Abstract
We prove an explicit formula of Atkinson type for the error term in the asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. To deal with the case when coefficients of the Dirichlet polynomial are complex, we apply the idea of the first author in his study on mean values of Dirichlet L-functions.
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References
Atkinson F.V., The mean-value of the Riemann zeta function, Acta Math., 1949, 81(1), 353–376
Balasubramanian R., Conrey J.B., Heath-Brown D.R., Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial, J. Reine Angew. Math., 1985, 357, 161–181
Hafner J.L., Ivić A., On the mean-square of the Riemann zeta-function on the critical line, J. Number Theory, 1989, 32(2), 151–191
Heath-Brown D.R., The mean value theorem for the Riemann zeta-function, Mathematika, 1978, 25(2), 177–184
Ishikawa H., A difference between the values of |L(1/2 + it, χ j)| and |L(1/2 + it, χ k)|. I, Comment. Math. Univ. St. Pauli, 2006, 55(1), 41–66
Ishikawa H., A difference between the values of |L(1/2 + it, χ j)| and |L(1/2 + it, χk)|. II, Comment. Math. Univ. St. Pauli, 2007, 56(1), 1–9
Ivić A., The Riemann Zeta-Function, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1985
Ivić A., Lectures on Mean Values of the Riemann Zeta Function, Tata Inst. Fund. Res. Lectures on Math. and Phys., 82, Springer, Berlin, 1991
Jutila M., Transformation formulae for Dirichlet polynomials, J. Number Theory, 1984, 18(2), 135–156
Jutila M., Lectures on a Method in the Theory of Exponential Sums, Tata Inst. Fund. Res. Lectures on Math. and Phys., 80, Springer, Berlin, 1987
Katsurada M., Matsumoto K., Asymptotic expansions of the mean values of Dirichlet L-functions, Math. Z., 1991, 208(1), 23–39
Katsurada M., Matsumoto K., A weighted integral approach to the mean square of Dirichlet L-functions, In: Number Theory and its Applications, Kyoto, November 10–14, 1997, Dev. Math., 2, Kluwer, Dordrecht, 1999, 199–229
Matsumoto K., Recent developments in the mean square theory of the Riemann zeta and other zeta-functions, In: Number Theory, Trends Math., Birkhäuser, Basel 2000, 241–286
Matsumoto K., On the mean square of the product of ζ(s) and a Dirichlet polynomial, Comment. Math. Univ. St. Pauli, 2004, 53(1), 1–21
Motohashi Y., A note on the mean value of the zeta and L-functions. I, Proc. Japan Acad. Ser. A Math. Sci., 1985, 61(7), 222–224
Motohashi Y., A note on the mean value of the zeta and L-functions. V, Proc. Japan Acad. Ser. A Math. Sci., 1986, 62(10), 399–401
Steuding J., On simple zeros of the Riemann zeta-function in short intervals on the critical line, Acta Math. Hungar., 2002, 96(4), 259–308
Titchmarsh E.C., The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford, 1951
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Ishikawa, H., Matsumoto, K. An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial. centr.eur.j.math. 9, 102–126 (2011). https://doi.org/10.2478/s11533-010-0085-5
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DOI: https://doi.org/10.2478/s11533-010-0085-5