Investigation about a statement equivalent to Riemann Hypothesis (RH) applied to Dirichlet primitive L functions

TL;DR

The paper proposes a RH-equivalence framework for Dirichlet primitive L-functions by analyzing the phase of the companion function ξ(s,χ) through an angular-momentum analogy. It builds a scaffold of lemmas relating ε- and t-variations, and leverages an Euler-product-based treatment of phase variations to argue that GRH holds for odd primitive L-functions for all t and q, up to a finite exceptional t-interval T_Asymp(α). It further extends the perspective toward all t for odd characters (and notes restricted applicability for even characters) in a subsequent section, suggesting a pathway to a full GRH proof contingent on controlling the analytic behavior near critical zeros. Overall, the work blends phase-analysis, Euler-product arguments, and asymptotic gamma-factor behavior to derive positivity properties that support RH-type conclusions for primitive L-functions, with explicit caveats and avenues for further rigorous validation.

Abstract

We try to apply a known equivalence, for RH about Riemann Z function, to Dirichlet L functions with primitive characters. The aim is to give a small contribution to the proof of the generalized version of Riemann Hypothesis (RH).

Figures (19)

• Figure 1: Derivative $\frac{\partial}{\partial \epsilon}$ of (\ref{['DerAngleDeT']}). With $\alpha=0$ even, $\alpha=1$ odd characters. The point of crossing with horizontal axis in case $\alpha=0$ is: $0.585 <t_{cross}<0.588$. The curves are independent from congruence modulus $q$.
• Figure 2: Plot of (\ref{['DerAngleDeT']}). For $q'>3$ we must shift the curve by the quantity $\frac{\ln(q'/\pi)-\ln(3/\pi)}{2}$. For $q \ge 11$ the curve is positive $\forall t$.Notice that the asymptotic result (\ref{['Asympt']} ) is well verified at least from $|t|>3$. Here $T_{Asymp}(\alpha) \approx 3$.
• Figure 3: The peaks symmetries of (\ref{['DerLAngleSuTL']} ) with $\epsilon=0$, respect to real axis, in not principal character case $q =5$ ( see tab. \ref{['tab1']}) , with $p_{max}=p^*=158 \times 10^6$ ( here are used the first 8868881 primes in (\ref{['DerLAngleSuTL']} ) ).In wolfram demostration project - Dirichlet L-Functions and their zeros we can compute, by zeroing real and imaginary parts, the zeros closer to $t=0$ of what here are called X1 and X3, the result is: $\pm9.443$, $\pm8.457$, $\pm6.184, \ \ \pm 4.133 \ $. Very comparable with results in figure.
• Figure 4: The peaks symmetries of (\ref{['DerLAngleSuTL']} ) with $\epsilon=0$, respect to real axis, in character case $q =5$, with $p_{max}=(p^*)^{3}=45.8 \times 10^6$ ( here are used the first 2763823 primes in (\ref{['DerLAngleSuTL']} ) ).The peaks are lower and fatter with respect to fig. \ref{['Simmetries']}, but, the inter-peaks behavior is much more close to $\ln\left( \sqrt{\frac{q t }{2 \pi}} \right)$, or, better, to (\ref{['DerAngleDeT']}), that follows different behavior close to $t=0$ if $\alpha=0$ (even chrs) or $\alpha=1$ (odd chrs).
• Figure 5: The peaks symmetries of (\ref{['DerLAngleSuTL']} ) with $\epsilon=0$, respect to real axis, in character case $q =3$ and $q =4$, with $p_{max}=(p^*)^{3}=158 \times 10^6$ ( here are used the first 8868881 primes).For $q=3$, and, $\chi_1(n)=(0,1,-1)$ or $\angle[\chi_1(n)]=(undefined, 0 , \pi)$ to comply with format of tab. \ref{['tab1']}. For $q=4$, and $\chi_1(n)=(0,1,0,-1)$, or, $\angle[\chi_1(n)]=(undefined, 0 , undefined, \pi)$ to comply with format of tab. \ref{['tab1']}. Notice close to $t=0$ the characteristic behavior of odd primitive L-functions. Compare with fig. \ref{['Q3']}. Until now the only example of even primitive L-function is phase variation of X2Q5 in fig. \ref{['Simmetries']}, and fig. \ref{['Q5-3Camp']}. In this case compare with fig. \ref{['Q5']}.