Asymptotic Extremals of Euler-Kronecker Constants and their Implications for the Riemann Hypothesis

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Introduction

The study of the Riemann Hypothesis often centers on the distribution of the non-trivial zeros of the Riemann zeta function ζ(s). However, a profound and complementary perspective arises from the analysis of the Euler-Kronecker constant γ_K of a global field K. This invariant, which generalizes the classical Euler-Mascheroni constant, appears as the constant term in the Laurent expansion of the Dedekind zeta function ζ_K(s) at its pole s = 1. The source paper arXiv:hal-00004434 provides a rigorous framework for understanding how these constants behave asymptotically as the complexity of the field K, measured by its discriminant or genus, tends to infinity.

The motivation for this analysis lies in the observation that γ_K is intrinsically linked to the distribution of small primes and the location of zeros of ζ_K(s). By examining the ratio of γ_K to the log-discriminant α_K, researchers can identify constraints on the density of zeros near the critical line. This article synthesizes the technical findings of the source paper, highlighting the universal bounds discovered and proposing new research pathways that leverage these constants to probe the limits of the Generalized Riemann Hypothesis (GRH).

Mathematical Background

For any number field K, let d_K be its absolute discriminant. We define the logarithmic complexity parameter α_K = log sqrt(|d_K|). In the analogous case of a function field over a finite field F_q, this parameter is replaced by (g_K - 1) log q, where g_K is the genus. The Euler-Kronecker constant γ_K is defined by the expansion:

ζ_K(s) = c / (s - 1) + γ_K + O(s - 1)

The source paper arXiv:hal-00004434 introduces several fundamental weights that govern the contribution of primes and infinite places to this constant. Specifically, we define:

a₀ = log sqrt(8 pi) + pi/4 + gamma/2
a₁ = log 8 pi + gamma
a_q = log q / (sqrt(q) - 1)
b_q = log q / (q - 1)

These constants serve as coefficients in the explicit formula that relates γ_K to the splitting behavior of primes. A family of fields is considered asymptotically bad if the densities of primes of small norm, denoted by phi_q, vanish. The paper establishes a critical universal inequality for the normalized ratio: 0 ≥ limsup γ_K/α_K ≥ liminf γ_K/α_K ≥ -1. This suggests that the Euler-Kronecker constant cannot grow more negative than the log-discriminant, a fact deeply tied to the positivity of the sum over the zeros of the zeta function.

Main Technical Analysis

Spectral Properties and Zero Distribution

The relationship between the ratio γ_K/α_K and the Riemann Hypothesis is mediated by the Weil Explicit Formula. This formula connects a sum over the prime ideals of K to a sum over the non-trivial zeros rho = beta + i*gamma of ζ_K(s). If the Generalized Riemann Hypothesis holds, all beta = 1/2. In this regime, the Euler-Kronecker constant is essentially a measure of the "vertical" density of these zeros and the "horizontal" splitting of primes.

The technical analysis in arXiv:hal-00004434 demonstrates that the lower bound of -1 for the ratio is approached when a large number of small primes split completely in the field extension. For example, in quadratic fields Q(sqrt(-D)), forcing the primes 2, 3, 5, 7, and 11 to split increases the negative contribution to γ_K. The paper provides explicit calculations for complex quadratic fields with large products of primes, such as Q(sqrt(-7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 79)), where the ratio γ_K/α_K is found to be approximately -0.1727.

The Role of Asymptotic Densities

The paper utilizes the Mestre-type inequalities to bound the functional involving phi_q (the density of primes of norm q). It is shown that the maximum of the functional is attained when phi_p^m = 0 for m > 1. This implies that the most significant downward pressure on the Euler-Kronecker constant comes from the first-degree prime ideals. The case where phi₇ = 2 * phi_C illustrates a specific configuration where the complex embeddings and the prime 7 contribute equally to the bound, highlighting a delicate balance between arithmetic and analytic invariants.

Furthermore, the comparison with the function field case over F_q is instructive. In that setting, the Riemann Hypothesis is a theorem, and the ratio γ_K/α_K is known to be bounded by the number of rational points on the corresponding curve. The number field results in arXiv:hal-00004434 mirror these geometric bounds, suggesting a unified theory of "arithmetic capacity" that limits how negative γ_K can become relative to the discriminant.

Novel Research Pathways

1. Stability of the Critical Line under Extremal Ratios

A promising research direction is to investigate whether a sequence of fields K_n that approaches the theoretical lower bound liminf γ_K/α_K = -1 must necessarily satisfy the Riemann Hypothesis for its low-lying zeros. This would involve proving a stability theorem: if the ratio is close to -1, then the zeros rho = beta + i*gamma must have beta values extremely close to 1/2. This could provide a way to verify the GRH for specific families of fields by calculating their Euler-Kronecker constants.

2. Dual Certificates for Zero-Free Regions

The constants a_q and b_q can be viewed as coefficients of a test function in the explicit formula. One could research the construction of "optimal" test functions that yield the sharpest possible lower bounds for γ_K/α_K. These functions would serve as dual certificates for zero-free regions near s = 1. By refining the Mestre inequalities mentioned in arXiv:hal-00004434, one might discover new constraints on the existence of Siegel zeros in quadratic extensions.

Computational Implementation

Wolfram Language

(* Section: Euler-Kronecker Ratio and Zeta Zero Analysis *)
(* Purpose: Calculate normalized gamma values and visualize critical line zeros *)

Module[{
  splitPrimes, alphaK, gammaK, ratio, zeros, p, 
  a0, aqFunction, testD
},
  (* Constants from hal-00004434 *)
  a0 = Log[Sqrt[8*Pi]] + Pi/4 + EulerGamma/2;
  aqFunction[q_] := Log[q]/(Sqrt[q] - 1);
  
  (* Define a set of small split primes for a hypothetical quadratic field *)
  splitPrimes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29};
  testD = Product[p, {p, splitPrimes}];
  
  (* Approximation of alpha_K = log(sqrt(D)) *)
  alphaK = 0.5 * Log[testD];
  
  (* Estimating gamma_K based on the sum of a_q for split primes *)
  (* This models the downward pressure on the constant *)
  gammaK = a0 - Total[Table[aqFunction[p], {p, splitPrimes}]];
  ratio = gammaK / alphaK;
  
  Print["Estimated alpha_K: ", alphaK];
  Print["Estimated gamma_K: ", gammaK];
  Print["Normalized Ratio (gamma_K/alpha_K): ", ratio];
  
  (* Visualize the distribution of the first 40 Zeta Zeros on the critical line *)
  zeros = Table[Im[ZetaZero[n]], {n, 1, 40}];
  Print[ListLinePlot[zeros, 
    PlotLabel -> "Imaginary Parts of Riemann Zeta Zeros", 
    AxesLabel -> {"n", "Im(rho)"}, 
    PlotStyle -> Thick, 
    GridLines -> Automatic]];
  
  (* Plotting the Zeta magnitude to show the critical line behavior *)
  Print[Plot[Abs[Zeta[1/2 + I*t]], {t, 0, 60}, 
    PlotLabel -> "|Zeta(1/2 + it)| Magnitude", 
    AxesLabel -> {"t", "|Zeta|"}, 
    ColorFunction -> "Rainbow"]];
]

Conclusions

The investigation into the Euler-Kronecker constant provided by arXiv:hal-00004434 offers a compelling arithmetic window into the Riemann Hypothesis. By establishing that the ratio γ_K/α_K is bounded between 0 and -1, the paper reinforces the idea that the distribution of primes and the distribution of zeros are two sides of the same coin. The most promising avenue for future work lies in the construction of extremal field families that test the limits of these bounds. Such research could eventually lead to a proof that the non-existence of zeros far from the critical line is a necessary condition for the observed bounds on γ_K.

References

arXiv:hal-00004434: On the Euler-Kronecker constant of a number field.
Ihara, Y. (2006). On the Euler-Kronecker constants of global fields and L-functions.
Zykin, A. (2010). On the asymptotic properties of zeta functions of number fields.
Mestre, J.-F. (1986). Formules explicites et minorations de discriminants.