21.12.01 · number-theory / prime-number-theorem

The von Mangoldt Function, the Chebyshev Psi Function, and the Logarithmic Derivative of Zeta

shipped3 tiersLean: none

Anchor (Master): Davenport 2000 *Multiplicative Number Theory* 3e §12-17 (the Hadamard product for $\xi(s)$, the partial-fraction expansion of $\zeta'/\zeta$, the explicit formula for $\psi(x)$ and the PNT contour proof); Riemann 1859 *Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse*; von Mangoldt 1895 *Crelle* 114 (the explicit formula); Edwards 1974 *Riemann's Zeta Function* (Academic Press) Ch. 3; Montgomery-Vaughan 2007 *Multiplicative Number Theory I* (Cambridge UP) Ch. 12

Intuition Beginner

The primes look random when you list them, but their counting function hides a startling regularity. To see it you first change what you count. Instead of giving each prime the weight $1$ , give the prime $p$ the weight $lo g p$ , and give each prime power $p, p^{2}, p^{3}, \dots$ that same weight $lo g p$ as well. Add these weights up to a cutoff $x$ and you get a running total called $ψ (x)$ . This weighted count grows almost exactly like $x$ itself, far more cleanly than the raw prime count does.

Why weight primes by their logarithm? Because logarithms turn the multiplicative world of factorisation into the additive world of sums, and the zeta function of Euler and Riemann is the bridge between the two. When you take the logarithm of Euler's product formula for zeta and then track how fast it changes, the prime-power weights $lo g p$ fall out automatically, one term for each prime power. So $ψ (x)$ is not an artificial bookkeeping device — it is exactly the quantity the zeta function knows how to compute.

Here is the payoff, which Riemann saw in 1859. The places where the zeta function equals zero act like the frequencies of a vibrating string. Each zero contributes a small oscillating wave to $ψ (x)$ , and when you add the main growth $x$ to all these waves, you recover the exact count of prime powers. The primes are not random; they are the sound of the zeros of one function.

Visual Beginner

Picture two staircases drawn on the same axes. The first is the weighted prime-power count $ψ (x)$ : it climbs from left to right, jumping by $lo g p$ each time $x$ passes a prime power. Laid over it is a smooth straight line of slope one, the line $y = x$ . The staircase hugs that line, wandering above and below by only a little. The small wandering is the combined effect of the zeros of the zeta function, each one adding a gentle wave.

$x$	$ψ (x)$ (approx.)	$ψ (x) / x$
$10$	$7.83$	$0.78$
$100$	$94.0$	$0.94$
$1000$	$993.0$	$0.99$
$10000$	$10013$	$1.00$

The ratio $ψ (x) / x$ creeps toward one. That single fact — that the weighted prime count is asymptotically just $x$ — is the prime number theorem, and the table is its numerical shadow.

Worked example Beginner

Build the weighted count by hand and watch the logarithmic derivative appear.

Step 1. List the prime powers up to $10$ : the powers of $2$ are $2, 4, 8$ ; the powers of $3$ are $3, 9$ ; then the single primes $5$ and $7$ . Each power of $2$ carries the weight $lo g 2$ , each power of $3$ carries $lo g 3$ , and $5$ and $7$ carry $lo g 5$ and $lo g 7$ .

Step 2. Add the weights: $$ \psi(10) = 3\log 2 + 2\log 3 + \log 5 + \log 7 = \log!\big(2^3 \cdot 3^2 \cdot 5 \cdot 7\big) = \log 2520 \approx 7.83. $$ So the weighted count at $10$ is $7.83$ , and the ratio to $x = 10$ is $0.78$ — already in the right ballpark.

Step 3. See where the weights come from. Euler's product writes zeta as a multiplication over primes. Taking the logarithm turns the product into a sum, and asking how fast that sum changes term by term hands back exactly the weight $lo g p$ attached to each prime power $p^{m}$ . The list of weights you just summed is the same list that the derivative of $lo g ζ$ produces.

What this tells us: $ψ (x)$ is the logarithm of the product of all prime powers up to $x$ , and this same list of weights is what you read off the zeta function by differentiating its logarithm. The weighted count and the zeta function are two views of one object.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $s = σ + i t$ is a complex variable, $p$ ranges over primes, $lo g$ is the natural logarithm, and $ζ$ is the Riemann zeta function of 21.03.01 and 06.01.16, with its Euler product $ζ (s) = \prod_{p} (1 - p^{- s})^{- 1}$ valid for $σ > 1$ . We follow Davenport ^{[Davenport §12]}.

Definition (von Mangoldt function). The von Mangoldt function $Λ : N \to R$ is $$ \Lambda(n) = \begin{cases} \log p & \text{if } n = p^m \text{ for a prime } p \text{ and integer } m \geq 1,\ 0 & \text{otherwise.}\end{cases} $$ It satisfies the convolution identity $lo g = Λ * 1$ , i.e. $\sum_{d ∣ n} Λ (d) = lo g n$ , established in 21.11.03.

Definition (Chebyshev psi function). For real $x \geq 1$ , the Chebyshev psi function is the summatory function of $Λ$ , $$ \psi(x) = \sum_{n \leq x} \Lambda(n) = \sum_{p^m \leq x} \log p. $$ It is the prime-counting weight natural to the analytic theory: the prime number theorem reads $ψ (x) \sim x$ , equivalent to $π (x) \sim x / lo g x$ by 21.11.03.

Definition (logarithmic derivative of zeta). For $σ > 1$ , where $ζ (s) \neq = 0$ , the logarithmic derivative is the meromorphic function $ζ^{'} (s) / ζ (s) = \frac{d}{d s} lo g ζ (s)$ . The central object of this unit is its negative, $- ζ^{'} (s) / ζ (s)$ .

The notation $Λ (n)$ , the summatory $ψ (x)$ , the variable $s = σ + i t$ , the critical zeros $ρ$ , and the symbol $Γ^{'} /Γ$ for the digamma function are recorded in _meta/NOTATION.md. A critical zero of $ζ$ is a zero $ρ$ with $0 < Re (ρ) < 1$ ; the elementary zeros are the points $s = - 2, - 4, - 6, \dots$ forced by the functional equation of 06.01.16.

Counterexamples to common slips

" $- ζ^{'} / ζ$ is entire because $ζ$ has a pole, not a zero, at $s = 1$ ." The logarithmic derivative has a pole wherever $ζ$ has either a zero or a pole. The simple pole of $ζ$ at $s = 1$ produces a simple pole of $ζ^{'} / ζ$ with residue $- 1$ , hence a pole of $- ζ^{'} / ζ$ with residue $+ 1$ . This $+ 1$ is the main term $x$ in the explicit formula.
"The Dirichlet series $\sum Λ (n) n^{- s}$ converges in the critical strip." It converges only for $σ > 1$ . The values of $- ζ^{'} / ζ$ inside the strip come from meromorphic continuation, not from the series; the contour proof of the prime number theorem moves the integration line left of $σ = 1$ precisely to access the continuation.
"Each critical zero contributes $lo g ρ$ to $ψ (x)$ ." Each simple zero $ρ$ contributes the term $- x^{ρ} / ρ$ to the explicit formula, an oscillating wave of size $x^{Re ρ}$ . The residue of $- ζ^{'} / ζ$ at a zero of multiplicity $m$ is $- m$ , not a logarithm.

Key theorem with proof Intermediate+

The signature identity of the unit expresses the logarithmic derivative of zeta as the Dirichlet series generated by the von Mangoldt function. It is the analytic bridge: the left side is a single complex-analytic object carrying the zeros and the pole of $ζ$ ; the right side is the prime-power weight whose partial sums are $ψ (x)$ .

Theorem (logarithmic derivative as a von Mangoldt series). For $Re (s) > 1$ , $$ -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^{s}}. $$

Proof. Fix $s$ with $σ = Re (s) > 1$ . By the Euler product of 21.03.01, $ζ (s) = \prod_{p} (1 - p^{- s})^{- 1}$ , with the product absolutely convergent and every factor non-zero, so $ζ (s) \neq = 0$ and $lo g ζ (s)$ is defined by the absolutely convergent sum $$ \log\zeta(s) = -\sum_p \log!\big(1 - p^{-s}\big) = \sum_p \sum_{m=1}^{\infty} \frac{p^{-ms}}{m}, $$ using the power series $- lo g (1 - z) = \sum_{m \geq 1} z^{m} / m$ for $∣ z ∣ = p^{- σ} < 1$ . Differentiating term by term — justified because the differentiated series $\sum_{p} \sum_{m} (- lo g p) p^{- m s}$ converges absolutely and uniformly on compact subsets of $σ > 1$ , where it is dominated by $\sum_{n} (lo g n) n^{- σ} < \infty$ — gives $$ \frac{\zeta'(s)}{\zeta(s)} = \frac{d}{ds}\log\zeta(s) = \sum_p \sum_{m=1}^{\infty} \frac{-\log p}{m} \cdot m, p^{-ms} = -\sum_p \sum_{m=1}^{\infty} (\log p), p^{-ms}. $$ Now collect by the integer $n = p^{m}$ . The coefficient of $n^{- s}$ is $lo g p$ when $n = p^{m}$ is a prime power and $0$ otherwise — exactly $Λ (n)$ . Hence $ζ^{'} (s) / ζ (s) = - \sum_{n} Λ (n) n^{- s}$ , and negating yields the claim. $□$

Bridge. This identity builds toward the prime number theorem and appears again in 21.11.04 Perron's formula, where integrating $- ζ^{'} (s) / ζ (s) \cdot x^{s} / s$ along a vertical line recovers $ψ (x)$ as a sum of residues. The foundational reason the identity matters is that it converts a question about the zeros of one analytic function into a statement about prime counts: this is exactly the move from the elementary $ψ (x) ≍ x$ of 21.11.03 to the analytic $ψ (x) \sim x$ , the bridge being the residue at $s = 1$ , which contributes the main term $x$ , while the critical zeros contribute the lower-order waves. The central insight is that $- ζ^{'} / ζ$ has a simple pole of residue $+ 1$ at $s = 1$ and a simple pole of residue $- 1$ at each simple zero $ρ$ ; putting these together through Perron's formula generalises Chebyshev's two-sided bound into the precise asymptotic, and the whole construction is dual to the Euler product, which is where the von Mangoldt weights were born.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Starting from $- ζ^{'} (s) / ζ (s) = \sum_{n} Λ (n) n^{- s}$ and the Dirichlet series $1/ ζ (s) = \sum_{n} μ (n) n^{- s}$ for $σ > 1$ ( $μ$ the Möbius function), prove the convolution identity $Λ = μ * lo g$ , i.e. $Λ (n) = \sum_{d ∣ n} μ (d) lo g (n / d)$ .

Hint

Write $- ζ^{'} (s) = \sum_{n} (lo g n) n^{- s}$ and multiply the two Dirichlet series $1/ ζ (s)$ and $- ζ^{'} (s)$ . Dirichlet-series multiplication is Dirichlet convolution of coefficients.

Answer

Differentiating $ζ (s) = \sum_{n} n^{- s}$ termwise gives $ζ^{'} (s) = - \sum_{n} (lo g n) n^{- s}$ , so $- ζ^{'} (s) = \sum_{n} (lo g n) n^{- s}$ , the Dirichlet series of the arithmetic function $lo g$ . Then $$ -\frac{\zeta'(s)}{\zeta(s)} = \frac{1}{\zeta(s)} \cdot \big(-\zeta'(s)\big) = \Big(\sum_n \mu(n) n^{-s}\Big)\Big(\sum_n (\log n) n^{-s}\Big) = \sum_n \Big(\sum_{d \mid n} \mu(d) \log(n/d)\Big) n^{-s}, $$ since the product of two Dirichlet series has coefficients given by the Dirichlet convolution. Matching coefficients with $- ζ^{'} / ζ = \sum_{n} Λ (n) n^{- s}$ gives $Λ (n) = \sum_{d ∣ n} μ (d) lo g (n / d) = (μ * lo g) (n)$ . Rubric: full credit for identifying $- ζ^{'} \leftrightarrow lo g$ , $1/ ζ \leftrightarrow μ$ , and reading off the convolution. This is the Möbius inversion of $lo g = Λ * 1$ .

Exercise 4 (medium, symbolic).

Show that $- ζ^{'} (s) / ζ (s)$ has a simple pole at $s = 1$ with residue $+ 1$ .

Hint

Near $s = 1$ , write $ζ (s) = \frac{1}{s - 1} + a_{0} + \dots$ (a simple pole, residue $1$ ). Compute $ζ^{'} (s)$ and form $ζ^{'} / ζ$ .

Answer

By 06.01.16, $ζ$ has a simple pole at $s = 1$ with residue $1$ : locally $ζ (s) = (s - 1)^{- 1} + a_{0} + a_{1} (s - 1) + \dots$ . Then $ζ^{'} (s) = - (s - 1)^{- 2} + a_{1} + \dots$ . Dividing, $$ \frac{\zeta'(s)}{\zeta(s)} = \frac{-(s-1)^{-2} + O(1)}{(s-1)^{-1} + O(1)} = \frac{-(s-1)^{-2}\big(1 + O((s-1)^2)\big)}{(s-1)^{-1}\big(1 + O(s-1)\big)} = -\frac{1}{s-1}\big(1 + O(s-1)\big). $$ So $ζ^{'} / ζ$ has a simple pole at $s = 1$ with residue $- 1$ , and $- ζ^{'} / ζ$ has residue $+ 1$ . Rubric: full credit for the Laurent expansion of $ζ$ , the resulting $- (s - 1)^{- 1}$ leading term in $ζ^{'} / ζ$ , and the sign flip. The residue $+ 1$ is the source of the main term $x$ in the explicit formula.

Exercise 5 (medium, symbolic).

Let $ρ$ be a zero of $ζ$ of multiplicity $m \geq 1$ (so $ζ (s) = (s - ρ)^{m} h (s)$ with $h$ holomorphic, $h (ρ) \neq = 0$ ). Compute the residue of $- ζ^{'} (s) / ζ (s)$ at $s = ρ$ .

Hint

Take the logarithmic derivative of $ζ (s) = (s - ρ)^{m} h (s)$ using $lo g ζ = m lo g (s - ρ) + lo g h (s)$ .

Answer

From $ζ (s) = (s - ρ)^{m} h (s)$ with $h (ρ) \neq = 0$ , $$ \frac{\zeta'(s)}{\zeta(s)} = \frac{m}{s - \rho} + \frac{h'(s)}{h(s)}, $$ and $h^{'} / h$ is holomorphic at $ρ$ . So $ζ^{'} / ζ$ has a simple pole at $ρ$ with residue $m$ , and $- ζ^{'} / ζ$ has residue $- m$ . For a simple zero ( $m = 1$ ) the residue of $- ζ^{'} / ζ$ is $- 1$ . Rubric: full credit for the additive splitting of the logarithmic derivative and the residue $- m$ . In the explicit formula this residue produces the term $- x^{ρ} / ρ$ per simple zero (the $1/ ρ$ comes from the Perron kernel $x^{s} / s$ evaluated at $s = ρ$ ).

Exercise 6 (medium, symbolic).

Verify the identity $- ζ^{'} (s) / ζ (s) = \sum_{n} Λ (n) n^{- s}$ at $s = 2$ to leading order by summing the first few terms of the right side and comparing with the known value $ζ^{'} (2) / ζ (2) \approx - 0.5699$ .

Hint

Compute $\sum_{n \leq 9} Λ (n) 2^{- n}$ ... no: the sum is $\sum_{n} Λ (n) n^{- 2}$ . Use $Λ (2) = Λ (4) = Λ (8) = lo g 2$ , $Λ (3) = Λ (9) = lo g 3$ , $Λ (5) = lo g 5$ , $Λ (7) = lo g 7$ .

Answer

The right side is $\sum_{n} Λ (n) n^{- 2}$ . The nonzero terms up to $n = 9$ are $$ \frac{\log 2}{4} + \frac{\log 3}{9} + \frac{\log 2}{16} + \frac{\log 5}{25} + \frac{\log 7}{49} + \frac{\log 2}{64} + \frac{\log 3}{81}. $$ Numerically: $0.1733 + 0.1221 + 0.0433 + 0.0644 + 0.0397 + 0.0108 + 0.0136 \approx 0.4672$ . Continuing the tail ( $n = 11, 13, 16, \dots$ ) adds roughly $0.10$ , converging to $\approx 0.5699$ , which matches $- ζ^{'} (2) / ζ (2)$ . Rubric: full credit for correctly assigning $Λ$ to prime powers, summing, and observing convergence toward the stated value. The slow tail reflects that the series converges but not rapidly at $s = 2$ .

Exercise 7 (hard, symbolic).

Assume the partial-fraction expansion of the logarithmic derivative (proved in the Master tier), $$ \frac{\zeta'(s)}{\zeta(s)} = B - \frac{1}{s - 1} + \sum_\rho \Big(\frac{1}{s - \rho} + \frac{1}{\rho}\Big) - \frac{1}{2}\frac{\Gamma'}{\Gamma}\Big(\frac{s}{2} + 1\Big), $$ where $ρ$ runs over the critical zeros and $B$ is a constant. Deduce the Perron-style heuristic explicit formula $ψ (x) = x - \sum_{ρ} \frac{x ^{ρ}}{ρ} - \frac{ζ ^{'} ( 0 )}{ζ ( 0 )} - \frac{1}{2} lo g (1 - x^{- 2})$ by formally summing residues of $- \frac{ζ ^{'} ( s )}{ζ ( s )} \frac{x ^{s}}{s}$ over the poles.

Hint

Perron's formula gives $ψ (x) = \frac{1}{2 π i} \int_{(c)} - \frac{ζ ^{'} ( s )}{ζ ( s )} \frac{x ^{s}}{s} d s$ . Shift the contour left and pick up a residue at each pole of the integrand: $s = 1$ (from the pole of $- ζ^{'} / ζ$ ), each $s = ρ$ , $s = 0$ (from $1/ s$ ), and the elementary zeros $s = - 2, - 4, \dots$ .

Answer

The integrand $F (s) = - \frac{ζ ^{'} ( s )}{ζ ( s )} \frac{x ^{s}}{s}$ has poles at: (i) $s = 1$ , where $- ζ^{'} / ζ$ has residue $+ 1$ and $x^{s} / s = x$ , giving residue $x$ ; (ii) each critical zero $s = ρ$ , where $- ζ^{'} / ζ$ has residue $- 1$ (simple zero) and $x^{s} / s = x^{ρ} / ρ$ , giving residue $- x^{ρ} / ρ$ ; (iii) $s = 0$ , where $1/ s$ has a simple pole and $- ζ^{'} (s) / ζ (s)$ is holomorphic, giving residue $- ζ^{'} (0) / ζ (0) = - lo g (2 π)$ ; (iv) each elementary zero $s = - 2 k$ , where $- ζ^{'} / ζ$ has residue $- 1$ and $x^{s} / s = x^{- 2 k} / (- 2 k)$ , giving residue $\sum_{k \geq 1} x^{- 2 k} / (2 k) = - \frac{1}{2} lo g (1 - x^{- 2})$ . Summing all residues (the contour at $Re (s) \to - \infty$ contributes nothing for $x > 1$ ): $$ \psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \log(2\pi) - \frac{1}{2}\log!\big(1 - x^{-2}\big). $$ Rubric: full credit for correctly listing the four pole types with their residues; the value $ζ^{'} (0) / ζ (0) = lo g (2 π)$ and the elementary-zero geometric series $\sum_{k} x^{- 2 k} / (2 k)$ may be cited. The sum over $ρ$ is conditionally convergent and paired as $ρ, \overset{ρ}{ˉ}$ .

Exercise 8 (hard, symbolic).

Show that the prime number theorem $ψ (x) \sim x$ is equivalent to the statement that every critical zero $ρ$ satisfies $Re (ρ) < 1$ — that is, $ζ (1 + i t) \neq = 0$ for all real $t$ — given the explicit formula $ψ (x) = x - \sum_{ρ} x^{ρ} / ρ + O (1)$ .

Hint

A zero $ρ$ contributes a term of size $∣ x^{ρ} / ρ ∣ = x^{Re ρ} /∣ ρ ∣$ . Compare the size of the zero-sum with the main term $x$ . What does $Re ρ = 1$ do?

Answer

Each term $x^{ρ} / ρ$ has modulus $x^{Re ρ} /∣ ρ ∣$ . If every critical zero has $Re (ρ) = β < 1$ , then $∣ x^{ρ} / ρ ∣ = x^{β} /∣ ρ ∣ = o (x)$ for each fixed $ρ$ , and (with the standard zero-density and convergence bookkeeping) the full sum $\sum_{ρ} x^{ρ} / ρ = o (x)$ , so $ψ (x) = x + o (x) \sim x$ . Conversely, suppose some zero had $Re (ρ_{0}) = 1$ , i.e. $ρ_{0} = 1 + i γ_{0}$ . Its term $x^{ρ_{0}} / ρ_{0} = x \cdot x^{i γ_{0}} / (1 + i γ_{0})$ has modulus $x /∣1 + i γ_{0} ∣$ , a fixed positive fraction of $x$ that oscillates in phase as $x^{i γ_{0}} = e^{i γ_{0} l o g x}$ . This term (together with its conjugate from $\overset{ρ}{ˉ}_{0}$ ) does not vanish relative to $x$ , so $ψ (x) / x$ does not tend to $1$ , contradicting the prime number theorem. Hence PNT forces $ζ (1 + i t) \neq = 0$ ; the zero-free line $Re (s) = 1$ and PNT are equivalent. Rubric: full credit for the size estimate $∣ x^{ρ} / ρ ∣ = x^{Re ρ} /∣ ρ ∣$ , the $o (x)$ argument when $Re ρ < 1$ , and the non-vanishing oscillation when $Re ρ = 1$ . This equivalence is the heart of the Hadamard-de la Vallée Poussin proof.

Lean formalization Intermediate+

Mathlib carries the von Mangoldt function, its key convolution identity, the L-series machinery, and an analytically continued Riemann zeta with a zero-free statement on the line $Re (s) = 1$ . The companion notes record the load-bearing declarations.

-- Operative imports: Mathlib.NumberTheory.VonMangoldt,
-- Mathlib.NumberTheory.LSeries.Dirichlet,
-- Mathlib.NumberTheory.LSeries.RiemannZeta
import Mathlib.NumberTheory.VonMangoldt
import Mathlib.NumberTheory.LSeries.Dirichlet
open ArithmeticFunction

-- The von Mangoldt function Λ; ψ(x) = ∑_{n ≤ x} Λ(n) is its summatory function.
#check (vonMangoldt : ArithmeticFunction ℝ)
-- vonMangoldt n = if IsPrimePow n then Real.log (minFac n) else 0

-- The convolution identity log = Λ * 𝟙, the foundation of Λ = μ * log:
#check @ArithmeticFunction.vonMangoldt_mul_zeta
-- vonMangoldt * ↑ζ = log
#check @ArithmeticFunction.log_eq_vonMangoldt_mul_zeta

-- The L-series of Λ equals -ζ'/ζ on Re(s) > 1 (the unit's key theorem):
#check @LSeries.vonMangoldt
-- LSeries (fun n => Λ n) s = - deriv riemannZeta s / riemannZeta s  (Re s > 1)

-- The analytically continued zeta and the zero-free line Re(s) = 1:
#check @riemannZeta_ne_zero_of_one_le_re
-- 1 ≤ s.re → s ≠ 1 → riemannZeta s ≠ 0

The gap documented in the unit metadata's Mathlib gap analysis is not the absence of $Λ$ or of $- ζ^{'} / ζ$ — Mathlib has both, and LSeries.vonMangoldt is exactly the key theorem above — but the absence of a single module assembling the Hadamard product for $ξ$ , the partial-fraction expansion of $ζ^{'} / ζ$ over the critical zeros, and the von Mangoldt explicit formula for $ψ (x)$ end to end, the chain that converts the logarithmic-derivative identity into the prime number theorem.

Advanced results Master

Theorem 1 (Hadamard product for the completed zeta). The completed zeta function $ξ (s) = \frac{1}{2} s (s - 1) π^{- s /2} Γ (s /2) ζ (s)$ is entire of order one ^{[Davenport §12]}, hence admits a Hadamard factorisation $$ \xi(s) = e^{A + Bs} \prod_\rho \Big(1 - \frac{s}{\rho}\Big) e^{s/\rho}, $$ the product over the critical zeros $ρ$ , with $A, B$ constants and the zeros paired as $ρ, \overset{ρ}{ˉ}$ to secure convergence. Because $ξ (s) = ξ (1 - s)$ and $ξ$ has no zeros outside the critical strip (the elementary zeros of $ζ$ are cancelled by the poles of $Γ (s /2)$ , the pole of $ζ$ at $s = 1$ by the factor $s - 1$ ), the zeros of $ξ$ are exactly the critical zeros of $ζ$ .

Theorem 2 (partial-fraction expansion of $ζ^{'} / ζ$ ). Taking the logarithmic derivative of the Hadamard product and of the definition of $ξ$ yields ^{[Davenport §12]} $$ \frac{\zeta'(s)}{\zeta(s)} = B - \frac{1}{s - 1} + \sum_\rho \Big(\frac{1}{s - \rho} + \frac{1}{\rho}\Big) - \frac{1}{2}\log\pi + \frac{1}{2}\frac{\Gamma'}{\Gamma}\Big(\frac{s}{2} + 1\Big). $$ This exhibits the entire pole structure: the pole at $s = 1$ from $ζ$ , one pole at each critical zero $ρ$ , and the elementary-zero contributions hidden in the digamma term $\frac{1}{2} Γ^{'} /Γ (s /2 + 1)$ , which has poles at the elementary zeros $s = - 2, - 4, \dots$ . The convergence-enforcing terms $1/ ρ$ are paired with $ρ \leftrightarrow \overset{ρ}{ˉ}$ .

Theorem 3 (von Mangoldt explicit formula). For $x > 1$ not a prime power ^{[von Mangoldt 1895]}, $$ \psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \log(2\pi) - \frac{1}{2}\log!\big(1 - x^{-2}\big), $$ the sum over critical zeros taken in order of increasing $∣ Im ρ ∣$ . The proof applies Perron's formula of 21.11.04 to $- ζ^{'} / ζ$ and shifts the contour left, collecting the residue $x$ at $s = 1$ , the residues $- x^{ρ} / ρ$ at the zeros, $- lo g (2 π) = - ζ^{'} (0) / ζ (0)$ at $s = 0$ , and the elementary-zero geometric series at $s = - 2 k$ . The main term $x$ is the prime number theorem; the zero-sum is the fluctuation.

Theorem 4 (zero-free region $\Rightarrow$ PNT with error). Combining the partial-fraction expansion with the de la Vallée Poussin zero-free region $Re (ρ) < 1 - c / lo g (∣ Im ρ ∣ + 2)$ ^{[Montgomery-Vaughan Ch. 6]} bounds the truncated zero-sum, yielding $$ \psi(x) = x + O!\big(x \exp(-c'\sqrt{\log x})\big). $$ Equivalently $π (x) = Li (x) + O (x exp (- c^{'} lo g x))$ . Under the Riemann hypothesis $Re (ρ) = \frac{1}{2}$ , the error sharpens to $ψ (x) = x + O (x^{1/2} (lo g x)^{2})$ , the best possible.

Theorem 5 (the explicit formula is reciprocal). Weil's explicit formula generalises Theorem 3 to a duality between a sum over primes (weighted by $Λ$ against a test function $g$ ) and a sum over zeros (the same test function's Mellin transform evaluated at the $ρ$ ), with an archimedean term from the $Γ$ -factor ^{[Davenport §17]}. The von Mangoldt formula is the case where the test function recovers $ψ (x)$ . This reciprocity — primes on one side, zeros on the other — is the structural template for the explicit formulas of all automorphic $L$ -functions.

Synthesis. The logarithmic derivative $- ζ^{'} / ζ$ is the foundational reason the prime number theorem is a theorem of complex analysis rather than of combinatorics. The central insight is that one analytic identity, $- ζ^{'} (s) / ζ (s) = \sum_{n} Λ (n) n^{- s}$ , converts the multiplicative data of the Euler product into the additive prime-power weight $ψ (x)$ , and this is exactly the bridge from the elementary $ψ (x) ≍ x$ of 21.11.03 to the analytic $ψ (x) \sim x$ : where Chebyshev squeezed $ψ$ between two multiples of $x$ using factorials, the contour integral of $- ζ^{'} / ζ$ pins the constant to exactly $1$ by reading the residue at the pole $s = 1$ . Putting these together, the explicit formula $ψ (x) = x - \sum_{ρ} x^{ρ} / ρ - \dots$ generalises the two-sided bound into an exact identity in which the zeros $ρ$ are dual to the primes, each zero an oscillating correction whose amplitude $x^{Re ρ}$ is governed by the zero-free region. The central insight recurs: the Hadamard product is dual to the Euler product — one factors $ξ$ over its zeros, the other factors $ζ$ over the primes — and the partial-fraction expansion of $ζ^{'} / ζ$ is the logarithmic-derivative shadow of both. The bridge is everywhere $- ζ^{'} / ζ$ : Riemann posited it, von Mangoldt made it rigorous, and the modern theory of $L$ -functions is the systematic exploitation of the same identification of primes with zeros.

Full proof set Master

Proposition 1 (the von Mangoldt identity). For $Re (s) > 1$ , $- ζ^{'} (s) / ζ (s) = \sum_{n} Λ (n) n^{- s}$ .

Proof. This is the Key theorem; we record the proof compactly. For $σ > 1$ the Euler product gives $lo g ζ (s) = \sum_{p} \sum_{m \geq 1} p^{- m s} / m$ , absolutely convergent. Differentiation, valid by local uniform convergence of $\sum_{p} \sum_{m} (- lo g p) p^{- m s}$ (dominated by $\sum_{n} (lo g n) n^{- σ} < \infty$ ), gives $ζ^{'} / ζ = - \sum_{p} \sum_{m} (lo g p) p^{- m s}$ . Reindexing by $n = p^{m}$ collects the coefficient $Λ (n)$ , so $- ζ^{'} / ζ = \sum_{n} Λ (n) n^{- s}$ . $□$

Proposition 2 (residues of $- ζ^{'} / ζ$ ). The function $- ζ^{'} / ζ$ is meromorphic on $C$ with a simple pole of residue $+ 1$ at $s = 1$ and a pole of residue $- m$ at each zero of $ζ$ of multiplicity $m$ (elementary or critical); it is holomorphic elsewhere.

Proof. If $ζ (s) = (s - s_{0})^{k} h (s)$ near $s_{0}$ with $h (s_{0}) \neq = 0$ and $k \in Z$ (with $k = - 1$ at the pole $s_{0} = 1$ , $k = m \geq 1$ at a zero of multiplicity $m$ , $k = 0$ elsewhere), then $ζ^{'} / ζ = k / (s - s_{0}) + h^{'} / h$ with $h^{'} / h$ holomorphic at $s_{0}$ . Hence $ζ^{'} / ζ$ has residue $k$ and $- ζ^{'} / ζ$ residue $- k$ . At $s_{0} = 1$ , $k = - 1$ gives residue $+ 1$ for $- ζ^{'} / ζ$ ; at a zero of multiplicity $m$ , $k = m$ gives residue $- m$ . Away from zeros and the pole, $k = 0$ and $- ζ^{'} / ζ$ is holomorphic. $□$

Proposition 3 ( $Λ$ is nonnegative and $ψ$ is increasing). $Λ (n) \geq 0$ for all $n$ , with $Λ (n) > 0$ exactly on prime powers; consequently $ψ (x) = \sum_{n \leq x} Λ (n)$ is a nondecreasing step function jumping by $lo g p$ at each prime power $p^{m} \leq x$ .

Proof. By definition $Λ (n) = lo g p \geq lo g 2 > 0$ when $n = p^{m}$ is a prime power and $Λ (n) = 0$ otherwise, so $Λ \geq 0$ with support exactly the prime powers. The summatory function $ψ (x) = \sum_{n \leq x} Λ (n)$ adds a nonnegative term as $x$ increases past each integer, hence is nondecreasing; it is constant between consecutive prime powers and jumps by $Λ (p^{m}) = lo g p$ at $x = p^{m}$ . $□$

Proposition 4 ( $ζ^{'} (0) / ζ (0) = lo g (2 π)$ ). The constant term at $s = 0$ in the explicit formula equals $lo g (2 π)$ .

Proof. From the functional equation $ζ (s) = 2^{s} π^{s - 1} sin (π s /2) Γ (1 - s) ζ (1 - s)$ of 06.01.16, take the logarithmic derivative: $$ \frac{\zeta'(s)}{\zeta(s)} = \log 2 + \log\pi + \frac{\pi}{2}\cot!\Big(\frac{\pi s}{2}\Big) - \frac{\Gamma'(1 - s)}{\Gamma(1 - s)} - \frac{\zeta'(1 - s)}{\zeta(1 - s)}. $$ Evaluate as $s \to 0$ . The cotangent term $\frac{π}{2} cot (π s /2) \sim 1/ s$ supplies the pole of $1/ s$ paired against the $1/ s$ from the Perron kernel; isolating the finite part, the regular terms give $lo g 2 + lo g π - Γ^{'} (1) /Γ (1) - ζ^{'} (1 - s) / ζ (1 - s) ∣_{s \to 0}$ . Using $Γ^{'} (1) = - γ$ and the Laurent data of $ζ$ at $s = 1$ (residue $1$ , constant term $γ$ , so $ζ^{'} / ζ \to$ a value cancelling the $γ$ terms), the finite part collapses to $lo g 2 + lo g π = lo g (2 π)$ . Thus the residue of $- ζ^{'} (s) / ζ (s) \cdot x^{s} / s$ at $s = 0$ is $- ζ^{'} (0) / ζ (0) = - lo g (2 π)$ . $□$

Connections Master

The von Mangoldt function and the convolution identity $lo g = Λ * 1$ are imported from 21.11.03, where the same $Λ$ drives the elementary factorial identity $lo g N! = \sum_{n} Λ (n) ⌊ N / n ⌋$ ; this unit is the analytic continuation of that story, replacing the factorial bookkeeping with the logarithmic derivative of $ζ$ , and Chebyshev's $ψ (x) ≍ x$ is sharpened here to $ψ (x) \sim x$ .
The Euler product, functional equation, and pole structure of $ζ$ used throughout are exactly those of 21.03.01 and 06.01.16; the logarithmic derivative reads the pole at $s = 1$ as the main term and the critical zeros as the fluctuations, so the analytic facts about $ζ$ established there become arithmetic facts about primes here.
The contour-integral mechanism that turns the identity $- ζ^{'} / ζ = \sum Λ (n) n^{- s}$ into the explicit formula is Perron's formula and Mellin inversion of 21.11.04; the convergence and truncation control needed to shift the contour past $s = 1$ relies on the Chebyshev-grade bound $ψ (x) = O (x)$ from 21.11.03.
The Hadamard product over the critical zeros is the entire-function factorisation theory of 06.01.16 applied to the completed zeta $ξ$ ; the order-one growth of $ξ$ and the pairing $ρ \leftrightarrow \overset{ρ}{ˉ}$ make the zero-sum in the explicit formula conditionally convergent.

Historical & philosophical context Master

Bernhard Riemann's 1859 memoir Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse ^{[Riemann 1859]} introduced the analytic continuation of $ζ (s)$ , the product over its critical zeros, and an explicit formula expressing the prime-counting function as a main term plus a sum over the zeros. Riemann stated the formula and sketched its derivation but did not supply complete proofs of the analytic estimates it required; several of his assertions, including the Hadamard product for an order-one entire function, awaited the function theory developed in the following decades.

Hans von Mangoldt's 1895 paper in Crelle's journal ^{[von Mangoldt 1895]} gave the first rigorous proof of Riemann's explicit formula, recast in terms of the weighted count $ψ (x)$ and the function $Λ (n)$ that now bears his name. The reformulation is decisive: $ψ (x)$ is the summatory function of $Λ$ , $Λ$ is the Dirichlet-series coefficient of $- ζ^{'} / ζ$ , and the contour integral of $- ζ^{'} / ζ$ produces $ψ$ as a sum of residues over the poles of $ζ^{'} / ζ$ . The prime number theorem followed in 1896, proved independently by Jacques Hadamard and Charles-Jean de la Vallée Poussin, each establishing the zero-free region $ζ (1 + i t) \neq = 0$ and inverting the logarithmic derivative by a contour integral ^{[Davenport §12]}. Davenport's Multiplicative Number Theory and Edwards's Riemann's Zeta Function are the canonical modern expositions of this chain.

Bibliography Master

@article{riemann1859,
  author  = {Riemann, Bernhard},
  title   = {Ueber die Anzahl der Primzahlen unter einer gegebenen Gr\"{o}sse},
  journal = {Monatsberichte der K\"{o}niglichen Preu\ss{}ischen Akademie der Wissenschaften zu Berlin},
  pages   = {671--680},
  year    = {1859}
}

@article{vonmangoldt1895,
  author  = {von Mangoldt, Hans},
  title   = {Zu Riemanns Abhandlung ``Ueber die Anzahl der Primzahlen unter einer gegebenen Gr\"{o}sse''},
  journal = {Journal f\"{u}r die reine und angewandte Mathematik (Crelle)},
  volume  = {114},
  pages   = {255--305},
  year    = {1895}
}

@book{davenport2000,
  author    = {Davenport, Harold},
  title     = {Multiplicative Number Theory},
  edition   = {3},
  series    = {Graduate Texts in Mathematics},
  volume    = {74},
  publisher = {Springer-Verlag},
  year      = {2000},
  note      = {Revised by H. L. Montgomery; \S12--17}
}

@book{edwards1974,
  author    = {Edwards, Harold M.},
  title     = {Riemann's Zeta Function},
  publisher = {Academic Press},
  year      = {1974},
  note      = {Chapter 3: the logarithmic derivative and the explicit formula}
}

@book{montgomeryvaughan2007,
  author    = {Montgomery, Hugh L. and Vaughan, Robert C.},
  title     = {Multiplicative Number Theory I: Classical Theory},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {97},
  publisher = {Cambridge University Press},
  year      = {2007}
}

Prerequisites

21.11.03
21.03.01
06.01.16

Tier anchors

beginner: Derbyshire 2003 *Prime Obsession* (Joseph Henry Press) Ch. 19-21 (the explicit formula for $\psi(x)$ described informally, the role of the zeros as wave corrections); 3Blue1Brown 'Pi hiding in prime regularities' for the spectral picture of the primes
intermediate: Davenport 2000 *Multiplicative Number Theory* 3e (Springer GTM 74) §12 (the logarithmic derivative $-\zeta'/\zeta(s) = \sum \Lambda(n) n^{-s}$, the partial-fraction expansion, the setup for the explicit formula); Apostol 1976 *Introduction to Analytic Number Theory* (Springer UTM) Ch. 11-13
master: Davenport 2000 *Multiplicative Number Theory* 3e §12-17 (the Hadamard product for $\xi(s)$, the partial-fraction expansion of $\zeta'/\zeta$, the explicit formula for $\psi(x)$ and the PNT contour proof); Riemann 1859 *Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse*; von Mangoldt 1895 *Crelle* 114 (the explicit formula); Edwards 1974 *Riemann's Zeta Function* (Academic Press) Ch. 3; Montgomery-Vaughan 2007 *Multiplicative Number Theory I* (Cambridge UP) Ch. 12

References

Davenport, H. — Multiplicative Number Theory · Springer Graduate Texts in Mathematics 74, 3rd edition (2000), revised by H. L. Montgomery. §12-17. Defines the von Mangoldt function $\Lambda$, proves the Dirichlet-series identity $-\zeta'(s)/\zeta(s) = \sum_{n} \Lambda(n) n^{-s}$ for $\operatorname{Re}(s) > 1$ from the Euler product, develops the Hadamard product for the completed zeta function $\xi(s)$, derives the partial-fraction expansion $\zeta'(s)/\zeta(s) = B - 1/(s-1) + \sum_\rho (1/(s-\rho) + 1/\rho) - \tfrac12 \Gamma'/\Gamma(s/2+1) + \ldots$, and proves the explicit formula $\psi(x) = x - \sum_\rho x^\rho/\rho - \log(2\pi) - \tfrac12 \log(1 - x^{-2})$ via Perron's formula and a contour integral, yielding the prime number theorem $\psi(x) \sim x$.
Riemann, B. — Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse · Monatsberichte der Königlichen Preussischen Akademie der Wissenschaften zu Berlin (1859), pp. 671-680. Introduces the analytic continuation and functional equation of $\zeta(s)$, the product over the critical zeros, and states (without full proof) the explicit formula connecting the prime-counting function to a sum over the zeros $\rho$.
von Mangoldt, H. — Zu Riemanns Abhandlung 'Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse' · Journal für die reine und angewandte Mathematik (Crelle) 114 (1895), 255-305. Gives the first rigorous proof of Riemann's explicit formula for $\psi(x)$ as a sum over the critical zeros, introducing the function $\Lambda(n)$ now named after him.
Edwards, H. M. — Riemann's Zeta Function · Academic Press (1974), Ch. 3. The logarithmic derivative, the Hadamard product, and the von Mangoldt explicit formula, developed from Riemann's memoir.
Montgomery, H. L. & Vaughan, R. C. — Multiplicative Number Theory I: Classical Theory · Cambridge Studies in Advanced Mathematics 97 (2007), Ch. 12. The zero-free region, the explicit formula with error term, and the contour-integral proof of the prime number theorem with the de la Vallée Poussin error.

Estimated time

beginner: 18m
intermediate: 46m
master: 90m