An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial

• Autorzy: Hideaki Ishikawa , Kohji Matsumoto
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Riemann zeta-function; Dirichlet polynomial; Atkinson formula

Kategorie tematyczne

• 11M41: Other Dirichlet series and zeta functions
• 11M06: ζ ( s ) and L ( s , χ )

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 1
• Strony: 102-126

Daty

wydano

2011-02-01

online

2010-12-30

Twórcy

autor

Hideaki Ishikawa

• Nagasaki University

autor

Kohji Matsumoto

• Nagoya University

Bibliografia

• [1] Atkinson F.V., The mean-value of the Riemann zeta function, Acta Math., 1949, 81(1), 353–376 http://dx.doi.org/10.1007/BF02395027
• [2] 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
• [3] 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 http://dx.doi.org/10.1016/0022-314X(89)90024-3
• [4] Heath-Brown D.R., The mean value theorem for the Riemann zeta-function, Mathematika, 1978, 25(2), 177–184 http://dx.doi.org/10.1112/S0025579300009414
• [5] 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
• [6] 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
• [7] Ivić A., The Riemann Zeta-Function, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1985
• [8] Ivić A., Lectures on Mean Values of the Riemann Zeta Function, Tata Inst. Fund. Res. Lectures on Math. and Phys., 82, Springer, Berlin, 1991
• [9] Jutila M., Transformation formulae for Dirichlet polynomials, J. Number Theory, 1984, 18(2), 135–156 http://dx.doi.org/10.1016/0022-314X(84)90049-0
• [10] 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
• [11] Katsurada M., Matsumoto K., Asymptotic expansions of the mean values of Dirichlet L-functions, Math. Z., 1991, 208(1), 23–39 http://dx.doi.org/10.1007/BF02571507
• [12] 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
• [13] 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
• [14] 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
• [15] 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 http://dx.doi.org/10.3792/pjaa.61.222
• [16] 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 http://dx.doi.org/10.3792/pjaa.62.399
• [17] 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 http://dx.doi.org/10.1023/A:1019767816190
• [18] Titchmarsh E.C., The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford, 1951

Identyfikatory

DOI

10.2478/s11533-010-0085-5