# Artin's Conjecture, Turing's Method, and the Riemann Hypothesis

@article{Booker2006ArtinsCT, title={Artin's Conjecture, Turing's Method, and the Riemann Hypothesis}, author={Andrew R. Booker}, journal={Experimental Mathematics}, year={2006}, volume={15}, pages={385 - 407} }

We present a group-theoretic criterion under which one may verify the Artin conjecture for some (nonmonomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to S 5 and A 5 representations. Under more general conditions, the technique allows for the possibility of verifying the Riemann hypothesis for Dedekind zeta functions of nonabelian extensions of ℚ. In addition, we discuss two methods for locating zeros of arbitrary L-functions. The… Expand

#### Topics from this paper

#### 44 Citations

A method for proving the completeness of a list of zeros of certain L-functions

- Mathematics, Computer Science
- Math. Comput.
- 2015

A general version of this method based on an explicit version of a theorem of Littlewood on the average of the argument of the Riemann zeta function is proved for an extension of the Selberg class including Hecke and Artin L-series, L-functions of modular forms, and, at least in the unramified case, automorphic L- Functions. Expand

The Artin conjecture for some $$S_5$$-extensions

- Mathematics
- 2011

We establish some new cases of Artin’s conjecture. Our results apply to Galois representations over $$\mathbf{Q }$$ with image $$S_5$$ satisfying certain local hypotheses, the most important of which… Expand

Computing degree-1 L-functions rigorously

- Computer Science, Mathematics
- 2011

A new, rigorous algorithm for efficiently and simultaneously computing many values of the Riemann zeta function on the critical line by exploiting the fast Fourier transform (FFT) and its implementation is described. Expand

Minor arcs for Goldbach's problem

- Mathematics
- 2012

The ternary Goldbach conjecture states that every odd number n>=7 is the sum of three primes. The estimation of sums of the form \sum_{p\leq x} e(\alpha p), \alpha = a/q + O(1/q^2), has been a… Expand

Numerical tests of modularity

- Mathematics
- 2005

We propose some numerical tests for identifying L-functions of automorphic representations of GL(r) over a number field. We then apply the tests to various conjectured automorphic L-functions,… Expand

An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups

- Mathematics
- Inventiones mathematicae
- 2019

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of… Expand

Computing L-Functions: A Survey

- Mathematics
- 2015

We survey a number of techniques for computing Lfunctions, including those of degree larger than 2. We discuss the computation of the Dirichlet coefficients using quite a variety of methods, for… Expand

A B\"ocherer-Type Conjecture for Paramodular Forms

- Mathematics
- 2010

In the 1980s B\"ocherer formulated a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a Siegel modular form F to the coefficients of F . He proved… Expand

Numerical computations with the trace formula and the Selberg eigenvalue conjecture

- Mathematics
- 2007

Abstract We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [M. N. Huxley, Introduction to Kloostermania, in: Elementary… Expand

Non-vanishing of Dirichlet L-functions at the Central Point

- Mathematics, Computer Science
- ANTS
- 2008

An efficient algorithm is given to compute the ordernχ of zero of L(s, χ) at s = 1/2, which enables us to efficiently compute nχ for L-functions with very large conductor near 1016 and it is proved that L(1/2), χ ≠ 0 for all real characters χ of modulus less than 1010. Expand

#### References

SHOWING 1-10 OF 45 REFERENCES

On computing artin l-functions in the critical strip

- Mathematics
- 1979

This paper gives a method for computing values of certain nonabelian Artin L-functions in the complex plane. These Artin L-functions are attached to irreducible characters of degree 2 of Galois… Expand

Artin’s conjecture for representations of octahedral type

- Mathematics
- 1981

Let L/F be a finite Galois extension of number fields. E. Artin conjectured that the Z-series of a nontrivial irreducible complex representation of Gal(L/F) is entire, and proved this for monomial… Expand

On Artin's L-Series with General Group Characters

- Mathematics
- 1947

where on the right side x is to be interpreted as a character of 5/9L IV. If K is an abelian field over F and if x is an irreducible character, then L(s, x, K/F) coincides with one of the ordinary… Expand

Zeros and poles of Artin L -series

- Mathematics
- 1989

Let E/F be a finite normal extension of number fields with Galois group G. For each virtual character χ of G, denote by L(s, χ) = L(s, χ, F) the Artin L-series attached to χ. It is defined for Re (s)… Expand

Numerical tests of modularity

- Mathematics
- 2005

We propose some numerical tests for identifying L-functions of automorphic representations of GL(r) over a number field. We then apply the tests to various conjectured automorphic L-functions,… Expand

Zeros of Dedekind zeta functions in the critical strip

- Mathematics, Computer Science
- Math. Comput.
- 1997

A computation which established the GRH to height 92 for cubic number fields for cubic and quartic fields with small discriminant is described, and Turing's criterion is generalized to prove that there is no zero off the critical line. Expand

Effective computation of Maass cusp forms

- Mathematics
- 2006

Author please provide the abstract. Please provide the abstract of this paper that should not exceed 150 words (including spaces) and citation free. 1 Preliminary The aim of this paper is to address… Expand

Numerical computations with the trace formula and the Selberg eigenvalue conjecture

- Mathematics
- 2007

Abstract We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [M. N. Huxley, Introduction to Kloostermania, in: Elementary… Expand

On the experimental verification of the artin conjecture for 2-dimensional odd galois representations over Q liftings of 2-dimensional projective galois representations over Q

- Mathematics
- 1994

1.1. Consider equivalence classes of 2-dimensional, irreducible, continuous, odd Galois representations over Q: ρ : Gal(Q/Q)→ GL2(C) with Artin conductor N ∈ N and determinant character det ρ = e.… Expand

On icosahedral Artin representations

- Mathematics
- 2001

If ρ : Gal(Qac/Q) → GL2(C) is a continuous odd irreducible representation with nonsolvable image, then under certain local hypotheses we prove that ρ is the representation associated to a weight 1… Expand