). D. R. Heath-Brown "Fractional moments of the Riemann zeta-function," J. London Math. Soc., 24 (2), 65-78 (1981). A.-Ivid, The Riemann Zeta-Function, New York, John Wiley and Sons (1985). Mo Jutila, "On the value distribution of the zeta-function on the critical line," Bull. London Math. Soc., 15, 513-518 (1983). E. Landau, Einf~hrung in die Elementare und Analytische Theorie der Algebraischen Zahlen und Ideale, New York, Chelsea (1949)o A. Laurincikas, "On moments of the Riemann zeta-function on the critical line," Mat~ Zametki (Russian), 39, 483-493 (1986). A. Laurincikas, "A limit theorem for the Riemann zeta-function on the critical line," Lithuanian Mathematical Journal (Russian), I) 27, 113-132; II) 27, 489-500. A. Laurincikas, "A limit theorem for Dirichlet L-functions on the critical line," Lithuanian Mathematical Journal (Russian), 27, 699-710 (1987). E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, Clarendon Press (1951). R. T. Turganaliev, "The asymptotic formula for fractional mean value moments of the zetafunction of Riemann," Trudy Mat. Inst. Steklova Akad. Nauk SSSR (Russian), 158, 203-226 (1981).
APPROXIMATION OF DISTRIBUTION DENSITIES OF SUMS OF RANDOM VARIABLES UDC 519.21
A. Karoblis and R. Sliesoraitiene
Let $i, $2,.~ and ~i, D2,...,Dn be two sequences of independent random variables. Let us assume that the random variables in each sequence are identically distributed with distribution functions F(x) and G(x), densities p(x) and q(x), and characteristic functions f(t) and g(t), respectively, +~
~
=
f x~d(F-OD(x)
is the pseudomoment of ~-th order,
--co +ao
%~= f x~dG(x) is
the v-th moment of the distribution G(x).
By Pn(X) and qn(x) we denote the densities of the distribution of the sums
and Z . =
S,= X ~J
~j. j=l
In o r d e r t o r a i s e t h e p r e c i s i o n of a p p r o x i m a t i o n of t h e unknown d e n s i t y Pn(X) by t h e d e n s i t y qn(X) t h e r e a r e c o n s t r u c t e d in [1] s p e c i a l f u n c t i o n s ( a s y m p t o t i c e x p a n s i o n s )
/=1 where the summation ~ * t=!
(l!)',(2!)k,...(aDk'kt!k,!. 9 .k.!
.---~- g,(x)
p=l
s'(m, p)n p,
is over all nonnegative integral solutions of the equations
k~+ 2k2 + . . . +sk,=l, kl+k~+..
9+k,=m
