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

Pour σ dans la bandre critique 1/2<σ<1, on note E σ (T) le terme d’erreur de la formule asymptotique de 1 T |ζ(σ+it)| 2 dt (pour T grand). C’est un analogue du terme d’erreur classique E(T) (=E 1/2 (T)). L’étude de E(T) a une longue histoire, mais celle de E σ (T) est assez récente. En particulier, lorsque 3/4<σ<1, on connaît peu d’informations sur E σ (T). Pour en gagner, nous étudions la moyenne 1 T E σ (u)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 σ (T) be the error term in the mean square formula of the Riemann zeta-function in the critical strip 1/2<σ<1. It is an analogue of the classical error term E(T). The research of E(T) has a long history but the investigation of E σ (T) is quite new. In particular there is only a few information known about E σ (T) for 3/4<σ<1. As an exploration, we study its mean value 1 T E σ (u)du. In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of T.

DOI : 10.5802/jtnb.553
Lau, Yuk-Kam 1

1 Department of Mathematics The University of Hong Kong Pokfulam Road, Hong Kong
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Lau, Yuk-Kam. 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. http://www.numdam.org/articles/10.5802/jtnb.553/

[1] 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 | Zbl

[2] 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 | Zbl

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

[4] D.R. Heath-Brown, K. Tsang, Sign Changes of E(T), Δ(x) and P(x). J. Number Theory 49 (1994), 73–83. | MR | Zbl

[5] A. Ivić, The Riemann Zeta-Function. Wiley, New York, 1985. | MR | Zbl

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

[7] 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 | Zbl

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

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

[10] I. Kiuchi, On an exponential sum involving the arithmetical function σ a (n). Math. J. Okayama Univ. 29 (1987), 193–205. | MR | Zbl

[11] I. Kiuchi, K. Matsumoto, The resemblance of the behaviour of the remainder terms E σ (t), Δ 1-2σ (x) and R(σ+it). 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 | Zbl

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

[13] Y.-K. Lau, On the limiting distribution of a generalized divisor problem for the case -1/2<a<0. Acta Arith. 98 (2001), 229–236. | EuDML | MR | Zbl

[14] Y.-K. Lau, On the error term of the mean square formula for the Riemann zeta-function in the critical strip 3/4<σ<1. Acta Arith. 102 (2002), 157–165. | EuDML | MR | Zbl

[15] Y.-K. Lau, K.-M. Tsang, Ω ± -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 | MR | Zbl

[16] 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 | Zbl

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

[18] 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 | Zbl

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

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

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

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

[23] K.-M. Tsang, Higher power moments of Δ(x), E(t) and P(x). Proc. London Math. Soc. (3) 65 (1992), 65–84. | MR | Zbl

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