Explicit zero-free regions for the ζ and L functions

1. Numerical verifications of the Generalized Riemann Hypothesis
Numerical verifications of the Riemann hypothesis for the Riemann ζ-function have been pushed extremely far. B. Riemann himself computed the first zeros. Concerning more recent published papers, we mention [Van de Lune et al., 1986 †Van de Lune, J., te Riele, H.J.J., & D.T.Winter. 1986
On the zeros of the Riemann zeta-function in the critical strip. IV
Math. Comp., 46(174), 667--681.
] who proved that
Theorem (1986)
Every zero ρ of ζ 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. satisfy Rρ=1/2.
The bound 545439823 is increased to 1000000000 in [Platt, 2011 †Platt, D.J. 2011
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)
T0=29538618432 is admissible in the theorem above.
And [Gourdon & Demichel, 2004 †Gourdon, X., & Demichel, P. 2004
The 1013 first zeros of the Riemann Zeta Function and zeros computations at very large height
http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf
] went one step further
Theorem (2004)
T0=2.4451012 is admissible in the theorem above.
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. 1979
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 L-function. Let us introduce some terminology. We say that a modulus q1 (i.e. an integer!) satisfies GRH(H) for some numerical value H when every zero ρ of the L-function associated to a primitive Dirichlet character of conductor q and whose real part lies within the critical line (i.e. has a real part lying inside the open interval (0,1)) and whose imaginary part is below, in absolute value, H, in fact satisfies Rρ=1/2. By employing an Euler-McLaurin formula, [Rumely, 1993 †Rumely, R. 1993
Numerical Computations Concerning the ERH
Math. Comp., 61, 415--440.
] has proved that
Theorem (1993)
  • Every q13 satisfies GRH(10000).
  • Every q belonging to one of the sets
    •   {k72}
    •   {k112,k non premier}
    •   {116,117,120,121,124,125,128,132,140,143,144,156,163,169,180,216,243,256,360,420,432}
    satisfies GRH(2500).
These computations have been extended by [Bennett, 2001 †Bennett, M. 2001
Rational approximation to algebraic numbers of small height: the Diophantine equation |axnbyn|=1
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 t-aspect and one in the q-aspect, as well as an approximate functionnal equation to prove via extremely rigorous computations that
Theorem (2011-2013)
Every modulus q400000 satisfies GRH(100000000/q).
We mention here the algorithm of [Omar, 2001 †Omar, S. 2001
Localization of the first zero of the Dedekind zeta function
Math. Comp., 70(236), 1607--1616.
] that enables one to prove efficiently that some L-functions have no zero within the rectangle 1/2σ1 et 2σ|t|=1 though this algorithm has not been put in practice. There are much better results concerning real zeros of Dirichlet L-functions associated to real characters.
2. Asymptotical zero-free regions
The first fully explicit zero free region for the Riemann zeta-function is due to [Rosser, 1938 †Rosser, J.B. 1938
The n-th prime is greater than nlogn
Proc. Lond. Math. Soc., II. Ser., 45, 21--44.
] in Lemma 19 (essentially with R0=19 in the notations below). This is next imporved upon in Theorem 1 of [Rosser & Schoenfeld, 1975 †Rosser, J.B., & Schoenfeld, L. 1975
Sharper bounds for the Chebyshev Functions ϑ(X) and ψ(X)
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 R0=9.646). The next step is in [Ramaré & Rumely, 1996 †Ramaré, O., & Rumely, R. 1996
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 {L de Dirichlet}
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 ζ de Riemann
Acta Arith., 117(4), 303--339.
], where it is proven that
Theorem (2002)
The Riemann ζ-function has no zeros in the region
Rs11R0log(|Is|+2)with R0=5.70175.
Concerning Dirichlet L-function, the first explicit zero-free region has been obtained in [McCurley, 1984c †McCurley, K.S. 1984c
Explicit zero-free regions for Dirichlet L-functions
J. Number Theory, 19, 7--32.
] by adaptating [Rosser & Schoenfeld, 1975 †Rosser, J.B., & Schoenfeld, L. 1975
Sharper bounds for the Chebyshev Functions ϑ(X) and ψ(X)
Math. Comp., 29(129), 243--269.
]. [Kadiri, 2002 †Kadiri, H. 2002
Une région explicite sans zéros pour les fonctions {L de Dirichlet}
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 {L}-functions
Being processed... http://arxiv.org/pdf/math.NT/0510570.
]) improves that into:
Theorem (2002)
The Dirichlet L-functions associated to a character of conductor q has no zero in the region:
Rs11R1log(qmax(1,|Is|))with R1=6.4355,
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").
Concerning the Vinogradov-Korobov zero-free region, [Ford, 2000 †Ford, K. 2000
Zero-free regions for the Riemann zeta function
Proceedings of the Millenial Conference on Number Theory, Urbana, IL.
] shows that
Theorem (2001)
The Riemann ζ-function has no zeros in the region
Rs1158(log|Is|)2/3(loglog|Is|)1/3(|Is|3).
Concerning the Dedekind ζ-function, see [Kadiri, 2012 †Kadiri, H. 2012
Explicit zero-free regions for Dedekind zeta functions
Int. J. Number Theory, 8(1), 125--147. http://dx.doi.org/10.1142/S1793042112500078.
].
3. Real zeros
[Rosser, 1949 †Rosser, J.B. 1949
Real roots of Dirichlet L-series
Bull. Amer. Math. Soc., 55, 906--913.
], [Rosser, 1950 †Rosser, J.B. 1950
Real roots of Dirichlet L-series
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.
],
4. Density estimates
[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 L-functions and an explicit formula for ψ(t,χ)
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.
].
5. Miscellanae
The LMFDB database contains the first zeros of many L-functions. A part of Andrew Odlyzko's website contains extensive tables concerning zeroes of the Riemann zeta function.

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