Zeros of Dirichlet L-series on the critical line

Peter J. Bauer

Acta Arithmetica (2000)

  • Volume: 93, Issue: 1, page 37-52
  • ISSN: 0065-1036

Abstract

top
Introduction. In 1974, N. Levinson showed that at least 1/3 of the zeros of the Riemann ζ-function are on the critical line ([19]). Today it is known (Conrey, [6]) that at least 40.77% of the zeros of ζ(s) are on the critical line and at least 40.1% are on the critical line and are simple. In [16] and [17], Hilano showed that Levinson's original result is also valid for Dirichlet L-series. This paper is a shortened version of parts of the dissertation [3], the full details of which may be found at http://www.math.uni-frankfurt.de/~pbauer/diss.ps. We shall prove a mean value theorem for Dirichlet L-series and use this for proving some corollaries concerning the distribution of the zeros of L-series - amongst other results we improve the above mentioned bounds for Dirichlet L-series.

How to cite

top

Peter J. Bauer. "Zeros of Dirichlet L-series on the critical line." Acta Arithmetica 93.1 (2000): 37-52. <http://eudml.org/doc/207398>.

@article{PeterJ2000,
abstract = { Introduction. In 1974, N. Levinson showed that at least 1/3 of the zeros of the Riemann ζ-function are on the critical line ([19]). Today it is known (Conrey, [6]) that at least 40.77% of the zeros of ζ(s) are on the critical line and at least 40.1% are on the critical line and are simple. In [16] and [17], Hilano showed that Levinson's original result is also valid for Dirichlet L-series. This paper is a shortened version of parts of the dissertation [3], the full details of which may be found at http://www.math.uni-frankfurt.de/~pbauer/diss.ps. We shall prove a mean value theorem for Dirichlet L-series and use this for proving some corollaries concerning the distribution of the zeros of L-series - amongst other results we improve the above mentioned bounds for Dirichlet L-series. },
author = {Peter J. Bauer},
journal = {Acta Arithmetica},
keywords = {-functions; zeros; Riemann's hypothesis},
language = {eng},
number = {1},
pages = {37-52},
title = {Zeros of Dirichlet L-series on the critical line},
url = {http://eudml.org/doc/207398},
volume = {93},
year = {2000},
}

TY - JOUR
AU - Peter J. Bauer
TI - Zeros of Dirichlet L-series on the critical line
JO - Acta Arithmetica
PY - 2000
VL - 93
IS - 1
SP - 37
EP - 52
AB - Introduction. In 1974, N. Levinson showed that at least 1/3 of the zeros of the Riemann ζ-function are on the critical line ([19]). Today it is known (Conrey, [6]) that at least 40.77% of the zeros of ζ(s) are on the critical line and at least 40.1% are on the critical line and are simple. In [16] and [17], Hilano showed that Levinson's original result is also valid for Dirichlet L-series. This paper is a shortened version of parts of the dissertation [3], the full details of which may be found at http://www.math.uni-frankfurt.de/~pbauer/diss.ps. We shall prove a mean value theorem for Dirichlet L-series and use this for proving some corollaries concerning the distribution of the zeros of L-series - amongst other results we improve the above mentioned bounds for Dirichlet L-series.
LA - eng
KW - -functions; zeros; Riemann's hypothesis
UR - http://eudml.org/doc/207398
ER -

References

top
  1. [1] T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986. 
  2. [2] P. J. Bauer, Über den Anteil der Nullstellen der Riemannschen Zeta-Funktion auf der kritischen Geraden, Diploma thesis, Frankfurt a.M., 1992. (Available at http://www.math.uni-frankfurt.de/ pbauer/diplom.ps.) 
  3. [3] P. J. Bauer, Zur Verteilung der Nullstellen der Dirichletschen L-Reihen, Dissertation, Frankfurt a.M., 1997. (Available at http://www.math.uni-frankfurt.de/ pbauer/diss.ps.) 
  4. [4] D. A. Burgess, On character sums and L-series, Proc. London Math. Soc. 12 (1962), 193-206. Zbl0106.04004
  5. [5] J. B. Conrey, Zeros of derivatives of Riemann's Xi-function on the critical line, J. Number Theory 16 (1983), 49-74. Zbl0502.10022
  6. [6] J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. Reine Angew. Math. 399 (1989), 1-26. Zbl0668.10044
  7. [7] J. B. Conrey and A. Ghosh, A simpler proof of Levinson's theorem, Math. Proc. Cambridge Philos. Soc. 97 (1985), 385-395. Zbl0554.10025
  8. [8] J. B. Conrey, A. Ghosh and S. M. Gonek, Mean values of the Riemann zeta-function with application to the distribution of zeros, in: Number Theory, Trace Formulas and Discrete Groups, K. E. Aubert et al. (eds.), Academic Press, 1989, 185-199. 
  9. [9] H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, 1980. Zbl0453.10002
  10. [10] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982), 219-288. 
  11. [11] J.-M. Deshouillers and H. Iwaniec, Power mean-values for Dirichlet's polynomials and the Riemann zeta-function II, Acta Arith. 43 (1984), 305-312. 
  12. [12] T. Estermann, On the representation of a number as the sum of two products, Proc. London Math. Soc. (2) 31 (1930), 123-133. Zbl56.0174.02
  13. [13] D. W. Farmer, Mean value of Dirichlet series associated with holomorphic cusp forms, J. Number Theory 49 (1994), 209-245. Zbl0817.11028
  14. [14] P. X. Gallagher, A large sieve density estimate near σ=1, Invent. Math. 11 (1970), 329-339. Zbl0219.10048
  15. [15] S. M. Gonek, Mean values of the Riemann zeta-function and its derivatives, ibid. 75 (1984), 123-141. Zbl0531.10040
  16. [16] T. Hilano, On the distribution of zeros of Dirichlet's L-function on the line σ=1/2, Proc. Japan Acad. 52 (1976), 537-540. Zbl0368.10031
  17. [17] T. Hilano, On the distribution of zeros of Dirichlet's L-function on the line σ=1/2 (II), Tokyo J. Math. 1 (1978), 285-304. Zbl0401.10051
  18. [18] G. Kolesnik, On the order of Dirichlet L-functions, Pacific J. Math. 82 (1979), 479-484. Zbl0423.10022
  19. [19] N. Levinson, More than one third of the zeros of Riemann's zeta-function are on σ=1/2, Adv. Math. 13 (1974), 383-436. Zbl0281.10017
  20. [20] H. L. Montgomery, Topics in Multiplicative Number Theory, Springer, 1971. Zbl0216.03501
  21. [21] K. Prachar, Primzahlverteilung, Springer, 1957. 
  22. [22] A. Selberg, On the zeros of Riemann's zeta-function, Skr. Norske Videnskaps-Akad. Oslo, I, 10 (1942), 1-59. 
  23. [23] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Clarendon Press, 1988. Zbl0042.07901

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.