Skip to main content Accessibility help
×
×
Home

Titchmarsh’s Method for the Approximate Functional Equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2}$ , $\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ , and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$

  • Jun Furuya (a1), T. Makoto Minamide (a2) and Yoshio Tanigawa (a3)

Abstract

Let $\unicode[STIX]{x1D701}(s)$ be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for $\unicode[STIX]{x1D701}^{2}(s)$ with error term $O(x^{1/2-\unicode[STIX]{x1D70E}}((x+y)/|t|)^{1/4}\log |t|)$ , where $-1/2<\unicode[STIX]{x1D70E}<3/2,x,y\geqslant 1,xy=(|t|/2\unicode[STIX]{x1D70B})^{2}$ . Later, in 1938, Titchmarsh improved the error term by removing the factor $((x+y)/|t|)^{1/4}$ . In 1999, Hall showed the approximate functional equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2},\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ , and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ (in the range $0<\unicode[STIX]{x1D70E}<1$ ) whose error terms contain the factor $((x+y)/|t|)^{1/4}$ . In this paper we remove this factor from these three error terms by using the method of Titchmarsh.

Copyright

Footnotes

Hide All

This work was supported by JSPS KAKENHI: 26400030, 15K17512 and 15K04778.

Footnotes

References

Hide All
[1] Akatsuka, H., Conditional estimates for error terms related to distribution of zeros of 𝜁 (s) . J. Number Theory 132(2012), 22422257. https://doi.org/10.1016/j.jnt.2012.04.012.
[2] Andrews, G. E., Askey, R., and Roy, R., Special functions . Encyclopedia of Mathematics and Its Applications, 71. Cambridge University Press, 1999.
[3] Aoki, M. and Minamide, M., A zero density estimate of the derivatives of the Riemann zeta function . JANTA 2(2012), 361375.
[4] Banerjee, D. and Minamide, M., On averages of the error term of a new kind of the divisor problem . J. Math. Anal. Appl. 438(2016), 533550. https://doi.org/10.1016/j.jmaa.2016.02.013.
[5] Berndt, B. C., The number of zeros for 𝜁(k)(s) . J. London Math. Soc. 2(1970), 577580. https://doi.org/10.1112/jlms/2.Part_4.577.
[6] Conrey, J. B., The fourth moment of derivatives of Riemann zeta-function . Quart. J. Oxford 39(1988), 2136. https://doi.org/10.1093/qmath/39.1.21.
[7] Furuya, J., Minamide, M., and Tanigawa, Y., Representations and evaluations of the error term in a certain divisor problem . Math. Slovaca 66(2016), 575582. https://doi.org/10.1515/ms-2015-0160.
[8] Furuya, J., Minamide, M., and Tanigawa, Y., On a restricted divisor problem . J. Indian Math. Soc. 83(2016), 269287.
[9] Gonek, S. M., Mean values of the Riemann zeta-function and its derivatives . Invent. Math. 75(1984), 123141. https://doi.org/10.1007/BF01403094.
[10] Hall, R. R., The behaviour of the Riemann zeta-function on the critical line . Mathematika 46(1999), 281313. https://doi.org/10.1112/S0025579300007762.
[11] Hardy, G. H. and Littlewood, J. E., The approximate functional equation in the theory of the zeta-functions, with applications to the divisor-problems of Dirichlet and Piltz . Proc. London Math. Soc. (2) 21(1923), 3974. https://doi.org/10.1112/plms/s2-21.1.39.
[12] Hardy, G. H. and Littlewood, J. E., The approximate functional equations for 𝜁(s) and 𝜁2(s) . Proc. London Math. Soc. (2) 29(1929), 8197. https://doi.org/10.1112/plms/s2-29.1.81.
[13] Heath-Brown, D. R., The fourth power moment of the Riemann zeta-function . Proc. London Math. Soc. (3) 38(1979), 385422. https://doi.org/10.1112/plms/s3-38.3.385.
[14] Ingham, A. E., Mean-value theorems in the theory of the Riemann zeta-function . Proc. London Math. Soc. 27(1926), 273300.
[15] Ivić, A., The Riemann zeta-function. Wiley, New York, 1985.
[16] Levinson, N. and Montgomery, H. L., Zeros of derivatives of the Riemann zeta-functions . Acta Math. 133(1974), 4965. https://doi.org/10.1007/BF02392141.
[17] Minamide, M., On the truncated Voronoï formula for the derivative of the Riemann zeta function . Indian J. Math. 55(2013), 325352.
[18] Speiser, A., Geometrisches zur Riemann Zetafuncktion . Math. Ann. 110(1934), 514521. https://doi.org/10.1007/BF01448042.
[19] Spira, R., Zero-free regions of 𝜁(k)(s) . J. London Math. Soc. 40(1965), 677682. https://doi.org/10.1112/jlms/s1-40.1.677.
[20] Spira, R., Another zero-free region for 𝜁(k)(s) . Proc. Amer. Math. Soc. 26(1970), 246247.
[21] Spira, R., Zeros of 𝜁 (s) and the Riemann hypothesis . Illinois J. Math. 17(1973), 147152.
[22] Titchmarsh, E. C., The approximate functional equation for 𝜁2(s) . Quart. J. Math. 9(1938), 109114.
[23] Titchmarsh, E. C., The theory of the Riemann zeta-function, Second edition. Revised by D. R. Heath-Brown, Oxford University Press, New York, 1986.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed