 A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma <1$
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 445-470.

Pour $\sigma$ dans la bandre critique $1/2<\sigma <1$, on note ${E}_{\sigma }\left(T\right)$ le terme d’erreur de la formule asymptotique de ${\int }_{1}^{T}{|\zeta \left(\sigma +it\right)|}^{2}\phantom{\rule{0.166667em}{0ex}}dt$ (pour $T$ grand). C’est un analogue du terme d’erreur classique $E\left(T\right)$ ($={E}_{1/2}\left(T\right)$). L’étude de $E\left(T\right)$ a une longue histoire, mais celle de ${E}_{\sigma }\left(T\right)$ est assez récente. En particulier, lorsque $3/4<\sigma <1$, on connaît peu d’informations sur ${E}_{\sigma }\left(T\right)$. Pour en gagner, nous étudions la moyenne ${\int }_{1}^{T}{E}_{\sigma }\left(u\right)\phantom{\rule{0.166667em}{0ex}}du$. Dans cet article, nous donnons une expression en série de type Atkinson et explorons quelques une des propriétés de la moyenne comme fonction en $T$.

Let ${E}_{\sigma }\left(T\right)$ be the error term in the mean square formula of the Riemann zeta-function in the critical strip $1/2<\sigma <1$. It is an analogue of the classical error term $E\left(T\right)$. The research of $E\left(T\right)$ has a long history but the investigation of ${E}_{\sigma }\left(T\right)$ is quite new. In particular there is only a few information known about ${E}_{\sigma }\left(T\right)$ for $3/4<\sigma <1$. As an exploration, we study its mean value ${\int }_{1}^{T}{E}_{\sigma }\left(u\right)\phantom{\rule{0.166667em}{0ex}}du$. In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of $T$.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.553
@article{JTNB_2006__18_2_445_0,
author = {Yuk-Kam Lau},
title = {A study of the mean value of the error term in the mean square formula of the {Riemann} zeta-function in the critical strip $3/4\le \sigma < 1$},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {445--470},
publisher = {Universit\'e Bordeaux 1},
volume = {18},
number = {2},
year = {2006},
doi = {10.5802/jtnb.553},
mrnumber = {2289433},
zbl = {05135398},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.553/}
}
Yuk-Kam Lau. A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma < 1$. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 445-470. doi : 10.5802/jtnb.553. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.553/

 P.M. Bleher, Z. Cheng, F.J. Dyson, J.L. Lebowitz, Distribution of the Error Term for the Number of Lattice Points Inside a Shifted Circle. Comm. Math. Phys. 154 (1993), 433–469. | MR 1224087 | Zbl 0781.11038

 J.L. Hafner, A. Ivić, On the Mean-Square of the Riemann Zeta-Function on the Critical Line. J. Number Theory 32 (1989), 151–191. | MR 1002469 | Zbl 0668.10045

 D.R. Heath-Brown, The Distribution and Moments of the Error Term in the Dirichlet Divisor Problem. Acta Arith. 60 (1992), 389–415. | MR 1159354 | Zbl 0725.11045

 D.R. Heath-Brown, K. Tsang, Sign Changes of $E\left(T\right)$, $\Delta \left(x\right)$ and $P\left(x\right)$. J. Number Theory 49 (1994), 73–83. | MR 1295953 | Zbl 0810.11046

 A. Ivić, The Riemann Zeta-Function. Wiley, New York, 1985. | MR 792089 | Zbl 0556.10026

 A. Ivić, Mean values of the Riemann zeta function. Lectures on Math. 82, Tata Instit. Fund. Res., Springer, 1991. | MR 1230387 | Zbl 0758.11036

 A. Ivić, K. Matsumoto, On the error term in the mean square formula for the Riemann zeta-function in the critical strip. Monatsh. Math. 121 (1996), 213–229. | MR 1383532 | Zbl 0843.11039

 M. Jutila, On the divisor problem for short intervals. Ann. Univ. Turkuensis Ser. A I 186 (1984), 23–30. | MR 748516 | Zbl 0536.10032

 M. Jutila, Transformation Formulae for Dirichlet Polynomials. J. Number Theory 18 (1984), 135–156. | MR 741946 | Zbl 0533.10034

 I. Kiuchi, On an exponential sum involving the arithmetical function ${\sigma }_{a}\left(n\right)$. Math. J. Okayama Univ. 29 (1987), 193–205. | MR 936745 | Zbl 0643.10032

 I. Kiuchi, K. Matsumoto, The resemblance of the behaviour of the remainder terms ${E}_{\sigma }\left(t\right)$, ${\Delta }_{1-2\sigma }\left(x\right)$ and $R\left(\sigma +it\right)$. Un Sieve Methods, Exponential Sums, and their Applications in Number Theory, G.R.H. Greaves et al.(eds.), London Math. Soc. LN 237, Cambridge Univ. Press (1997), 255–273. | MR 1635770 | Zbl 0933.11043

 Y.-K. Lau, On the mean square formula for the Riemann zeta-function on the critical line. Monatsh. Math. 117 (1994), 103–106. | EuDML 178639 | MR 1266776 | Zbl 0794.11036

 Y.-K. Lau, On the limiting distribution of a generalized divisor problem for the case $-1/2. Acta Arith. 98 (2001), 229–236. | EuDML 278921 | MR 1829624 | Zbl 1059.11056

 Y.-K. Lau, On the error term of the mean square formula for the Riemann zeta-function in the critical strip $3/4<\sigma <1$. Acta Arith. 102 (2002), 157–165. | EuDML 278362 | MR 1889626 | Zbl 0987.11053

 Y.-K. Lau, K.-M. Tsang, ${\Omega }_{±}$-results of the Error Term in the Mean Square Formula of the Riemann Zeta-function in the Critical Strip. Acta Arith. 98 (2001), 53–69. | EuDML 279369 | MR 1831456 | Zbl 0972.11078

 Y.-K. Lau, K.-M. Tsang, Moments of the probability density functions of error terms in divisor problems. Proc. Amer. Math. Soc. 133 (2005), 1283–1290. | MR 2111933 | Zbl 02140111

 K. Matsumoto, The mean square of the Riemann zeta-function in the critical strip. Japan J. Math. 15 (1989), 1–13. | MR 1053629 | Zbl 0684.10035

 K. Matsumoto, Recent Developments in the Mean Square Theory of the Riemann Zeta and Other Zeta-Functions. In Number Theory, Trends Math., Birhkäuser, Basel, (2000), 241–286. | MR 1764806 | Zbl 0959.11036

 K. Matsumoto, T. Meurman, The mean square of the Riemann zeta-function in the critical strip III. Acta Arith. 64 (1993), 357–382. | EuDML 206557 | MR 1234967 | Zbl 0788.11035

 K. Matsumoto, T. Meurman, The mean square of the Riemann zeta-function in the critical strip II. Acta Arith. 68 (1994), 369–382. | EuDML 206666 | MR 1307453 | Zbl 0812.11049

 T. Meurman, On the mean square of the Riemann zeta-function. Quart. J. Math. Oxford (2) 38 (1987), 337–343. | MR 907241 | Zbl 0624.10032

 T. Meurman, The mean square of the error term in a generalization of Dirichlet’s divisor problem. Acta Arith. 74 (1996), 351–364. | EuDML 206858 | MR 1378229 | Zbl 0848.11043

 K.-M. Tsang, Higher power moments of $\Delta \left(x\right)$, $E\left(t\right)$ and $P\left(x\right)$. Proc. London Math. Soc. (3) 65 (1992), 65–84. | MR 1162488 | Zbl 0757.11030