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On the zeros of the Riemann zeta-function in the critical strip. IV

Math. Comp., 46(174), 667--681.] who proved that

Theorem (1986)

The boundEvery zeroρ ofζ that have a real part between 0 and 1 and an imaginary part not more, in absolute value, than≤T0=545439823 are in fact on the critical line, i.e. satisfyRρ=1/2 .

Computing degree 1 L-function rigorously

Ph.D. thesis, Mathematics. arXiv:1305.3087.]. Between these two results, [Wedeniwski, 2002 †Wedeniwski, S. 2002

On the Riemann hypothesis

http://www.zetagrid.net/zeta/announcements.html] announced that, he and many collaborators proved, using a network method:

Theorem (2002)

And [Gourdon & Demichel, 2004 †Gourdon, X., & Demichel, P. 2004T0=29538618432 is admissible in the theorem above.

The

http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf] went one step further

Theorem (2004)

These two last announcements have not been subject to any academic papers. One of the key ingredient is an explicit Riemann-Siegel formula due to [Gabcke, 1979 †Gabcke, W. 1979T0=2.445⋅1012 is admissible in the theorem above.

Neue Herleitung und explizite Restabschaetzung der Riemann-Siegel-Formel

Ph.D. thesis, Mathematisch-Naturwissenschaftliche Fakultät der Georg-August-Universität zu Göttingen.] (the preprint of Gourdon mentionned above gives a version of Gabcke's result) and such a formula is missing in the case of Dirichlet

Numerical Computations Concerning the ERH

Math. Comp., 61, 415--440.] has proved that

Theorem (1993)

These computations have been extended by [Bennett, 2001 †Bennett, M. 2001

- Every
q≤13 satisfiesGRH(10000) .- Every
q belonging to one of the setssatisfies

{k≤72} {k≤112,k non premier} {116,117,120,121,124,125,128,132,140,143,144,156,163,169,180,216,243,256,360,420,432} GRH(2500) .

Rational approximation to algebraic numbers of small height: the Diophantine equation

J. reine angew. Math., 535, 1--49.] by using Rumely's programm. [Platt, 2011 †Platt, D.J. 2011

Computing degree 1 L-function rigorously

Ph.D. thesis, Mathematics. arXiv:1305.3087.] and [Platt, 2013 †Platt, D.J. 2013

Numerical computations concerning the GRH

Ph.D. thesis. http://arxiv.org/abs/1305.3087.] use two fast Fourier transforms, one in the

Theorem (2011-2013)

We mention here the algorithm of [Omar, 2001 †Omar, S. 2001Every modulusq≤400000 satisfiesGRH(100000000/q) .

Localization of the first zero of the Dedekind zeta function

Math. Comp., 70(236), 1607--1616.] that enables one to prove efficiently that some

The

Proc. Lond. Math. Soc., II. Ser., 45, 21--44.] in Lemma 19 (essentially with

Sharper bounds for the Chebyshev Functions

Math. Comp., 29(129), 243--269.] by using a device of [Stechkin, 1970 †Stechkin, S.B. 1970

Zeros of Riemann zeta-function

Math. Notes, 8, 706--711.] (getting essentially

Primes in arithmetic progressions

Math. Comp., 65, 397--425.] where the second order term is improved upon, relying on [Stechkin, 1989 †Stechkin, S.B. 1989

Rational inequalities and zeros of the Riemann zeta-function

Trudy Mat. Inst. Steklov (english translation: in Proc. Steklov Inst. Math. 189 (1990)), 189, 127--134.]. The latest result comes from [Kadiri, 2002 †Kadiri, H. 2002

Une région explicite sans zéros pour les fonctions {

Ph.D. thesis, Université Lille 1. http://tel.ccsd.cnrs.fr/documents/archives0/00/00/26/95/index_fr.html.] and later [Kadiri, 2005 †Kadiri, H. 2005

Une région explicite sans zéros pour la fonction

Acta Arith., 117(4), 303--339.], where it is proven that

Theorem (2002)

Concerning DirichletThe Riemannζ -function has no zeros in the regionRs≥1−1R0log(|Is|+2)with R0=5.70175.

Explicit zero-free regions for Dirichlet

J. Number Theory, 19, 7--32.] by adaptating [Rosser & Schoenfeld, 1975 †Rosser, J.B., & Schoenfeld, L. 1975

Sharper bounds for the Chebyshev Functions

Math. Comp., 29(129), 243--269.]. [Kadiri, 2002 †Kadiri, H. 2002

Une région explicite sans zéros pour les fonctions {

Ph.D. thesis, Université Lille 1. http://tel.ccsd.cnrs.fr/documents/archives0/00/00/26/95/index_fr.html.] (cf also [Kadiri, 2009 †Kadiri, H. 2009

An explicit zero-free region for the Dirichlet {

Being processed... http://arxiv.org/pdf/math.NT/0510570.]) improves that into:

Theorem (2002)

Concerning the Vinogradov-Korobov zero-free region, [Ford, 2000 †Ford, K. 2000The DirichletL -functions associated to a character of conductorq has no zero in the region:to the exception of at most one of them which would hence be attached to a real-valued character. This exceptional one would have at most one zero inside the forbidden region (and which is loosely called a "Siegel zero").Rs≥1−1R1log(qmax(1,|Is|))with R1=6.4355,

Zero-free regions for the Riemann zeta function

Proceedings of the Millenial Conference on Number Theory, Urbana, IL.] shows that

Theorem (2001)

Concerning the DedekindThe Riemannζ -function has no zeros in the regionRs≥1−158(log|Is|)2/3(loglog|Is|)1/3(|Is|≥3).

Explicit zero-free regions for Dedekind zeta functions

Int. J. Number Theory, 8(1), 125--147. http://dx.doi.org/10.1142/S1793042112500078.]. [Rosser, 1949 †Rosser, J.B. 1949

Real roots of Dirichlet

Bull. Amer. Math. Soc., 55, 906--913.], [Rosser, 1950 †Rosser, J.B. 1950

Real roots of Dirichlet

J. Res. Nat. Bur. Standards, 505--514.], [Chua, 2005 †Chua, Kok Seng. 2005

Real zeros of Dedekind zeta functions of real quadratic field

Math. Comput., 74(251), 1457--1470.], [Watkins, 2004 †Watkins, M. 2004

Real zeros of real odd Dirichlet L-functions

Math. Comp., 73(245), 415--423. http://www.math.psu.edu/watkins/papers.html.], [Chen & Wang, 1989 †Chen, Jingrun, & Wang, Tianze. 1989

On the distribution of zeros of Dirichlet L-functions

J. Sichuan Univ., Nat. Sci. Ed., 26.], [Liu & Wang, 2002 †Liu, Ming-Chit, & Wang, Tianze. 2002

Distribution of zeros of Dirichlet

Acta Arith., 102(3), 261--293.], [Kadiri & Ng, 2012 †Kadiri, H., & Ng, N. 2012

Explicit zero density theorems for Dedekind zeta functions

J. Number Theory, 132(4), 748--775. http://dx.doi.org/10.1016/j.jnt.2011.09.002.]. The LMFDB database contains the first zeros of many

Last updated on July 14th, 2012, by Olivier Ramaré