In-Depth Analysis: Trivial Zeros and the Riemann Hypothesis Statement

1. Overview of the Equations

Equation 15 — The Trivial Zeros

$$\zeta(-2n) = 0, \quad n = 1, 2, 3, \ldots$$

That is, the Riemann zeta function vanishes at $s = -2, -4, -6, -8, \ldots$ These are called the trivial zeros because their existence follows immediately ("trivially") from the functional equation.

Equation 16 — The Riemann Hypothesis (RH)

$$\zeta(\rho) = 0, \; \rho \text{ non-trivial} \implies \operatorname{Re}(\rho) = \frac{1}{2}$$

All non-trivial zeros of $\zeta(s)$ lie on the critical line $\sigma = \operatorname{Re}(s) = \frac{1}{2}$.

2. Historical Background

Bernhard Riemann (1826–1866)

Both statements originate in Riemann's single, revolutionary 8-page paper:

B. Riemann, "Über die Anzahl der Primzahlen unter einer gegebenen Größe" ("On the Number of Primes Less Than a Given Magnitude"), Monatsberichte der Berliner Akademie, November 1859.

This was Riemann's only paper on number theory, written for his induction into the Berlin Academy of Sciences. In it, Riemann:

1. Extended Euler's real-variable zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ to a complex variable $s = \sigma + it$, performing analytic continuation to all of $\mathbb{C}$ except a simple pole at $s = 1$.

2. Derived the functional equation: $$\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s)\, \zeta(1-s)$$

3. Observed (Equation 15's origin) that the factor $\sin(\pi s / 2)$ vanishes at $s = -2, -4, -6, \ldots$, forcing zeros there.

4. Stated (Equation 16's origin), almost as an aside, the famous conjecture:

"...es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen..."

("...it is very probable that all roots are real. One would certainly wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this...")

Riemann phrased it in terms of the roots of $\xi(t)$ being real, which is equivalent to the modern statement $\operatorname{Re}(\rho) = 1/2$.

Why the Trivial Zeros Are "Trivial"

In the functional equation, for $s = -2n$ ($n \geq 1$):

• $\sin(\pi s/2) = \sin(-n\pi) = 0$
• $\Gamma(1-s) = \Gamma(1+2n)$ is finite
• $\zeta(1-s) = \zeta(1+2n)$ is finite and nonzero

Hence $\zeta(-2n) = 0$. Note $s = 0$ is not a zero: the zero of $\sin(\pi s/2)$ is cancelled by the pole of $\zeta(1-s)$ at $s=0$, giving $\zeta(0) = -1/2$.

An elegant connection to Bernoulli numbers (due to Euler's work, made rigorous by Riemann's continuation):

$$\zeta(-n) = -\frac{B_{n+1}}{n+1}$$

Since odd-indexed Bernoulli numbers $B_3, B_5, B_7, \ldots$ all vanish, we get $\zeta(-2n) = 0$.

Key Historical Milestones on Equation 16

3. Structure of the Zero Set (the logic of Equation 16)

The precise logical content, as your notes state:

1. Euler product $\zeta(s) = \prod_p (1 - p^{-s})^{-1}$ converges for $\sigma > 1$, and a convergent product of nonzero factors is nonzero ⟹ no zeros in $\sigma > 1$.
2. Functional equation maps $\sigma > 1$ to $\sigma < 0$; the only zeros created there come from $\sin(\pi s/2)$ ⟹ zeros in $\sigma < 0$ are exactly $s = -2, -4, -6, \ldots$
3. All remaining zeros lie in the critical strip $0 \leq \sigma \leq 1$.
4. Symmetries: zeros come in quadruples $$\{\rho,\; 1-\rho,\; \bar{\rho},\; 1-\bar{\rho}\}$$ from $s \mapsto 1-s$ (functional equation) and $s \mapsto \bar{s}$ (real Dirichlet coefficients, i.e., $\overline{\zeta(s)} = \zeta(\bar{s})$). The quadruple collapses to a pair $\{\rho, \bar\rho\}$ if and only if $\operatorname{Re}(\rho) = 1/2$.

RH asserts this collapse always happens. This symmetry argument is why the critical line is so natural: it is the unique fixed line of the symmetry $s \mapsto 1 - \bar{s}$.

Critical caveat for your paper: No zero-free region of the form $\sigma < 1 - \varepsilon$ (fixed $\varepsilon > 0$) is known. The best known zero-free regions (Vinogradov–Korobov, 1958) shrink toward $\sigma = 1$ as $|t| \to \infty$: $$\sigma \geq 1 - \frac{c}{(\log|t|)^{2/3}(\log\log|t|)^{1/3}}$$

4. Key Research Papers and Links

1. Riemann's original 1859 paper (German original + English translation by Wilkins): https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf (Clay Mathematics Institute hosts the translation)

2. Clay Millennium Problem — Official Problem Description by E. Bombieri: https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf

3. G. H. Hardy (1914), "Sur les zéros de la fonction $\zeta(s)$ de Riemann," Comptes Rendus de l'Académie des Sciences 158, 1012–1014. (Infinitely many zeros on the line.)

4. A. Selberg (1942), "On the zeros of Riemann's zeta-function," Skr. Norske Vid. Akad. Oslo I 10, 1–59.

5. N. Levinson (1974), "More than one third of zeros of Riemann's zeta-function are on σ = 1/2," Advances in Mathematics 13, 383–436. https://doi.org/10.1016/0001-8708(74)90074-7

6. J. B. Conrey (1989), "More than two fifths of the zeros of the Riemann zeta function are on the critical line," J. reine angew. Math. 399, 1–26. https://www.degruyter.com/document/doi/10.1515/crll.1989.399.1/html

7. Bui, Conrey, Young (2011), "More than 41% of the zeros of the zeta function are on the critical line," Acta Arithmetica 150, 35–64. arXiv: https://arxiv.org/abs/1002.4127

8. D. Platt & T. Trudgian (2021), "The Riemann hypothesis is true up to $3 \cdot 10^{12}$," Bulletin of the LMS 53(3), 792–797. arXiv: https://arxiv.org/abs/2004.09765

9. Survey: J. B. Conrey (2003), "The Riemann Hypothesis," Notices of the AMS 50(3), 341–353. https://www.ams.org/notices/200303/fea-conrey-web.pdf

10. Book reference: H. M. Edwards, Riemann's Zeta Function (Dover, 2001) — contains a full translation and line-by-line analysis of the 1859 paper.

5. Computational Examples in Python

Install dependency: pip install mpmath matplotlib numpy

Example 1: Verifying the Trivial Zeros (Equation 15)

from mpmath import mp, zeta, bernoulli, mpf

mp.dps = 30  # 30 decimal places of precision

print("=== Equation 15: Trivial zeros zeta(-2n) = 0 ===\n")
print(f"{'s':>6} | {'zeta(s)':>35}")
print("-" * 45)

for n in range(0, 7):
    s = -2 * n
    val = zeta(s)
    print(f"{s:>6} | {mp.nstr(val, 25):>35}")

# Note: zeta(0) = -1/2 is NOT a zero (pole/zero cancellation)

print("\n=== Cross-check via Bernoulli numbers: zeta(-n) = -B_{n+1}/(n+1) ===\n")
for n in range(1, 10):
    lhs = zeta(-n)
    rhs = -bernoulli(n + 1) / (n + 1)
    match = abs(lhs - rhs) < mpf('1e-25')
    print(f"zeta(-{n}) = {mp.nstr(lhs, 12):>18}   "
          f"-B_{n+1}/{n+1} = {mp.nstr(rhs, 12):>18}   match: {match}")

Expected output: zeta(-2), zeta(-4), ... all print 0.0, while odd negative integers give nonzero rational-related values (e.g., zeta(-1) = -1/12).

Example 2: Locating Non-Trivial Zeros on the Critical Line (Equation 16)

from mpmath import mp, zetazero, zeta

mp.dps = 25

print("=== First 10 non-trivial zeros (via mpmath.zetazero) ===\n")
print(f"{'n':>3} | {'Re(rho)':>10} | {'Im(rho)':>28} | |zeta(rho)|")
print("-" * 70)

for n in range(1, 11):
    rho = zetazero(n)          # n-th zero in the upper half-plane
    residual = abs(zeta(rho))  # should be ~0
    print(f"{n:>3} | {mp.nstr(rho.real, 8):>10} | "
          f"{mp.nstr(rho.imag, 22):>28} | {mp.nstr(residual, 3)}")

Expected output: every zero has Re(rho) = 0.5 exactly, with imaginary parts 14.1347..., 21.0220..., 25.0108..., 30.4249..., 32.9350..., ... (these are the famous Gram/Titchmarsh values, first computed by Riemann himself and later by Gram in 1903).

Example 3: Verifying the Quadruple Symmetry {ρ, 1−ρ, ρ̄, 1−ρ̄}

from mpmath import mp, zeta, zetazero, mpc

mp.dps = 25

rho = zetazero(1)  # first zero: 0.5 + 14.1347...i
quadruple = [rho, 1 - rho, rho.conjugate(), (1 - rho).conjugate()]

print("=== Quadruple symmetry check for the first zero ===\n")
for z in quadruple:
    print(f"s = {mp.nstr(z, 15):>40}   |zeta(s)| = {mp.nstr(abs(zeta(z)), 3)}")

# Since Re(rho) = 1/2, the quadruple collapses:
# 1 - rho == conj(rho)  and  conj(1 - rho) == rho
print("\nCollapse check (RH case, Re = 1/2):")
print("1 - rho == conj(rho)? ->", mp.nstr(abs((1 - rho) - rho.conjugate()), 3))

Example 4: Visualizing |ζ(1/2 + it)| — Zeros as Dips to Zero

import numpy as np
import matplotlib.pyplot as plt
from mpmath import mp, zeta

mp.dps = 15
t_vals = np.linspace(0, 60, 2500)
abs_zeta = [float(abs(zeta(complex(0.5, t)))) for t in t_vals]

known_zeros = [14.134725, 21.022040, 25.010858, 30.424876,
               32.935062, 37.586178, 40.918719, 43.327073,
               48.005151, 49.773832]

plt.figure(figsize=(12, 5))
plt.plot(t_vals, abs_zeta, 'b-', lw=1)
for tz in known_zeros:
    plt.axvline(tz, color='red', ls='--', alpha=0.4)
plt.xlabel('t')
plt.ylabel('|ζ(1/2 + it)|')
plt.title('|ζ(s)| on the critical line — zeros (red) are where it touches 0')
plt.grid(alpha=0.3)
plt.tight_layout()
plt.savefig('critical_line.png', dpi=150)
plt.show()

Example 5: Counting Zeros with the Riemann–von Mangoldt Formula

The number of zeros with $0 < \operatorname{Im}(\rho) \leq T$ satisfies:

$$N(T) = \frac{T}{2\pi}\log\frac{T}{2\pi} - \frac{T}{2\pi} + \frac{7}{8} + O(\log T)$$

(Stated by Riemann in 1859, proved by von Mangoldt in 1905.)

from mpmath import mp, zetazero, log, pi, mpf

mp.dps = 20

def N_asymptotic(T):
    T = mpf(T)
    return T / (2 * pi) * log(T / (2 * pi)) - T / (2 * pi) + mpf(7) / 8

print("=== Riemann–von Mangoldt zero-counting formula ===\n")
print(f"{'T':>6} | {'Actual N(T)':>12} | {'Formula':>12} | {'Error':>8}")
print("-" * 50)

for T in [50, 100, 200, 500]:
    # count actual zeros with imaginary part <= T
    n, count = 1, 0
    while True:
        if zetazero(n).imag > T:
            break
        count += 1
        n += 1
    approx = N_asymptotic(T)
    print(f"{T:>6} | {count:>12} | {mp.nstr(approx, 8):>12} | "
          f"{mp.nstr(abs(count - approx), 3):>8}")

Expected output: e.g., N(100) = 29 actual zeros vs. formula ≈ 28.13 — the error stays $O(\log T)$, exactly as the theorem predicts.

6. Suggested Framing for Your Paper

• Equation 15 is a theorem — a direct, complete consequence of the functional equation (Riemann 1859, made fully rigorous by Hadamard's 1893 theory of entire functions).
• Equation 16 is a conjecture — the deepest open problem in mathematics, unresolved for 165+ years despite: infinitely many zeros proven on the line (Hardy 1914), ≥41% proven on the line (Bui–Conrey–Young 2011), and numerical verification through $|t| \leq 3 \times 10^{12}$ (Platt–Trudgian 2021).
• The structural gap to emphasize: all known results are "positive proportion on the line" or "finite-height verification" — nobody has excluded zeros from any fixed vertical strip $\sigma \in (1/2, 1)$. This is the barrier any proof must break, and it is a powerful organizing theme for a research paper.