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

Abstract

We present a group-theoretic criterion under which one may verify the Artin conjecture for some (non-monomial) 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 non-abelian extensions of Q.In addition, we discuss two methods for locating zeros of arbitrary L-functions. The first uses the explicit formula and techniques developed in [BS05] for computing with trace formulae. The second method generalizes that of Turing for verifying the Riemann hypothesis. In order to apply it we develop a rigorous algorithm for computing general L-functions on the critical line via the Fast Fourier Transform.Finally, we present some numerical results testing Artin's conjecture for S 5 representations, and the Riemann hypothesis for Dedekind zeta functions of S 5 and A 5 fields.1 Reading Turing's paper on the subject, which was his last, one marvels at what he accomplished with the limited computational resources of the day. His method was truly ahead of its time.

Ranks of quadratic twists of elliptic curves

“…We also note that for the congruent number curve we can restrict to u > v > 0 and to u, v of opposite parity, both of these due to the 2-torsion. 4 The method of computation then proceeds by looping over 1 ≤ v < u ≤ L up to some limit L, and for each u/v computing the associated d-value as indicated above. It is at this point that the methods that are used tend to vary.…”

Isolating some non-trivial zeros of zeta

“…That due to Odlyzko and Schönhage [19] [10]. Booker's algorithm for generic L-functions is described in [6]. When applied to ζ, it has the same asymptotic time complexity as Odlyzko-Schönhage but requires space O(t).…”

“…We extended MPFI in the obvious way to handle complex valued intervals as rectangles and our working precision was 300 bits. For steps (1) to (6) we used N = 2 15 with B = 5, 476 as the width of our window and A = N/B. However, by exploiting Whittaker-Shannon up-sampling once again we can increase N to 2 20 during the Discrete Fourier Transform at stage (6), almost for free, keeping B the same.…”

“…Booker provides bounds for these quantities for the general (non-windowed) Lfunction case in [6]. However, we must accommodate the inclusion of the Gaussian and we can also exploit our knowledge of the specific analytic properties of ζ.…”

Computing L-Functions: A Survey

“…I mention the available public data. 1 • M. Rubinstein's C++ program lcalc, which can compute values of L-functions, make large tables of zeros, and so on. The program uses C++ language double, so is limited to 15 decimal digits, but is highly optimized, hence very fast, and used in most situations.…”