ON THE FUNCTION $E_{\sigma}(T)$

Kohji MATSUMOTO ( $\Psirightarrow\wedge$ k $\not\equiv$fl $\sim$-)

Deparmient of Mathematics, Faculty of Educauon,Iwate University, Ueda, Morioka 020, Japm

The error term function for the mean square of the Riemannzeta-function $\zeta(s)$ in the strip $-

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First we list up the results proved by the author[28].

(A) The explicit formula of Atkinson-type for $E_{\sigma}(T)$ $(- < \sigma

(1.1) $E_{\sigma}(T)

$f(x_{n})>cg(x_{n})$ $($ resp. $f(x_{n})

had been obtained. The author gave a talk on this result at NihonUniversity, in front of Ivi\v{c}. This is the result mentioned in theNotes of Chapter 2 of Ivi\v{c}[8].

The improved form (1.7) is proved in Matsumoto-Meurman$[31,\mathbb{I}]$ . In the same paper, the conjecture(1.8) $F_{\sigma}(T)\sim B(\sigma)^{2}T$ $(- < \sigma

The case $\sigma=\frac{1}{2}$ is classical, md has been smdied extensively from$1920s$. It is easy to show that

$\lim_{\sigmaarrow 1/2}E_{\sigma}(T)=E(T)$

( $(2_{:}3)$ of [8]). On the other hand, a deep smdy of ffie case $\sigma=1$was first canied out by a $j$oint work of Balasubramanian, Ivi\v{c} andRamachandra$[2](1992)$ ; they proved the asymptotic formula

$\int_{1}^{T}1\zeta(1+it)1^{2}dt=\zeta(2)T-\pi\log T+R(\tau)$

with $R(T)=O((\log T)^{2/3}(\log\log T)^{1/3})$ , md also obtmed mem valueresults on $R(T)$ . The hmit of $E_{\sigma}(T)$ $(as \sigmaarrow 1)$ is connectedwith $R(T)$ by the fonnula

$\lim_{\sigmaarrow 1- 0}\{\zeta(2\sigma)T+(2\pi)^{2\sigma-1}\frac{\zeta(2-2\sigma)}{2-2\sigma}(T^{2- 2\sigma}-1)\}=\zeta(2)T-\pi\log T$ ,

as is shown in Ivi\v{c}[9]. On various related mean values on thelin$e\sigma=1$ , see Ivi\v{c}[ll], Nakaya[40] [41], md Balasubraimanian-Ivi\v{c}-Ramachandra[3].

The $re\ovalbox{\tt\small REJECT} g$ smp $\frac{3}{4}\leq\sigma

“averaging $\dagger$ ’ idea of Meunnm [33]; the same idea was alsoinvented, independently, by Motohashi $[$34,$N][36]$ . Second,Preissmann’s technique[43] of using Montgomery-Vaughm’sinequality; this idea was originaUy $in\alpha oduced$ by Preissmann[42].Ivi\v{c} also gave the same result as in [43] (independently, butinspired by [42] $)$ in his talk at Vancouver symposium in 1989, andin his lecmre note ((2.100) of [8]).

$\cdot$

The third tool is ffie meanvalue theorem of $D\ddot{m}chlet$ polynomials.

A digressive talk. Preissmam[43] was published in 1993, butthe preprint had akeady been completed around 1988. Thisdelay is because [43] was submitted to J. Number Theory, and wasleft there (on editor’s desk?) three years long. FinaUyPreissmam found mother place to publish. It is $we\mathbb{I}$-knownthat J. Number Theory causes mmy srilar $\alpha oubles$ . Forexmple, Matsumoto$- Meurmm[31,II]$ was submitted to J. NumberTheory in March 1992, but there was no correspondence from theeditors. Memmm wrote a letter of inquiry in March 1993, butno mswer again. And finaUy, as the response to the author’srecent inquiry(November 1993), they replied $\dagger$We have no recordof your paper”.

We can obseive that (2.1) and (2.2) establish clearly thesingular property of $E_{\sigma}(T)$ on the line $\sigma=\frac{3}{4}$ , which was firstpointed out by the author[28]. In fact, the coefficient

$c_{1}( \sigma)=\frac{2(2\pi)^{2\sigma-3/2}\zeta^{2}(3/2)}{5-4\sigma\zeta(3)}\zeta(\frac{5}{2}-2\sigma)\zeta(\frac{1}{2}+2\sigma)$

of the main term in (C) is divergent when $\sigmaarrow\frac{3}{4}-0$ , and on $\sigma=\frac{3}{4}$the figure of the main term is changed as, in (2.1).

We do not know how to extend the conjecmre (1.8) to theregion $\frac{3}{4}\leq\sigma

may hold for $\frac{3}{4}

to the author, Launn\v{c}ikas wrote that his error term cm be madeas $\alpha\min(\ell_{T}/2,\log T)\log T)$ , hence in case $\ell_{T}\cong const.$ , it cm $b$reduced to $O(\log T)$ . (Note that his result is proved under theadditional condition $l_{T}\leq c\log T.$ ) His main concem is ffie case$f_{T}arrow\infty-$ as $rarrow\infty$ , because the motivauon of his work lies in hisresearches on the $value- dist\dot{n}bution$ of $\zeta(s)$ .

Generalizauons of Atkmson’s method to $D\ddot{m}chlet$ $L$-funcuonswere studied by Meunnan md Motohashi in the middle of $1980s$.Meuran[32] proved the Atkmson-type formula for

$\sum_{\chi m}J_{q^{0}}^{T}|L(\frac{1}{2}+it,\chi)|^{2}dt$,

where $L(s.\chi)$ is the Dmchlet L-funcuon associated with the$D\ddot{m}chlet$ character $\chi$ , md the summation mns over $aU$characters $\chi$ of modulus $q$ . Motohashi$[34,II][34_{2}III]\alpha eated$the mem square of individual L-functions (The details are givenin [35] $)$ . Recently, Launn\v{c}ikas[27] obtained an analogue ofMeurm’s result[32] near the criucal lin$e$ .

Motohashi$[34_{2}I]$ discovered that Atkmson’s method cm bemodified so as to be useful for the smdy of the sum $\chi mdq\sum 1L(s,\chi)1^{2}$ ,

md this idea has been developed and deepened by Katsurada-Matsumoto[17][18] md Katsurada$[16_{2} I\prod[16,\Pi\prod$ (The resultsproved in $[16_{2}I\mathbb{I}]$ are amounced in Katsurada[15]; see also hissummarizing $ar\mathfrak{a}cle[16_{2}N])$ . In this case Atkinson’s method iseffective not only in the critical smp, but also on the whole plane.For instance, one of the results in [18] is the asymptotic expansionof $\sum_{\chi m\alpha 1q}|L\langle 1.\chi)|^{2}$ with respect to $q$ , which is far better than the

former results (going back to Paley md Selberg; the hitherto bestresult was due to Zhang). Katsurada-Matsumoto[19] [20][21]found that ffie same method cm be applied to the (discrete andalso conmuous) mean squares of Humitz zeta-funcuons $\zeta(s,\alpha)$with respect to the parameter $\alpha$ . For the details, seeKatsurada’s arucle in the present Proceedings.

Motohashi’s very important works (partly with Ivi\v{c}) on thefourth power mean of $\zeta(s)$ and additive divisor problem can alsobe regarded as a variant of Atkinson’s method (see his expository

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paper[39] $)$ . In $1980s$, there were two main $s$treams in memvalue theory of $\zeta(s)$ ; Atkmson’s $meffiod_{2}$ and applicauons ofKuznetsov’s $\alpha ace$ formula. Motohashi’s theory is a splendidcombinauon of these two pnnciples. RecenUy, in the ffame ofhis theory, Motohashi considered mean squares of other types of$D\ddot{m}chlet$ senes. See [38] and his arucle in the presmtProceedings.

Motohashi nouced that the criucal propeny of the lme $\sigma=\frac{3}{4}$also appears in the fomh power moment simation. See alsoIvi\v{c}[12].

Lastly in this secuon, we mention Kiuchi$\iota_{S}$ recent results, inwhich some singular simauon again appears on the line $\sigma=\frac{3}{4}$ .The mean square of the error tem in the approrimate funcuonalequauon of $\zeta^{2}(s)$ was first considered by Kiuchi-Matsumoto[25](1992), and then smdied further by Kiuchi[22], Ivi\v{c}[10] and so on.(Note that these works are based on Motohashi’s aformentionedwork on the Riemam-Siegel-type formula for $\zeta^{2}(s).)$ Recently,Kiuchi[23] discovered that ffie main teim of this mean squarechanges its figure at $\sigma=\frac{3}{4}$ , in the same mamer as in the case of$E_{\sigma}(T)$ . (Quite recently, the author reflned the result of Kiuchi[23]in case $-

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In the classical case of $\sigma=\frac{1}{2}$ , the best known $\Omega- res$ults for$E(D$ are due to Hafner- Ivi\v{c}[4], which assert

(4. 1) $E(T)=\Omega_{+}(T^{1/4}(\log T)^{1l4}(\log\log T)^{(3+\log 4)/4}\exp(-c_{3}\sqrt{\log\log\log T}))$

(4.2) $E(D=\Omega_{-}$ $(T^{1/4}\exp$( $c_{4}(\log\log T)^{1/4}$(loglog log $T)^{-3/4}$ )$)$ .

How about $E_{\sigma}(T)^{7}$ We already menuoned that in the stnp$-< \sigma

Since Ivi\v{c} knew the result (4.5) in the preprint of $[31,III]$ , heclaimed repeatedly in his letters that $m$ improvement of $\Omega_{-}-$result is surely possible too, md it should be done by Meurmm orthe author. This pressure of Ivi\v{c} was the initial driving force ofthe $\Omega_{-}$ -part of the joint research[13].

In the region $\frac{3}{4}

THEOREM B. $Ie \tau\frac{3}{4}\leq\sigma

(5.6) $E_{\sigma}(T)\ll T^{9(1-\sigma)/(23-*r)+\epsilon}$ $( \frac{3}{4}\leq\sigma

my $\epsilon>0$ , then (5.8) would follow from Theorems A md C. Butthe latter conjecmre would even lead the Iindel\"of hypothesis$\zeta(\frac{1}{2}+tt)

hence (5.10) contradicts with the Lindel\"of hypothesis, thereforewith the Riemam hypothesis. Nevertheless, it is perhaps not awise way to throw over (5.10) immediately. It may beperitted to say that the certainty of the Lindel\"of hypothesis(5.11) in case $\frac{1}{2}\leq\sigma

151-191.[5] D.R.Heath-Brown, The mean value theorem for the $Ri\ovalbox{\tt\small REJECT}$

zeta-function, Mathemauka 25 (1978)‘ 177-184.[6] M.N.Huxley, $E\psi onenual$ sums and the Rienrami zeta-function

N, Proc. London Math. Soc. (3)66 (1993) 1-40.[7] A. Ivi\v{c}, On some integrals involving the mem square formula

for the Riemann zeta-function, Publ. Inst. Math. (Belgrade) 46(60) (1989) 33-42.

[8] A. Ivi\v{c}, Mean values of the Riemam zeta funcuon, Lecmre NoteSer. 82, Tata Inst. Fund. ${\rm Res}.$ , Bombay, dismbuted by Spnnger,1991.

$\mathfrak{s}\prime s\acute{e}\dot{m}naire$ d\’e ?h\’eorie des Nombres, Paris, $1990- 1991^{\dagger\dagger}$ , S.David(ed.), Birkh\"auser, 1993, pp. 115-125.

[10] A Ivi\v{c}, Power moments of the error term in the approrimatefuncuonal equauon for $\zeta^{2}(s)$ , Acta Nith. 65. (1993) 137-145.

[11] A Ivi\v{c}, The moments of the zeta-funcuon on the line $\sigma=1$ ,Nagoya Maffi. J., to appear.

[12] A Ivi\v{c}, Some problems on mean values of the Riemann zeta-fmcuon, prepnnt.

[13] A Ivi\v{c} and K.Matsumoto, On the error term in the memsquare fomiula for the Riemam zeta-funcuon in the criticalsmp, preprint (flrst version, 1993).

[14] M.Juola, Riemam’s zeta-fmction and the divisor problm,Ark. Mat. 21 (1983) 75-96.

[15] M.Katsurada, Asymptouc expansions of the mean square of$D\ddot{m}chlet$ L-functions, Proc. Japan Acad. 68A (1992) 219-222.

[16] M.Katsurada, Asymptouc expansions of the mean values ofDmchlet L-functions II, in $l\uparrow Analyuc$ Number Theory mdRelated $Topics”$ , K.Nagasaka (ed.), World Scienoflc, 1993, pp.61-71; III, Manuscnpta Math., to appear; N, $S\overline{u}rikais$ekiKenkyiTsho K\={o}ky\={u}roku 837, Kyoto Univ. (1993) 72-83.

[17] M.Katsurada and K.Matsumoto, Asymptotic expansions of themean values of $D\ddot{m}chlet$ L-funcuons, Math. Z. 208 (1991) 23-39.

[18] M.Katsurada md K.Matsumoto, The mean values of DmchletL-funcuons at integer points and class numbers of cyclotomicfields, Nagoya Math. J., to appear.

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[19] M.Katsurada md K.Matsumoto, Discrete mean values ofHuiwitz zeta-fmcuons, Proc. Japm Acad. 69A (1993) 164-169.

[20] M.Katsurada md K.Matsumoto, $E\varphi licit$ fonnulas andasymptotic $\infty ansions$ for cemin mean square of Hurwitz zeta-fmcuons, Proc. Japm Acad. 69A (1993) 303-307.

[21] M.Katsurada and K.Matsumoto, $E\varphi licit$ fomiulas andasymptotic expansions for certain mem square of Hurwitz zeta-funcuons I, preprmt.

[22] I.Kiuchi, An improvement on the mem value formula for theapprorimate fmcuonal equauon of ffie square of the $Ri\ovalbox{\tt\small REJECT}$zeta-funcuon, J. Number Theory 45 (1993) 312-319.

[23] I.Kiuchi, The mean value fomiula for the approrimatefunctional equauon of $\zeta^{2}(s)$ in the criucal smp, Arch. Math., toappear.

[24] I.Kiuchi, An integral involving the enior ter of the meansquare for the Riemam zeta-funcuon in the critical smp, Math.J. Okayama Univ., to appear.

[25] I.Kiuchi and K.Matsumoto, Mem value results for theapprorimate $f\iota mcuonal$ equauon of ffie square of the $Ri\ovalbox{\tt\small REJECT}$zeta-function, Acta Arith. 61 (1992) 337-345.

[26] ALamn\v{c}ikas, The Atkinson fonnula near the criucal line,in “New Trends in Probab. and Statist.”, F.Schweiger and E.Mmstavi\v{c}ius (eds.), $VSP/TEV$, 1992, pp.335-354.

[27] A.Laum\v{c}ikas, The Atkmson formula for L-funcuons near thecriucal lme, Iiet. Mat. $\ovalbox{\tt\small REJECT} ys$ , to appear.

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[30] K.Matsumoto, The mean square of the Riemann zeta-fmctionin the smp $-

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[33] T.Meurman, On the mean square of the Riemam zeta-function, $Qum$. J. Math. Oxford (2)38 (1987) 337-343.

[34] Y.Motohashi, A note on the mean value of the zeta md L-functions I, Proc. Japm Acad. $61A$ (1985) 222-224; II, ibid.$61A$ (1985) 313-316; III, ibid. $62A$ (1986) 152-154; IV, ibid.$62A$ (1986), 311-313.

[35] Y.Motohashi, On the mean square of L-functions, unpublishedmnuscript, 1986 (originaUy intended to include in [36]; nowwe can $caU$ it as “a missing chapter”).

[36] Y.Motohashi, Lecmres on the Riemann-Siegel formula, UlamSeminar, Dept. of Math., Colorado Univ., Boulder, 1987.

[37] Y.Motohashi, The mean square of $\zeta(s)$ off the critical line,unpublished mmuscript, 1990.

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[39] Y.Motohashi, The $Ri\ovalbox{\tt\small REJECT}$ zeta-funcuon and the non-Euclidem Laplacian (in Japanese), S\={u}gaku 45 (1993) 221-243;English $\alpha ansl.:$ S\={u}gaku Expositions, to appear.

[40] H.Nakaya, The negative power moment of the Rientann zeta-function on the line $\sigma=1,$ preprint.

[41] H.Nakaya, Some mean squares in connection with $\zeta(1+it)$ ,Proc. Japan Acad. $69A$ (1993) 275-277.

[42] $E.Preiss’\ovalbox{\tt\small REJECT}$ , Sur la moyenne quadratique du tere de restedu probl\‘eme du cercle, C. R. Acad. Sci. Paris 306 (1988) 151-154.

[43] E.Preissntann, Sur la moyenne de la fonction z\^eta, in AnalyucNumber Theory and Related Topics $\uparrow’$ , K.Nagasaka (ed.), WorldScimofic, 1993, pp. 119-125.

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