On mean values of some zeta-functions in the critical strip

Aleksandar Ivić

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 1, page 163-178
  • ISSN: 1246-7405

Abstract

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For a fixed integer k ≥ 3 , and fixed 1 2 < σ < 1 we consider ∫ 1 T ζ ( σ + i t ) 2 k d t = ∑ n = 1 ∞ d k 2 ( n ) n - 2 σ T + R ( k , σ ; T ) , where R ( k , σ ; T ) = 0 ( T ) ( T → ∞ ) is the error term in the above asymptotic formula. Hitherto the sharpest bounds for R ( k , σ ; T ) are derived in the range min ( β k , σ k * ) < σ < 1 . We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

How to cite

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Ivić, Aleksandar. "On mean values of some zeta-functions in the critical strip." Journal de théorie des nombres de Bordeaux 15.1 (2003): 163-178. <http://eudml.org/doc/249093>.

@article{Ivić2003,
abstract = {For a fixed integer $k \ge 3$, and fixed $\frac\{1\}\{2\} &lt; \sigma &lt; 1$ we consider\begin\{equation*\} \int ^T\_1 \left| \zeta (\sigma + it)\right|^\{2k\} dt = \sum ^ \infty \_\{n=1\} d^2\_k (n)n^\{-2 \sigma \} T + R (k, \sigma ; T),\end\{equation*\}where $R(k, \sigma ; T) = 0(T) (T \rightarrow \infty )$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R(k, \sigma ; T)$ are derived in the range min $(\beta _k, \sigma ^\ast _k) &lt; \sigma &lt; 1$. We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.},
author = {Ivić, Aleksandar},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {mean-value; Riemann zeta-function},
language = {eng},
number = {1},
pages = {163-178},
publisher = {Université Bordeaux I},
title = {On mean values of some zeta-functions in the critical strip},
url = {http://eudml.org/doc/249093},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Ivić, Aleksandar
TI - On mean values of some zeta-functions in the critical strip
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 1
SP - 163
EP - 178
AB - For a fixed integer $k \ge 3$, and fixed $\frac{1}{2} &lt; \sigma &lt; 1$ we consider\begin{equation*} \int ^T_1 \left| \zeta (\sigma + it)\right|^{2k} dt = \sum ^ \infty _{n=1} d^2_k (n)n^{-2 \sigma } T + R (k, \sigma ; T),\end{equation*}where $R(k, \sigma ; T) = 0(T) (T \rightarrow \infty )$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R(k, \sigma ; T)$ are derived in the range min $(\beta _k, \sigma ^\ast _k) &lt; \sigma &lt; 1$. We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.
LA - eng
KW - mean-value; Riemann zeta-function
UR - http://eudml.org/doc/249093
ER -

References

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  1. [1] R. Balasubramanian, A. Ivić, K. Ramachandra, An application of the Hooley-Huxley contour. Acta Arith.65 (1993), 45-51. Zbl0781.11035MR1239242
  2. [2] K. Chandrasekharan, R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. of Math.76 (1962), 93-136. Zbl0211.37901MR140491
  3. [3] K. Chandrasekharan, R. Narasimhan, The approximate functional equation for a class of zeta-functions. Math. Ann.152 (1963), 30-64. Zbl0116.27001MR153643
  4. [4] A. Ivić, The Riemann zeta-function, John Wiley & Sons, New York, 1985. Zbl0556.10026MR792089
  5. [5] A. Ivić, Large values of certain number-theoretic error terms. Acta Arith.56 (1990), 135-159. Zbl0659.10053MR1075641
  6. [6] A. Ivić, On zeta-functions associated with Fourier coefficients of cusp forms. Proceedings of the Amalfi Conference on Analytic Number Theory (Amalfi, September 1989), Università di Salerno, Salerno1992, pp. 231-246. Zbl0787.11035MR1220467
  7. [7] A. Ivić, Some problems on mean values of the Riemann zeta-function. J. Théor. Nombres Bordeaux8 (1996), 101-123. Zbl0858.11045MR1399949
  8. [8] A. Ivić, On some conjectures and results for the Riemann zeta-function and Hecke series. Acta Arith.99 (2001), no. 2, 115-145. Zbl0984.11043MR1847617
  9. [9] A. Ivić, M. Ouellet, Some new estimates in the Dirichlet divisor problem. Acta Arith.52 (1989), 241-253. Zbl0619.10041MR1031337
  10. [10] 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. Zbl0843.11039MR1383532
  11. [11] A. Ivić, K. Matsumoto, Y. Tanigawa, On Riesz means of the coefficients of the Rankin-Selberg series. Math. Proc. Camb. Phil. Soc.127 (1999), 117-131. Zbl0958.11065MR1692491
  12. [12] S. Kanemitsu, A. Sankaranarayanan, Y. Tanigawa, A mean value theorm for Dirichlet series and a general divisor problem, to appear. Zbl1022.11047
  13. [13] K. Matsumoto, The mean values and the universality of Rankin-Selberg L-functions. Proc. Turku Symposium on Number Theory in Memory of K. Inkeri, May 31-June 4, 1999, Walter de Gruyter, Berlin etc., 2001, in print. Zbl0972.11075MR1822011
  14. [14] A. Perelli, General L-functions. Ann. Mat. Pura Appl.130 (1982), 287-306. Zbl0485.10030MR663975
  15. [15] R.A. Rankin, Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions. II, The order of Fourier coefficients of integral modular forms. Math. Proc. Cambridge Phil. Soc.35 (1939), 357-372. Zbl0021.39202
  16. [16] R.A. Rankin, Modular Forms. Ellis Horwood Ltd., Chichester, England, 1984. Zbl0546.00010MR803359
  17. [17] H.-E. Richert, Über Dirichletreihen mit Funktionalgleichung. Publs. Inst. Math. (Belgrade) 11 (1957), 73-124. Zbl0082.05802MR92816
  18. [18] A. Selberg, Old and new conjectures and results about a class of Dirichlet series. Proc. Amalfi Conf. Analytic Number Theory, eds. E. Bombieri et al., Università di Salerno, Salerno, 1992, pp. 367-385. Zbl0787.11037MR1220477
  19. [19] E.C. Titchmarsh, The theory of the Riemann zeta-function, 2nd edition.Oxford University Press, Oxford, 1986. Zbl0601.10026MR882550
  20. [20] Zhang Wenpeng, On the divisor problem. Kexue Tongbao33 (1988), 1484-1485. MR969977

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