The Dirichlet Series Definition of the Riemann Zeta Function: An In-Depth Analysis

$$\zeta(s) = \sum_{n=1}^{\infty} n^{-s}, \qquad s = \sigma + it$$

1. Historical Origins

1.1 Before Riemann: Euler's Foundational Work (1730s–1740s)

Although the equation bears Riemann's name, the series itself was studied intensively by Leonhard Euler (1707–1783) more than a century before Riemann. Euler treated the series only for real values of $s$.

• 1735 — The Basel Problem. Euler achieved international fame by solving the famous Basel Problem (posed by Pietro Mengoli in 1650 and attacked unsuccessfully by the Bernoullis), showing:

$$\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

• 1737 — The Euler Product. In his paper "Variae observationes circa series infinitas" (presented to the St. Petersburg Academy), Euler proved the profound identity connecting the series to the prime numbers:

$$\sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \qquad \sigma > 1$$

This is the true birth of analytic number theory: the zeta function became a bridge between analysis (infinite series) and arithmetic (prime numbers). As a corollary, Euler gave a new proof of the infinitude of primes, and even showed $\sum_p 1/p$ diverges.

• Euler also computed $\zeta(2k)$ for all positive even integers in terms of Bernoulli numbers:

$$\zeta(2k) = \frac{(-1)^{k+1} B_{2k} (2\pi)^{2k}}{2(2k)!}$$

1.2 Dirichlet (1837): Why It's Called a "Dirichlet Series"

Peter Gustav Lejeune Dirichlet (1805–1859) generalized the series to what we now call Dirichlet series:

$$\sum_{n=1}^{\infty} \frac{a_n}{n^s}$$

In his landmark 1837 paper proving that every arithmetic progression $a, a+d, a+2d, \dots$ with $\gcd(a,d)=1$ contains infinitely many primes, Dirichlet introduced Dirichlet L-functions $L(s,\chi) = \sum \chi(n) n^{-s}$. The zeta function is the special case $\chi \equiv 1$. Dirichlet, like Euler, worked with real $s$ only.

1.3 Riemann (1859): The Revolutionary Step

Bernhard Riemann (1826–1866) published his only paper on number theory — a mere 8 pages — in November 1859, upon his election to the Berlin Academy of Sciences:

"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.

Riemann's decisive innovation was to treat $s$ as a complex variable $s = \sigma + it$. In this single paper he:

1. Defined $\zeta(s)$ by the Dirichlet series for $\text{Re}(s) > 1$ and by the Euler product.
2. Proved the analytic continuation of $\zeta(s)$ to all of $\mathbb{C}$ except a simple pole at $s = 1$.
3. Derived the functional equation relating $\zeta(s)$ and $\zeta(1-s)$.
4. Connected the zeros of $\zeta$ to the distribution of primes via an explicit formula.
5. Stated, almost in passing, the Riemann Hypothesis: all non-trivial zeros lie on the critical line $\sigma = 1/2$ — remarking "it is very probable... Certainly one would wish for a stricter proof here; I have though, after some fleeting futile attempts, provisionally put aside the search for this."

The notation $s = \sigma + it$ used in your equation is itself Riemann's original notation, preserved by tradition ever since — a rare case in mathematics where 19th-century notation survives unchanged.

2. Mathematical Analysis of the Three Steps

2.1 Step 1 — Absolute Convergence for $\sigma > 1$

The key observation is that the modulus of each term depends only on the real part of $s$:

$$|n^{-s}| = |e^{-s \ln n}| = e^{-\sigma \ln n} = n^{-\sigma}$$

(The factor $e^{-it\ln n}$ lies on the unit circle and contributes nothing to the modulus.) The integral comparison test then bounds the partial sums:

$$\sum_{n=1}^N n^{-\sigma} \le 1 + \int_1^N x^{-\sigma}\,dx = 1 + \frac{1 - N^{1-\sigma}}{\sigma - 1} \xrightarrow{N\to\infty} 1 + \frac{1}{\sigma - 1}$$

Note the bound $\zeta(\sigma) \le \frac{\sigma}{\sigma-1}$, which shows explicitly how the series "blows up" like $\frac{1}{\sigma - 1}$ as $\sigma \to 1^+$ — foreshadowing the simple pole at $s=1$ with residue 1:

$$\lim_{s \to 1} (s-1)\zeta(s) = 1$$

2.2 Step 2 — Uniform Convergence and Analyticity

On any half-plane $\sigma \ge 1 + \delta$ ($\delta > 0$), the Weierstrass M-test applies with $M_n = n^{-(1+\delta)}$, since $|n^{-s}| \le n^{-(1+\delta)}$ uniformly and $\sum M_n < \infty$.

The conclusion of analyticity rests on a classical theorem of Weierstrass (sometimes called the Weierstrass convergence theorem): a locally uniform limit of holomorphic functions is holomorphic, and the series may be differentiated term by term. Hence for $\sigma > 1$:

$$\zeta'(s) = -\sum_{n=1}^{\infty} \frac{\ln n}{n^s}$$

Important subtlety: the convergence is not uniform on the open half-plane $\sigma > 1$ itself — only on closed sub-half-planes. This is why the $\delta$ is essential.

2.3 Step 3 — Divergence at $s = 1$ (Harmonic Series)

The block-grouping argument (Cauchy condensation, but historically due to Nicole Oresme, c. 1350 — one of the oldest results in analysis):

$$\underbrace{\frac{1}{2}}_{\ge 1/2} + \underbrace{\frac{1}{3} + \frac{1}{4}}_{\ge 1/2} + \underbrace{\frac{1}{5} + \cdots + \frac{1}{8}}_{\ge 1/2} + \cdots$$

Each dyadic block $\left(\frac{1}{2^k+1} + \cdots + \frac{1}{2^{k+1}}\right)$ contains $2^k$ terms each $\ge 2^{-(k+1)}$, hence sums to $\ge \frac{1}{2}$. The partial sums thus grow like $\frac{1}{2}\log_2 N$, i.e., logarithmically — consistent with the more precise asymptotic:

$$\sum_{n \le N} \frac{1}{n} = \ln N + \gamma + O(1/N)$$

where $\gamma \approx 0.5772$ is the Euler–Mascheroni constant. In fact, the Laurent expansion of $\zeta$ at its pole is:

$$\zeta(s) = \frac{1}{s-1} + \gamma + \sum_{k=1}^{\infty} \frac{(-1)^k \gamma_k}{k!}(s-1)^k$$

where $\gamma_k$ are the Stieltjes constants.

2.4 What Lies Beyond: Analytic Continuation

The Dirichlet series is only the starting definition. The zeros relevant to the Riemann Hypothesis lie in the critical strip $0 \le \sigma \le 1$, where the series diverges. The function there is defined by analytic continuation, e.g., via the alternating (Dirichlet eta) series, valid for $\sigma > 0$:

$$\zeta(s) = \frac{1}{1 - 2^{1-s}} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s}$$

or Riemann's own contour-integral / theta-function methods, leading to the functional equation:

$$\zeta(s) = 2^s \pi^{s-1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma(1-s)\, \zeta(1-s)$$

This is a critical point for your paper: a common misconception is that RH concerns zeros of the series itself — it does not; the series never vanishes (and doesn't even converge) where the non-trivial zeros live.

3. Connection to the Riemann Hypothesis

• Euler product ⟹ no zeros for $\sigma > 1$ (a convergent product of nonzero factors is nonzero).
• Hadamard & de la Vallée Poussin (1896) independently proved $\zeta(1+it) \ne 0$, which suffices to prove the Prime Number Theorem: $\pi(x) \sim x/\ln x$.
• RH is equivalent to the optimal error term (von Koch, 1901):

$$\pi(x) = \operatorname{Li}(x) + O\!\left(\sqrt{x}\,\ln x\right)$$

• Hardy (1914) proved infinitely many zeros lie on the critical line; Levinson (1974) got more than 1/3 of zeros; Conrey (1989) more than 2/5; Pratt–Robles–Zaharescu–Zeindler (2019) pushed past 41.7%.

4. Key Papers and References (with Links)

Primary Sources

1. Riemann, B. (1859), "Über die Anzahl der Primzahlen unter einer gegebenen Größe." - Original German scan and English translation hosted by the Clay Mathematics Institute: https://www.claymath.org/collections/riemanns-1859-manuscript/ - English translation (D. R. Wilkins): https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf

2. Euler, L. (1737), "Variae observationes circa series infinitas" — the Euler product. - The Euler Archive (E072): https://scholarlycommons.pacific.edu/euler-works/72/

3. Euler, L. (1740), "De summis serierum reciprocarum" (Basel problem, E041): - https://scholarlycommons.pacific.edu/euler-works/41/

Landmark Research Papers

4. Hardy, G. H. (1914), "Sur les zéros de la fonction ζ(s) de Riemann," Comptes Rendus 158, 1012–1014 — infinitely many zeros on the critical line.

5. Conrey, J. B. (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

6. Bombieri, E. (2000), "Problems of the Millennium: The Riemann Hypothesis" — official Clay Institute problem statement (excellent survey): - https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf

7. Conrey, J. B. (2003), "The Riemann Hypothesis," Notices of the AMS 50(3), 341–353 — highly readable survey, ideal for a research paper: - https://www.ams.org/notices/200303/fea-conrey-web.pdf

8. Platt, D. & Trudgian, T. (2021), "The Riemann hypothesis is true up to 3·10¹²," Bull. LMS 53(3) — state-of-the-art numerical verification: - arXiv preprint: https://arxiv.org/abs/2004.09765

Standard Books

9. Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, 2nd ed. (revised by D. R. Heath-Brown), Oxford, 1986 — the technical reference.
10. Edwards, H. M., Riemann's Zeta Function, Dover, 2001 — includes a full annotated translation of Riemann's 1859 paper; strongly recommended for the historical portion of your paper.
11. Ivić, A., The Riemann Zeta-Function: Theory and Applications, Dover, 2003.

5. Numerical Illustration in Python

Since you may want computational examples in your paper:

import mpmath as mp

mp.mp.dps = 30  # 30 decimal digits

# Verify Euler: zeta(2) = pi^2/6
print(mp.zeta(2))            # 1.6449340668...
print(mp.pi**2 / 6)          # 1.6449340668...

# Behavior of the partial Dirichlet series near the pole s = 1
for sigma in [2, 1.5, 1.1, 1.01, 1.001]:
    print(f"sigma={sigma:6}:  zeta = {mp.zeta(sigma)},  1/(s-1) = {1/(sigma-1)}")

# First non-trivial zero on the critical line (found via mpmath)
rho1 = mp.zetazero(1)
print(rho1)                  # 0.5 + 14.1347251417...j
print(abs(mp.zeta(rho1)))    # ~ 1e-30  (numerically zero)

6. Summary for Your Paper

If you'd like, I can provide the same in-depth treatment for the natural follow-up equations — the Euler product, the functional equation, the Riemann–Siegel formula, or Riemann's explicit formula for $\pi(x)$ — each of which would strengthen your paper's progression from definition to hypothesis.