Asymptotic Bounds of Euler-Kronecker Constants and Their Connection to the Riemann Hypothesis

Introduction

The study of prime number distribution and the zeros of the Riemann zeta function is a cornerstone of analytic number theory. A pivotal quantity in this landscape is the Euler-Kronecker constant, denoted as gamma_K. In the source paper arXiv:hal-00004434v2, M.A. Tsfasman explores the asymptotic behavior of these constants in global fields, providing a rigorous framework that links the arithmetic of field extensions to the analytic properties of their associated zeta functions.

The Riemann Hypothesis (RH) posits that all non-trivial zeros of the Riemann zeta function zeta(s) lie on the critical line Re(s) = 1/2. For an arbitrary algebraic number field K, the Dedekind zeta function zeta_K(s) takes its place. The Generalized Riemann Hypothesis (GRH) similarly suggests that the non-trivial zeros of zeta_K(s) lie on the critical line. The Euler-Kronecker constant emerges from the Laurent expansion of the logarithmic derivative of zeta_K(s) at s = 1. Specifically, it is defined by the relation: zeta_K'(s) / zeta_K(s) = -1/(s-1) + gamma_K + O(s-1).

This article connects the results of arXiv:hal-00004434v2 to the Riemann Hypothesis by interpreting these constants as renormalized log-derivatives whose behavior is sensitive to low-lying zeros. We examine how the distribution of zeros controls the admissible asymptotic splitting data and how extremal families serve as a testbed for RH-strengthened inequalities.

Mathematical Background

To analyze these constants asymptotically, we define a scaling parameter alpha_K. In the number field case, alpha_K = log sqrt(|d_K|), where d_K is the absolute discriminant. In the function field case over a finite field F_q, the corresponding measure is alpha_K = (g_K - 1) log q, where g_K is the genus.

The core analysis involves several auxiliary constants that capture the contributions of various places:

• a_0 = log sqrt(8 pi) + pi/4 + gamma/2
• a_1 = log 8 pi + gamma
• a_q = log q / (sqrt(q) - 1)
• b_q = log q / (q - 1)

These constants appear in the Basic Inequality derived by Tsfasman and Vladut, which bounds the sum of the contributions of primes of various norms. For an asymptotically exact family of fields, the distribution of primes is described by measures phi_q, representing the limit of the ratio of the number of places of norm q to the log-discriminant. The paper establishes the central inequality: 0 ≥ limsup gamma_K / alpha_K ≥ liminf gamma_K / alpha_K ≥ -1.

Main Technical Analysis

Explicit Formula and the Prime-Zero Duality

At the heart of the theory is a family-level explicit formula. By choosing a nonnegative test function and applying the explicit formula to zeta_K(s), one normalizes by alpha_K to pass to limits. The sign of the zero-side term is the crucial factor: under RH, it becomes nonnegative because the transform of the test function is evaluated on zeros constrained to the critical line Re(s) = 1/2.

Without RH, zeros drifting toward s = 1 would create negative contributions that destroy the clean inequality. This explains why arXiv:hal-00004434v2 distinguishes between conditional results (assuming GRH) and unconditional variants. The coefficient (sqrt(q)-1)^-1 in a_q reflects that the critical line is at distance 1/2 from s = 1, essentially acting as an analytic fingerprint of the RH-predicted geometry of zeros.

Spectral Properties and Field Towers

The gamma_K / alpha_K ratio measures the "gravitational pull" of zeros toward the point s = 1. Zeros closer to the critical line contribute significantly to the sum. The source paper constructs specific field families to test the theoretical limits of this ratio. A classic example involves quadratic fields Q(sqrt(D)) forming infinite class field towers. By ensuring many small primes (2, 3, 5, 7) split completely, we maximize the phi_q terms, driving gamma_K to its most negative values.

For example, the field Q(sqrt(-5.7.11.13.17.19.23.29.31.37)) yields a ratio of approximately -0.17849. In the function field case, where Weil's RH is a theorem, the lower bound is attained by optimal families reaching the Drinfeld-Vladut bound. This serves as a template: if one had an RH-quality inequality and optimal towers in the number field case, one could hope for sharp extremal values.

Novel Research Pathways

1. Extremal Linear Programming for Splitting Invariants

Future research could treat the invariants (phi_R, phi_C, phi_q) as variables in a linear programming problem. Under GRH, the Basic Inequality yields constraints that define a feasible polytope. By solving for the extremal limits of gamma_K / alpha_K and comparing them to values realized by explicit class field towers, we can identify how much "slack" remains in the analytic bounds and whether improved constructions would effectively test the sharpness of RH-strengthened inequalities.

2. Spectral Rigidity of Euler-Kronecker Extremizers

One could propose a "low-lying zeros vs. Euler-Kronecker" principle. If the normalized constants approach the GRH-linear-programming lower bound, the family must exhibit generic low-lying zero statistics compatible with RH. This transforms the inequality framework into a spectral diagnostic: extremal behavior in the prime-splitting data would force a specific zero distribution, providing a new way to characterize the critical line conjecture.

3. Non-Abelian Towers and Sieve Refinements

While the source paper focuses on quadratic fields, the behavior of gamma_K / alpha_K in non-abelian towers remains largely unexplored. Constructing towers using p-adic representations might allow for denser prime splitting than abelian ones, potentially pushing the ratio closer to the GRH bound of -1. Additionally, integrating these bounds into the weights of a Selberg-type sieve could lead to improved explicit bounds for the prime ideal theorem.

Computational Implementation

The following Wolfram Language implementation demonstrates how to calculate the log-derivative of the zeta function near s = 1 and compute the Euler-Kronecker constant for a model quadratic field, illustrating the sensitivity of these values to zero distribution.

(* Section: Euler-Kronecker Constant Analysis *)
(* Purpose: Compare log-derivative of Zeta near 1 to prime sums and compute gamma_K *)

Module[{s = 1.05, nZeros = 100, primeCut = 1000, disc, chi, L1, Lp1, gammaK, ratio, alphaK},
  
  (* 1. Approximate zeta'/zeta near 1 using primes *)
  primes = Prime[Range[PrimePi[primeCut]]];
  primeSum = -N@Total[Table[Log[p]/(p^s - 1), {p, primes}]];
  
  (* 2. True value of zeta'/zeta at s *)
  zetaLogDeriv = N[ZetaPrime[s]/Zeta[s], 20];
  
  Print["s = ", s];
  Print["True zeta'/zeta: ", zetaLogDeriv];
  Print["Prime-sum approximation: ", primeSum];

  (* 3. Quadratic field example: Q(sqrt(D)) *)
  (* D = -5*7*11*13*17*19*23*29*31*37 from hal-00004434v2 *)
  disc = -510510; 
  chi[n_] := KroneckerSymbol[disc, n];
  
  (* gamma_K = EulerGamma + (L'/L)(1, chi) *)
  L1 = DirichletL[chi, 1];
  Lp1 = DirichletLPrime[1, {disc, _}];
  
  gammaK = N[EulerGamma + Lp1/L1, 20];
  alphaK = Log[Sqrt[Abs[disc]]];
  ratio = gammaK / alphaK;

  Print["Discriminant: ", disc];
  Print["Euler-Kronecker gamma_K: ", gammaK];
  Print["Ratio gamma_K / alpha_K: ", ratio];
  
  (* Plotting the ratio behavior for small discriminants *)
  Plot[EulerGamma + DirichletLPrime[1, {x, _}]/DirichletL[1, {x, _}], {x, -100, -1},
    PlotLabel -> "gamma_K Variation vs Discriminant", 
    AxesLabel -> {"D", "gamma_K"}]
]

Conclusions

The analysis of Euler-Kronecker constants in arXiv:hal-00004434v2 constitutes a vital bridge between arithmetic geometry and analytic number theory. By establishing that the ratio gamma_K / alpha_K is bounded between -1 and 0 under GRH, Tsfasman and Vladut have provided a fundamental constraint on the behavior of global fields. The existence of optimal families where the ratio approaches these limits suggests that gamma_K is an essential descriptor of a field's arithmetic density. Further exploration of non-abelian towers and high-precision spectral analysis remains the most promising avenue for testing the limits of the Riemann Hypothesis.

References

• arXiv:hal-00004434v2: M.A. Tsfasman, "Asymptotic upper and lower bounds for the Euler-Kronecker constant."
• Tsfasman, M.A., Vladut, S.G. "Infinite Global Fields and the Generalized Brauer-Siegel Theorem." Moscow Math. J. 2 (2002).
• Ihara, Y. "On the Euler-Kronecker constants of global fields and primes with small norms." Progress in Mathematics, Vol. 253 (2006).