On Some Mean Value Results for the Zeta-Function and a Divisor Problem

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1 Filomat 3:8 (26), DOI.2298/FIL6835I Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: On Some Mean Value Results for the Zeta-Function and a Divisor Problem Aleksandar Ivić a a Serbian Academy of Sciences and Arts, SANU Knez Mihailiva 35, Beograd, Serbia Abstract. Let (x) denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be (x) (x) + 2 (2x) (4x). We show that 2 +H ( t ) ζ( 2 + it) 2 dt H /6 log 7/2 ( 2/3+ε H H() ), and obtain asymptotic formulae for (t) ζ( 2 + it) 2 dt 9/8 (log ) 5/2, ( ( t ))2 ζ( 2 + it) 2 dt, ( ( t ))3 ζ( 2 + it) 2 dt. he importance of the -function comes from the fact that it is the analogue of E(), the error term in the mean square formula for ζ( 2 + it) 2. We also show, if E () E() (/()), E (t)e j (t) ζ( 2 + it) 2 dt j,ε 7/6+j/4+ε (j, 2, 3).. Introduction As usual, let (x) : d(n) x(log x + 2γ ) (x 2) (.) denote the error term in the classical Dirichlet divisor problem. Also let n x E() : ζ( 2 + it) 2 dt ( log ( ) ) + 2γ ( 2) (.2) 2 Mathematics Subject Classification. Primary M6, N37 Keywords. Dirichlet divisor problem, Riemann zeta-function, integral of the error term, mean value estimates Received: 2 June 24; Accepted: 2 August 24 Communicated by Hari M. Srivastava address: aivic_2@yahoo.com (Aleksandar Ivić)

2 A. Ivić / Filomat 3:8 (26), denote the error term in the mean square formula for ζ( 2 + it). Here d(n) is the number of all positive divisors of n, ζ(s) is the Riemann zeta-function, and γ Γ () is Euler s constant. Long ago F.V. Atkinson [] established a fundamental explicit formula for E() (see also [5, Chapter 5] and [7, Chapter 2]), which indicated a certain analogy between (x) and E(). However, in this context it seems that instead of the error-term function (x) it is more exact to work with the modified function (x) (see M. Jutila [2], [3] and. Meurman [4]), where (x) : (x) + 2 (2x) 2 (4x) 2 ( ) n d(n) x(log x + 2γ ), (.3) since it turns out that (x) is a better analogue of E() than (x). Namely, M. Jutila (op. cit.) investigated both the local and global behaviour of the difference n 4x and in particular in [3] he proved that E (t) : E(t) ( t ), (.4) +H (E (t)) 2 dt ε H /3 log 3 + +ε ( H ). (.5) Here and later ε denotes positive constants which are arbitrarily small, but are not necessarily the same ones at each occurrence, while a(x) ε b(x) (same as a(x) O ε (b(x))) means that the a(x) Cb(x) for some C C(ε) >, x x. he significance of (.5) is that, in view of (see e.g., [5, Chapter 5]) ( (t)) 2 dt A 3/2, E 2 (t) dt B 3/2 (A, B >, ), (.6) it transpires that E (t) is in the mean square sense of a lower order of magnitude than either (t) or E(t). We also refer the reader to the review paper [8] of K.-M. sang on this subject. 2. Statement of Results Mean values (or moments) of ζ( 2 + it) represent one of the central themes in the theory of ζ(s), and they have been studied extensively. here are two monographs dedicated solely to them: the author s [7], and that of K. Ramachandra [7]. We are interested in obtaining mean value results for (t) and ζ( 2 + it) 2, namely how the quantities in question relate to one another. Our results are as follows. heorem. For 2/3+ε H H() we have +H ( t ) ζ( 2 + it) 2 dt H /6 log 7/2. (2.) Remark. If one uses the first formula in (.6), the classical bound (see e.g., [5, Chapter 4]) and the Cauchy-Schwarz inequality for integrals, one obtains ζ( 2 + it) 4 dt log 4 (2.2) ( t ) ζ( 2 + it) 2 dt 5/4 log 2, which is considerably poorer than (2.) for H, thus showing that this bound of heorem is non-trivial.

3 heorem 2. If γ is Euler s constant and A. Ivić / Filomat 3:8 (26), C : 2ζ4 (3/2) 3 ζ(3) 2 3 d 2 (n)n 3/ , (2.3) n then ( ( t ))2 ζ( 2 + it) 2 dt C 4π 3/2( log 2 + 2γ 2 ) + Oε ( 7/2+ε ). (2.4) 3 Remark 2. Note that (2.4) is a true asymptotic formula (7/2 3/2 - /2). It would be interesting to analyze the error term in (2.4) and see how small it can be, i.e., to obtain an omega-result (recall that f (x) Ω( (x)) means that f (x) o( (x)) does not hold as x ). heorem 3. For some explicit constant D > we have ( ( t ))3 ζ( 2 + it) 2 dt D 7/4( log + 2γ 4 ) + Oε ( 27/6+ε ). (2.5) 7 Remark 3. Like (2.4), the formula in (2.5) is also a true asymptotic formula (27/6 7/4 - /6). Moreover, the main term is positive, which shows that, in the mean, ( t ) is more biased towards positive values. In the most interesting case when H, heorem can be improved. Indeed, we have heorem 4. We have and (2.6) remains true if (t) is replaced by (t), (t/()) or (t/()). (t) ζ( 2 + it) 2 dt 9/8 (log ) 5/2, (2.6) Remark 3. he presence of ( t ) instead of the more natural (t) in (2.), (2.4) and (2.5) comes from the defining relation (.4). It would be interesting to see what could be proved if in the integrals in (2.4) and (2.5) one had (t) (or (t)) instead of of ( t ). Remark 4. In the case of (2.) (when H ), heorem 4 answers this question. However, obtaining a short interval result for (t) ζ( 2 + it) 2 is not easy. he method of proof of heorem 4 cannot be easily adapted to yield the analogues of (2.4) and (2.5) for (t) in place of (t/()). here are some other integrals which may be bounded by the method used to prove previous theorems. For example, one such result is heorem 5. For j, 2, 3 we have E (t)e j (t) ζ( 2 + it) 2 dt j,ε 7/6+j/4+ε. (2.7) 3. he Necessary Lemmas In this section we shall state some lemmas needed for the proof of our theorems. he proofs of the theorems themselves will be given in Section 4. he first lemma embodies some bounds for the higher moments of E (). Lemma. We have E (t) 3 dt ε 3/2+ε, (3.)

4 and also A. Ivić / Filomat 3:8 (26), E (t) 5 dt ε 2+ε, (3.2) (E (t)) 4 dt ε 7/4+ε. (3.3) he author proved (3.) in [8, Part IV], and (3.2) in [8, Part II]. he bound (3.3) follows from (3.) and (3.2) by the Cauchy-Schwarz inequality for integrals. For the mean square of E(t) we need a more precise formula than (.6). his is Lemma 2. With C given by (2.3) we have E 2 (t) dt C 3/2 + R(), R() O( log 4 ). (3.4) he first result on R() is due to D.R. Heath-Brown [2], who obtained R() O( 5/4 log 2 ). he sharpest known result at present is R() O( log 4 ), due independently to E. Preissmann [6] and the author [7, Chapter 2]. For the mean square of E (t) we have a result which is different from (3.4). his is Lemma 3. We have (E (t)) 2 dt 4/3 P 3 (log ) + O ε ( 7/6+ε ), (3.5) where P 3 (y) is a polynomial of degree three in y with positive leading coefficient, and all its coefficients may be evaluated explicitly. his formula was proved by the author in [9]. It sharpens (.4) when H. It seems likely that the error term in (3.5) is O ε ( +ε ), but this seems difficult to prove. Lemma 4. We have ζ( 2 + it) 4 dt Q 4 (log ) + O( 2/3 log 8 ), (3.6) where Q 4 (x) is an explicit polynomial of degree four in x with leading coefficient /( 2 ). his result was proved first (with error term O( 2/3 log C )) by Y. Motohashi and the author []. he value C 8 was given by Y. Motohashi in his monograph [5]. Lemma 5. For N x we have (x) π 2 x 4 ( ) n d(n)n 3 4 cos(4π nx 4 π) + O ε(x 2 +ε N 2 ). (3.7) n N he expression for (x) (see [5, Chapter 5]) is the analogue of the classical truncated Voronoï formula for (x) (ibid. Chapter 3), which is the expression in (3.7) without ( ) n. where Lemma 6. We have E(t) ζ( 2 + it) 2 dt π ( log + 2γ ) + U(), (3.8) U() O( 3/4 log ), U() Ω ± ( 3/4 log ).

5 A. Ivić / Filomat 3:8 (26), he asymptotic formula (3.8) is due to the author [6]. Here the symbol f (x) Ω ± ( (x)) has its standard meaning, namely that both lim sup x f (x)/ (x) > and lim inf x f (x)/ (x) < holds. Lemma 7. We have E 3 (t) dt C 7/4 + O ε ( 5/3+ε ), (3.9) E 4 (t) dt C O ε ( 23/2+ε ), (3.) where C, C 2 are certain explicit, positive constants. hese asymptotic formulae are due to P. Sargos and the author []. Lemma 8. We have d 2 (n) π x 2 log3 x + O(x log 2 x). (3.) his is a well-known elementary formula; see e.g., page 4 of [5]. Lemma 9. For real k [, 9] the limits n x E k : lim k/4 E(t) k dt exist. his is a result of D.R. Heath-Brown [4]. he limits of moments without absolute values also exist when k, 3, 5, 7 or 9. Lemma. For 4 A 2 we have with some positive constant C(A). ζ( 2 + it) A dt A + 8 (A 4) log C(A) (3.2) hese are at present the strongest upper bounds for moments of ζ( 2 + it) for the range in question. hey follow by convexity from the fourth moment bound (2.2) and the twelfth moment ζ( 2 + it) 2 dt 2 log 7 of D.R. Heath-Brown [3] (see e.g., [5, Chapter 8] for more details). 4. Proofs of the heorems We begin with the proof of (2.). We start from +H E (t) ζ( 2 + it) 2 dt { +H +H } /2 (E (t)) 2 dt ζ( 2 + it) 4 dt ( H /3 log 3 H log 4 ) /2 H /6 log 7/2. (4.)

6 A. Ivić / Filomat 3:8 (26), Here we assumed that 2/3+ε H H() and used (3.6) of Lemma 4, (.5) and the Cauchy-Schwarz inequality for integrals. On the other hand, by the defining relation (.4) we have +H E (t) ζ( 2 + it) 2 dt +H +H E(t) ζ( 2 + it) 2 dt ( t ) ζ( 2 + it) 2 dt. Using (3.8) of Lemma 6 and (4.), we obtain then from (4.2) +H +H ( t ) ζ( 2 + it) 2 dt E (t) ζ( 2 + it) 2 dt +H E(t) ζ( 2 + it) 2 dt O(H /6 log 7/2 ) + πt ( log t + 2γ ) +H + O( 3/4 log ) O(H /6 log 7/2 ) + O(H log ) + O( 3/4 log ) H /6 log 7/2, since 2/3+ε H H(). his completes the proof of heorem. he proof of heorem 2 is somewhat more involved. It suffices to consider the integral from to 2, and then at the end of the proof to replace by 2 j and sum the resulting expressions when j, 2,.... First, by squaring (.4), we have (4.2) 2 2 (E (t)) 2 ζ( 2 + it) 2 dt 2 2 (E(t)) 2 ζ( 2 + it) 2 dt E(t) ( t ) 2 ζ( 2 + ( it) 2 dt + 4π ( 2 t )) 2 ζ( 2 + it) 2 dt. he expression in the middle of the right-hand side of (4.3) equals, on differentiating (.2), 2 2 E(t) ( t )( t log + 2γ + E (t) ) dt 2 2 E(t) ( t )( t log + 2γ) dt + J(), say, where (4.3) (4.4) 2 J() : 2 E(t) ( t ) E (t) dt. (4.5) o bound J() we use Lemma 5 with N N(), N, where N will be determined a little later. he error term in (3.7) trivially makes a contribution which is 2 ε E(t) ( ζ( 2 + it) 2 + log ) /2+ε N /2 ε 7/4+ε N /2 (4.6) on using the second formula in (.6), (2.2) and the Cauchy-Schwarz inequality for integrals. here remains the contribution of a multiple of 2 ( E 2 (t) ) t /4 ( ) n d(n)n 3/4 cos( 8πnt π/4) dt. n N

7 A. Ivić / Filomat 3:8 (26), his is integrated by parts. he integrated terms are /2 N /3 log, by using the standard estimate E() /3 (see e.g., [5, Chapter 5]) and trivial estimation. he main contribution comes from the differentiation of the sum over n. Its contribution will be, with n K meaning that K < n K 2K, 2 /4 E 2 (t) ( ) n d(n)n /4 exp(i 8πnt) dt 2 /4 E 4 (t) dt n N n N 2 ( ) n d(n)n /4 exp(i /2 2 8πnt) dt n N 2 3/4 d 2 (n)n /2 dt + log 2 max K N m n K K ε /2 3/4 (N /2 log 3 + /2+ε N) /2 5/4 N /4 log 3/2 m n for ε N N() ε. Here we used the standard first derivative test (see e.g., Lemma 2. of [6]) for exponential integrals, Lemma 7, (3.) and K m K 3/2 log K. n m n m n K n K m K,m n From (4.6) and (4.7) we see that the right choice for N should be if we have 7/4 N /2 5/4 N /4, N 2/3, and with this choice of N we obtain /2 N /3 4/36 (4/36 < 7/2), and /2 (4.7) J() ε 7/2+ε. (4.8) In view of (.4), the formula (4.3) and the bound (4.8) give 4π ( ( t ))2 ζ( 2 + it) 2 dt O ε ( 7/2+ε ) E(t) ( t )( t 2 log + 2γ) dt E 2 (t) ζ( 2 + it) 2 dt (E (t)) 2 ζ( 2 + it) 2 dt O ε ( 7/2+ε ) + 2I I 2 + I 3, say. On using (3.3) of Lemma, (2.2) and the Cauchy-Schwarz inequality we obtain (4.9) Further we have 2I I E 2 (t) 2 I 3 { 2 2 /2 (E (t)) 4 dt ζ( 2 dt} + it) 4 ε /8+ε. E(t) ( E(t) E (t) )( log t + 2γ) dt I 2 { 2 ( log t } + 2γ) ζ( 2 + it) 2 dt E(t)E (t) ( log t + 2γ) dt E 2 (t) ( log t + 2γ E (t) ) dt 2 2 E(t)E (t) ( log t + 2γ) dt.

8 A. Ivić / Filomat 3:8 (26), he last integral is, by Lemma 2, Lemma 3 and the Cauchy-Schwarz inequality for integrals, log { 2 On the other hand, 2 E 2 (t) ( log t + 2γ E (t) ) dt 2 2 E 2 (t) ( log t + 2γ) dt 3 E3 (t) E 2 (t) ( log t + 2γ) dt + O(). 2 /2 E 2 (t) dt (E (t)) dt} 2 7/2 log 5/2. 2 (4.) o evaluate the last integral in (4.) we use Lemma 6 and integration by parts. his shows that the integral in question is ( Ct 3/2 + R(t) )( log t + 2γ) 2 Ct 3/2( log t + 2γ) 2 Ct 3/2( log t + 2γ 2 3 It transpires from (4.9) and (4.) that 4π which gives at once (2.4) of heorem 2. ( Ct /2 + R(t) ) dt t +O( log5 ) Ct3/2 ) 2 + O( log5 ). ( ( t ))2 ζ( 2 + it) 2 dt Ct 3/2( ) log t + 2γ O ε( 7/2+ε ), We turn now to the proof of heorem 3. he basic idea is analogous to the one used in the proof of heorem 2, so that we shall be relatively brief. he integral in (2.5) equals /(8π 3 ) times { E 3 (t) 3E (t)e 2 (t) + 3(E (t)) 2 E(t) (E (t)) 3} ζ( 2 + it) 2 dt. (4.) he main term in (2.5) comes from E 3 (t) ζ( 2 + it) 2 dt C 7/4( log + 2γ) C 7/4( log + 2γ 4 7 E 3 (t) ( log t + 2γ E (t) ) dt C t 3/4 dt + O ε ( 5/3+ε ) ) + Oε ( 5/3+ε ), where (3.9) of Lemma 7 was used. By Hölder s inequality for integrals, (3.3) of Lemma and (3.2) of Lemma (with A 5) we obtain ( ) 3/5 ( (E (t)) 3 ζ( 2 + it) 2 dt E (t) 5 dt ζ( 2 + it) 5 dt ε 6/5+9/2+ε 33/2+ε. ) 2/5

9 A. Ivić / Filomat 3:8 (26), Similarly we obtain (E (t)) 2 E(t) ζ( 2 + it) 2 dt ( ) 3/8 ( ) /8 ( E (t) 6/3 dt E 8 (t) dt ζ( 2 + it) 4 dt ε 3 8 (2+ 9 ) ε 5 3 +ε, where we used (3.2) and the fact that E () /3, which follows from the definition of E and the classical estimates (x) x /3, E() /3. Finally, by using (3.3), Lemma 9 with k 8 and (2.2), we obtain E (t)e 2 (t) ζ( 2 + it) 2 dt ( ) /4 ( ) /4 ( E (t) 4 dt E 8 (t) dt ζ( 2 + it) 4 dt ε 7/6+3/4+/2+ε 27/6+ε. Since 27/ > 5/3 > 33/2.65, we obtain easily the assertion of heorem 3. We shall prove now (2.6) of heorem 4. We suppose t 2 and take N in (3.7) of Lemma 5. his holds both for (x) and (x), and one can see easily that the proof remains valid if we have an additional factor of /() in the argument of or (or any constant c >, for that matter). hus we start from (t) t/4 π 2 t/4 π 2 d(n)n 3/4 cos(4π nt π/4) + O ε ( ε ) n + G<n + O ε( ε ), ) /2 ) /2 (4.2) say, where ε G G() ε, and G will be determined a little later. he error term in (4.2) makes a contribution of O ε ( +ε ) to (2.6). We have 2 t /4 d(n)n 3/4 cos(4π nt π/4) ζ( 2 + it) 2 dt 2 I + I 2, t /4( log t + 2γ + E (t) ) d(n)n 3/4 cos(4π nt π/4) dt say. By the first derivative test 2 I : t /4( log t + 2γ) d(n)n 3/4 cos(4π nt π/4) dt /4 log d(n)n 3/4 /2 n /2 3/4 log, since n d(n)n α converges for α >. he integral I 2, namely I 2 : 2 t /4 E (t) d(n)n 3/4 cos(4π nt π/4) dt (4.3)

10 A. Ivić / Filomat 3:8 (26), is integrated by parts. he integrated terms are trivially O(), and there remains 2 4 t 3/4 E(t) d(n)n 3/4 cos(4π nt π/4) dt + 2 t /4 E(t) d(n)n /4 sin(4π nt π/4) dt. (4.4) Both integrals in (4.4) are estimated analogously, and clearly it is the latter which is larger. By the Cauchy- Schwarz inequality for integrals it is /4 (J J 2 ) /2, where J : J 2 : 2 2 d(n)n /4 e 4πi nt E 2 (t) dt 3/2, 2 dt on using Lemma 2 in bounding J 2. Using the first derivative test and (3.) of Lemma 8, we find that 2 J d 2 (n)n /2 d(m)d(n) + e 4πi( m n) t dt (mn) /4 m G /2 log 3 + /2 d(m)d(n) (mn) /4 m n. m When n/2 < m 2n the contribution of the last double sum is ε /2 n ε /2 n /2 n/2<m 2n,m n m n ε /2+ε G. If m n/2 then m n n /2, and when m > 2n it is m /2. hus the total contribution of the double sum above is certainly We infer that In a similar vein it is found that 2 t /4 G<n ε /2+ε G G /2 log 3 ( ε G G() ε ). /4 (J J 2 ) /2 /4 (G /2 log 3 3/2 ) /2 G /4 (log ) 3/2. d(n)n 3/4 cos(4π nt π/4) ζ( 2 + it) 2 dt 2 /4 d(n)n 3/4 e 4πi nt 2 2 dt ζ( 2 + it) 4 dt 3/4 log 2 G<n 2 n>g d 2 (n) n 3/2 + G<m n /2 d(m)d(n) m n) t) e4πi( (mn) 3/4 dt ε 3/4 log 2 { G /2 log 3 + /2+ε} /2 5/4 G /4 (log ) 7/2. /2

11 We finally infer that, for ε G G() ε, 2 A. Ivić / Filomat 3:8 (26), (t) ζ( 2 + it) 2 dt G /4 (log ) 3/2 + 5/4 G /4 (log ) 7/2 9/8 (log ) 5/2 with the choice G /2 log 4. his leads to (2.6) on replacing by 2 j and adding the resulting estimates. Corollary. We have Namely E (t) ζ( 2 + it) 2 dt 9/8 (log ) 5/2. (4.5) 2 E (t) ζ( 2 + it) 2 dt 2 { E(t) (t/()) } ζ( 2 + it) 2 dt. he integral with is 9/8 (log ) 5/2 by heorem 4. here remains 2 by (3.8) of Lemma 6. his gives E(t) ζ( 2 + it) 2 dt π ( log 2 π + 2γ ) + O( 3/4 log ) O( log ) 2 E (t) ζ( 2 + it) 2 dt 9/8 (log ) 5/2. o complete the proof of (4.5), again one replaces by 2 j and adds the resulting estimates. It remains to prove (2.7) of heorem 5 (the bound (4.5) gives a result when j ). he proof is analogous to the proofs given before, so we shall be brief. We have 2 2 E (t)e j (t) ζ( 2 + it) 2 dt E (t)e j (t) ( log t + 2γ + E (t) ) dt I + I, say. By the Cauchy-Schwarz inequality for integrals, Lemma 3 and Lemma 9 (with k 2j), it follows that I : 2 E (t)e j (t) ( log t + 2γ) dt log { 2 (E (t)) 2 dt 2 E 2j (t) dt }/2 log ( 4/3 log 3 +j/2 ) /2 7/6+j/4 log 5/2. On the other hand, by (.4) we have I : 2 2 E (t)e j (t)e (t) dt 2 E j+ (t)e (t) dt ( t ) E j (t)e (t) dt. (4.6)

12 Note that 2 A. Ivić / Filomat 3:8 (26), E j+ (t)e (t) dt j+2 Ej+2 (t) 2 O((j+2)/3) ), and (j + 2)/3 < 7/6 + j/4 for < j 6. By using (3.7) of Lemma 4 it is seen that the last integral in (4.6) is a multiple of 2 t /4 n N( ) n d(n)n 3/4 cos( 8πnt π/4)e j (t)e (t) dt + J(), (4.7) say, where ε N N() ε. Using Lemma 9 we have 2 J() ε /2+ε N /2 E j (t) E (t) dt { 2 ε /2+ε N /2 E 2j (t) dt 2 ( log 2 + ζ( 2 + it) 4) dt} /2 ε /2+ε N /2( +j/2 log 4 ) /2 3/2+j/4+ε N /2. he remaining integral in (4.7) is again integrated by parts. he major contribution will come from a multiple of 2 E j+ (t)t /4 n N( ) n d(n)n /4 sin( 8πnt π/4) dt /4{ 2 E 2j+2 (t) dt 2 ( ) n d(n)n /4 e i 8πnt 2 } /2 dt n N /4{ +(j+)/2 N /2 log 3 } /2 3/4+(j+)/4 N /4 log 3/2, where Lemma 9 was used with k 2j he choice N 2/3 gives 3/4+(j+)/4 N /4 3/2+j/4+ε N /2 7/6+j/4, as asserted by heorem 5. he bound in (2.7) is an expected one, since (in the mean square sense) E (t) is of the order t /6 log 3/2 t, E(t) is of the order t /4, and ζ( 2 + it) 2 is of logarithmic order. However, by Hölder s inequality for integrals (2.7) does not follows directly, since it would require the yet unknown estimates for higher moments of ζ( 2 + it). References [] F.V. Atkinson, he mean value of the Riemann zeta-function, Acta Math. 8(949), [2] D.R. Heath-Brown, he distribution and moments of the error term in the Dirichlet divisor problems, Acta Arith. 6(992), [3] D.R. Heath-Brown, he mean value theorem for the Riemann zeta-function, Mathematika 25(978), [4] D.R. Heath-Brown, he twelfth power moment of the Riemann zeta-function, Quart. J. Math. (Oxford) 29(978), [5] A. Ivić, Mean values of the Riemann zeta-function, LN s 82, ata Inst. of Fundamental Research, Bombay, 99 (distr. by Springer Verlag, Berlin etc.). [6] A. Ivić, On some integrals involving the mean square formula for the Riemann zeta-function, Publications Inst. Math. (Belgrade) 46(6) (989), [7] A. Ivić, On the mean square of the zeta-function and the divisor problem, Annales Acad. Scien. Fennicae Mathematica 23(27), -9. [8] A. Ivić, On the Riemann zeta-function and the divisor problem, Central European J. Math. (2)(4) (24), -5, II, ibid. (3)(2)(25), 23-24, III, Annales Univ. Sci. Budapest, Sect. Comp. 29(28), 3-23, and IV, Uniform Distribution heory (26), [9] A. Ivić, he Riemann zeta-function, John Wiley & Sons, New York, 985 (2nd ed. Dover, Mineola, New York, 23). [] A. Ivić and Y. Motohashi, On the fourth power moment of the Riemann zeta-function, J. Number heory 5(995), [] A. Ivić and P. Sargos, On the higher power moments of the error term in the divisor problem, Illinois J. Math. 8(27),

13 A. Ivić / Filomat 3:8 (26), [2] M. Jutila, On a formula of Atkinson, opics in classical number theory, Colloq. Budapest 98, Vol. I, Colloq. Math. Soc. János Bolyai 34(984), [3] M. Jutila, Riemann s zeta-function and the divisor problem, Arkiv Mat. 2(983), and II, ibid. 3(993), 6-7. [4]. Meurman, A generalization of Atkinson s formula to L-functions, Acta Arith. 47(986), [5] Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge University Press, Cambridge, 997. [6] E. Preissmann, Sur la moyenne quadratique du terme de reste du probléme du cercle, C. R. Acad. Sci. Paris Sér. I Math. 36(988), no. 4, [7] K. Ramachandra, On the mean-value and omega-theorems for the Riemann zeta-function, LN s 85, ata Inst. of Fundamental Research (distr. by Springer Verlag, Berlin etc.), Bombay, 995. [8] K.-M. sang, Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function, Sci. China Math. 53(2), no. 9,