# Zeros of Dirichlet L-series on the critical line

Acta Arithmetica (2000)

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

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topPeter 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

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- [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.)
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- [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] G. Kolesnik, On the order of Dirichlet L-functions, Pacific J. Math. 82 (1979), 479-484. Zbl0423.10022
- [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
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