21.11.05 · number-theory / dirichlet-series-arithmetic-functions

The Selberg-Delange Method

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Anchor (Master): Selberg 1954 *J. Indian Math. Soc.* 18, 83 (the note on the number of primes in short intervals and the Sathe-Selberg theorem); Delange 1954 *Ann. Sci. ENS* 71, 213 (the general Tauberian theorem for zeta^z G(s)); Tenenbaum 2015 *Introduction to Analytic and Probabilistic Number Theory* §II.5-6; Montgomery-Vaughan 2007 *Multiplicative Number Theory I* §II.5; Hankel 1864 *Z. Math. Phys.* 9, 1 (the loop-contour integral for 1/Gamma)

Intuition Beginner

Many counting questions in number theory come down to adding up a multiplicative function: a rule $f$ that assigns a number to each integer in a way that respects factorisation. The simplest example is the rule that gives every integer the value $1$ ; adding it up to a cutoff $x$ just counts the integers, and the answer grows like $x$ . But richer rules grow differently. The rule that counts how many distinct primes divide each integer, or that weights an integer by something depending on its prime factors, produces running totals that grow like $x$ multiplied by a power of $ln x$ .

The Selberg-Delange method is a machine for finding exactly which power of $ln x$ appears, and the constant in front. The key observation is that the packaged version of $f$ — its Dirichlet series — often looks like the Riemann zeta function raised to a complex power, times a tame correction. The zeta function has a single trouble spot at $s = 1$ , and raising it to a power $z$ turns that spot into a branch point, like the tip of a spiral staircase in the complex plane.

Why does this matter? Once you know the running total is steered by a branch point of strength $z$ , a contour wrapped tightly around that point reads off the answer: the total is about $C \cdot x \cdot (ln x)^{z - 1}$ , with the constant built from the reciprocal of the Gamma function at $z$ . This single template explains a whole family of counting laws at once.

Visual Beginner

A spiral staircase in the complex plane sitting above the point $s = 1$ . Walking a full loop around that point does not bring you back to the same height: the function $ζ (s)^{z}$ changes value because of the power $z$ . To read off the running total, a contour is wrapped tightly around the staircase axis, hugging the cut that runs left from $s = 1$ .

   imaginary
     axis
      ^
      |        Hankel keyhole contour
      |        wraps the branch point at s = 1
      |             ___
      |            /   \
======|===========(  o  )=========> real axis
      |   <--cut-- \___/   s = 1
      |        (zeta(s)^z multivalued here)
      |
   strength of the branch point = z
   running total  ~  C * x * (log x)^(z - 1)

The loop hugging the cut is the heart of the method: it converts the branch point of strength $z$ into the factor $(ln x)^{z - 1}$ , exactly as a single pole would have produced a clean power of $x$ .

Worked example Beginner

Take the rule $f (n) = 1$ for every integer, the simplest multiplicative function. Its packaged form is the Riemann zeta function, which is $ζ (s)^{z}$ with $z = 1$ . Let us check what the method predicts and compare with the obvious answer.

Step 1. The method says the running total up to $x$ is about $C \cdot x \cdot (ln x)^{z - 1}$ . With $z = 1$ the exponent $z - 1$ equals $0$ , so $(ln x)^{0} = 1$ , and the prediction collapses to $C \cdot x$ .

Step 2. The constant for $z = 1$ comes out as $1/Γ (1)$ , and $Γ (1) = 1$ , so $C = 1$ . The prediction is simply $x$ .

Step 3. Check directly: adding $f (n) = 1$ over the integers up to $x$ counts them, giving about $x$ . The two answers agree.

Step 4. Now nudge the strength to $z = 2$ , the rule whose packaged form is $ζ (s)^{2}$ . The exponent $z - 1$ becomes $1$ , so the running total grows like $x \cdot ln x$ — faster than $x$ by one logarithm. This is the running total of the divisor counts, which indeed grows like $x ln x$ .

What this tells us: the strength $z$ of the branch point at $s = 1$ is the dial that sets the power of $ln x$ in the running total. Turning the dial from $1$ to $2$ adds exactly one factor of $ln x$ .

Check your understanding Beginner

Formal definition Intermediate+

Let $f$ be a multiplicative arithmetic function with Dirichlet series $F (s) = \sum_{n = 1}^{\infty} f (n) n^{- s}$ , absolutely convergent for $Re (s) > 1$ . Write $σ = Re (s)$ . The Selberg-Delange method applies when $F (s)$ factors, in a neighbourhood of the line $Re (s) = 1$ , as a complex power of the zeta function times an analytic correction. ^{[Tenenbaum §II.5]}

Definition (Selberg-Delange class). Fix $z \in C$ . The function $f$ belongs to the class $D (z)$ if its Dirichlet series admits a representation $$ F(s) = \zeta(s)^{z}, G(s), \qquad \operatorname{Re}(s) > 1, $$ where $ζ (s)^{z} := exp (z lo g ζ (s))$ is defined by the principal branch of $lo g ζ (s)$ for $Re (s) > 1$ , and $G (s)$ continues holomorphically to a region $Re (s) \geq 1 - c / (1 + ∣ Im (s) ∣)$ (a standard zero-free strip) with $G (s)$ bounded and $G (1) \neq = 0$ there. The function $ζ (s)^{z}$ inherits from the simple pole of $ζ$ at $s = 1$ a branch point of exponent $- z$ : locally $ζ (s)^{z} = (s - 1)^{- z} H (s)$ with $H$ holomorphic and non-vanishing at $s = 1$ , where $H (s) = ((s - 1) ζ (s))^{z}$ .

Definition (branch cut and Hankel contour). Because $(s - 1)^{- z}$ is multivalued for $z \in / Z$ , fix the branch cut along the half-line $(- \infty, 1]$ and take the principal branch of $(s - 1)^{- z}$ on the slit plane $C ∖ (- \infty, 1]$ . The Hankel contour $H$ is the positively oriented loop that comes in from $- \infty$ below the cut, circles the branch point $s = 1$ at radius $r \to 0$ , and returns to $- \infty$ above the cut. The reciprocal Gamma function has the loop representation $$ \frac{1}{\Gamma(z)} = \frac{1}{2\pi i} \int_{\mathcal{H}} e^{t}, t^{-z}, dt, $$ valid for all $z \in C$ 06.01.15, which is the analytic source of the constant in the main term.

Definition (main-term asymptotic). For $f \in D (z)$ with $Re (z) > 0$ , the Selberg-Delange asymptotic for the summatory function is $$ \sum_{n \le x} f(n) = x,(\log x)^{z - 1} \sum_{j=0}^{N-1} \frac{\lambda_j(z)}{(\log x)^{j}} + O!\left( x,(\log x)^{\operatorname{Re}(z) - N - 1} \right), $$ where the leading coefficient is $λ_{0} (z) = G (1) /Γ (z)$ , and the higher $λ_{j} (z)$ are the Taylor coefficients of $H (s) G (s) /Γ (z + \dots)$ at $s = 1$ . The case $z = 1$ , $G \equiv 1$ recovers $\sum_{n \leq x} 1 \sim x$ ; the case $z = 2$ , $G (s) = 1$ , $F (s) = ζ (s)^{2}$ recovers Dirichlet's $\sum_{n \leq x} d (n) \sim x lo g x$ .

Counterexamples to common slips

The exponent of $lo g x$ is $z - 1$ , not $z$ . A single factor of $ζ (s)$ (the case $z = 1$ ) gives a summatory function $\sim x$ , with $(lo g x)^{0}$ . The logarithmic saving relative to the bare $x$ comes only from the excess of the branch-point exponent above $1$ .
The correction $G (1)$ must be non-zero and finite on the line. If $G (1) = 0$ the leading coefficient $λ_{0} = G (1) /Γ (z)$ vanishes and the genuine asymptotic order drops; if $G$ has a pole on $Re (s) = 1$ the method does not apply without first removing it. The factorisation $F = ζ^{z} G$ is meaningful only when $G$ is analytic and non-degenerate across the branch point.
The contour must hug the cut, not encircle a pole. For $z \in / Z_{> 0}$ there is no pole to pick up by the residue theorem; the value is produced by the discontinuity of $(s - 1)^{- z}$ across $(- \infty, 1]$ . Treating $s = 1$ as a pole and applying a naive residue gives the wrong constant (it omits the $1/Γ (z)$ factor entirely).

Key theorem with proof Intermediate+

The signature result is the leading-order Selberg-Delange asymptotic, derived by deforming the truncated Perron contour 21.11.04 into a Hankel loop around the branch point at $s = 1$ . This is the step that converts the analytic input " $F = ζ^{z} G$ " into the arithmetic output " $\sum_{n \leq x} f (n) ≍ x (lo g x)^{z - 1}$ ". ^{[Montgomery-Vaughan §II.5]}

Theorem (Selberg-Delange, leading term). Let $f \in D (z)$ with $Re (z) > 0$ , so that $F (s) = ζ (s)^{z} G (s)$ with $G$ holomorphic, bounded, and non-vanishing on a zero-free strip $Re (s) \geq 1 - c / (1 + ∣ Im (s) ∣)$ . Then $$ \sum_{n \le x} f(n) = \frac{G(1)}{\Gamma(z)}, x, (\log x)^{z - 1} ,\big(1 + o(1)\big) \qquad (x \to \infty). $$

Proof. By the truncated Perron formula 21.11.04, for $c = 1 + 1/ lo g x$ and a height $T$ to be chosen, $$ \sum_{n \le x}{}' f(n) = \frac{1}{2\pi i} \int_{c - iT}^{c + iT} F(s), \frac{x^s}{s}, ds + R(x, T), $$ with $R (x, T)$ controlled by the standard error bound, negligible for $T$ a fixed power of $lo g x$ once the strip estimate on $F$ is used. Substitute $F (s) = ζ (s)^{z} G (s)$ and write the local factorisation $ζ (s)^{z} = (s - 1)^{- z} H (s)$ with $H (s) = ((s - 1) ζ (s))^{z}$ holomorphic and $H (1) = 1$ (since $(s - 1) ζ (s) \to 1$ as $s \to 1$ ).

Deform the vertical segment leftward to the boundary of the zero-free strip. Because $G$ and $H$ are holomorphic there and $ζ$ has no zeros, the only obstruction to free deformation is the branch cut of $(s - 1)^{- z}$ along $(- \infty, 1]$ . The contour collapses onto a Hankel loop $H$ encircling $s = 1$ and hugging the cut, plus contributions from the truncation that are absorbed into the error. Thus $$ \sum_{n \le x}{}' f(n) = \frac{1}{2\pi i} \int_{\mathcal{H}} (s-1)^{-z} H(s) G(s), \frac{x^s}{s}, ds + (\text{smaller}). $$ On the loop, set $s = 1 + w / lo g x$ , so $x^{s} = x \cdot e^{w}$ and $d s = d w / lo g x$ . Near the branch point $H (s) G (s) / s \to H (1) G (1) /1 = G (1)$ , and $(s - 1)^{- z} = (lo g x)^{z} w^{- z}$ . Hence $$ \frac{1}{2\pi i}\int_{\mathcal{H}} (s-1)^{-z} \frac{H(s)G(s)}{s} x^s, ds = G(1), x, (\log x)^{z-1} \cdot \frac{1}{2\pi i}\int_{\mathcal{H}'} e^{w}, w^{-z}, dw ,\big(1 + o(1)\big), $$ where $H^{'}$ is the image Hankel contour in the $w$ -plane. By the Hankel representation of the reciprocal Gamma function 06.01.15, $\frac{1}{2 π i} \int_{H^{'}} e^{w} w^{- z} d w = 1/Γ (z)$ . Collecting, $$ \sum_{n \le x}{}' f(n) = \frac{G(1)}{\Gamma(z)}, x, (\log x)^{z-1},\big(1 + o(1)\big), $$ and replacing the half-weighted sum by the ordinary one changes nothing at this order. $□$

Bridge. This derivation builds toward the Landau-Selberg-Sathe count of integers with exactly $k$ prime factors, where setting $z$ to a formal variable and reading off Taylor coefficients in $z$ appears again in 21.11.04 through the same Perron contour now wrapped on a branch point instead of slid past a pole. The foundational reason the method works is that a branch point of exponent $- z$ contributes $(lo g x)^{z - 1} /Γ (z)$ exactly as a simple pole contributes a clean power of $x$ ; this is exactly the analytic continuation of the residue calculus to non-integer order. The central insight is that the Hankel contour is the geometric shadow of $1/Γ (z)$ , so the constant in every mean-value law is a Gamma value — putting these together, the Selberg-Delange template generalises Perron's formula from poles to branch points, and is dual to the Dirichlet-series packaging that produced $F = ζ^{z} G$ in the first place.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Let $r_{2} (n)$ count representations of $n$ as an ordered sum of two squares, and recall that the count of $n \leq x$ representable as a sum of two squares, weighted to be multiplicative, has Dirichlet series of the form $ζ (s)^{1/2} G (s)$ with $G$ analytic and non-zero at $s = 1$ . Determine the power of $lo g x$ in the count and explain the appearance of $1/Γ (1/2)$ .

Hint

Here $z = 1/2$ . Recall $Γ (1/2) = π$ .

Answer

With $z = 1/2$ , the exponent of $lo g x$ is $z - 1 = - 1/2$ , so the summatory function behaves like $C x (lo g x)^{- 1/2} = C x / lo g x$ . The constant carries the factor $1/Γ (1/2) = 1/ π$ , so $$ #{n \le x : n \text{ a sum of two squares}} \sim \frac{K, x}{\sqrt{\log x}}, $$ the Landau-Ramanujan law, with $K = B / π$ for the Landau-Ramanujan constant $B = G (1) / 2$ packaged into $G (1)$ . The non-integer exponent $z = 1/2$ is exactly why a half-power of $lo g x$ and a $π$ appear: a genuine branch point, not a pole. $□$

Exercise 5 (medium, symbolic).

Verify the Hankel normalisation in the case $z = 1$ : show that $\frac{1}{2 π i} \int_{H} e^{w} w^{- 1} d w = 1$ , consistent with $1/Γ (1) = 1$ .

Hint

For $z = 1$ the integrand $e^{w} / w$ has an honest simple pole at $w = 0$ ; the Hankel loop degenerates to a small circle around it.

Answer

When $z = 1$ , $w^{- z} = w^{- 1}$ is single-valued, so the branch cut disappears and the Hankel contour may be collapsed to a small positively oriented circle around $w = 0$ . The integrand $e^{w} / w$ has a simple pole at $w = 0$ with residue $e^{0} = 1$ . By the residue theorem, $$ \frac{1}{2\pi i}\int_{\mathcal{H}} \frac{e^w}{w}, dw = \operatorname{Res}_{w=0}\frac{e^w}{w} = 1 = \frac{1}{\Gamma(1)}. $$ The integer case is precisely where the branch point degenerates into a pole and the Selberg-Delange constant reduces to an ordinary residue. $□$

Exercise 6 (medium, multiple choice).

Which feature of $G (s)$ in the factorisation $F (s) = ζ (s)^{z} G (s)$ is required for the leading Selberg-Delange constant $G (1) /Γ (z)$ to be a genuine nonzero asymptotic constant?

A. $G$ has a simple pole at $s = 1$ .

B. $G$ is holomorphic and non-vanishing at $s = 1$ .

C. $G$ vanishes at $s = 1$ .

D. $G$ equals $ζ (s)$ .

Hint

The constant is $G (1) /Γ (z)$ ; for it to be finite and nonzero, what must $G (1)$ be?

Answer

B. The leading constant is $G (1) /Γ (z)$ , which is finite and nonzero precisely when $G$ is holomorphic at $s = 1$ (so $G (1)$ is finite) and $G (1) \neq = 0$ . Feedback-correct: a pole in $G$ (A) would change the branch-point structure and the exponent of $lo g x$ ; a zero (C) would kill the leading term and demote the asymptotic order; D would merely add $1$ to $z$ , not provide an admissible correction. The whole point of separating $ζ (s)^{z}$ from $G (s)$ is that all the singular behaviour sits in the $ζ^{z}$ factor while $G$ stays tame.

Exercise 7 (hard, symbolic).

Derive the Landau-Selberg-Sathe asymptotic for $π_{k} (x) = # {n \leq x : Ω (n) = k}$ (or $ω (n) = k$ ) by introducing the generating Dirichlet series $F (s, z) = \sum_{n} z^{Ω (n)} n^{- s}$ , showing it lies in $D (z)$ , applying the leading theorem, and extracting the coefficient of $z^{k}$ .

Hint

The Euler product of $F (s, z)$ is $\prod_{p} (1 - z p^{- s})^{- 1}$ , which equals $ζ (s)^{z}$ times an Euler product $G (s, z)$ convergent for $Re (s) > 1/2$ . Then $\sum_{n \leq x} z^{Ω (n)} \sim \frac{G ( 1 , z )}{Γ ( z )} x (lo g x)^{z - 1}$ ; expand in powers of $z$ .

Answer

The completely-multiplicative weight $z^{Ω (n)}$ has Euler product $$ F(s,z) = \prod_p \frac{1}{1 - z p^{-s}} = \zeta(s)^z \prod_p \left(1 - z p^{-s}\right)^{-1}\left(1 - p^{-s}\right)^{z} = \zeta(s)^z, G(s,z), $$ where $G (s, z) = \prod_{p} (1 - z p^{- s})^{- 1} (1 - p^{- s})^{z}$ converges absolutely for $Re (s) > 1/2$ (the $p^{- s}$ terms cancel to order $p^{- 2 s}$ ), so $G (\cdot, z)$ is holomorphic and non-vanishing at $s = 1$ . Hence $f = z^{Ω (\cdot)} \in D (z)$ and the leading theorem gives $$ \sum_{n \le x} z^{\Omega(n)} \sim \frac{G(1,z)}{\Gamma(z)}, x,(\log x)^{z - 1}. $$ Now $π_{k} (x) = [z^{k}] \sum_{n \leq x} z^{Ω (n)}$ , the coefficient of $z^{k}$ . The dominant contribution as $x \to \infty$ comes from the saddle near $z \approx k / lo g lo g x$ ; a Cauchy-coefficient extraction of $\frac{G ( 1 , z )}{Γ ( z )} x (lo g x)^{z - 1}$ yields $$ \pi_k(x) \sim \frac{x}{\log x}, \frac{(\log\log x)^{k - 1}}{(k - 1)!}, G!\left(1, \tfrac{k-1}{\log\log x}\right), $$ and for fixed $k$ the correction $G (1, o (1)) \to G (1, 0) = 1$ , giving the Landau-Selberg-Sathe theorem $$ \pi_k(x) \sim \frac{x}{\log x}, \frac{(\log\log x)^{k-1}}{(k-1)!}. $$ The Poisson-shaped distribution in $k$ with mean $lo g lo g x$ is the analytic root of the Erdős-Kac theorem. $□$

Exercise 8 (hard, symbolic).

Justify the contour deformation at the heart of the method: explain why, for $F (s) = ζ (s)^{z} G (s)$ with $G$ holomorphic on a zero-free strip, the truncated Perron vertical segment may be replaced by the Hankel loop around $s = 1$ with an error of smaller order, and identify what controls the size of that error.

Hint

The integrand is holomorphic on the slit strip except along the cut $(- \infty, 1]$ . Bound the side and bottom segments of the deformed contour using the strip bound on $G$ and the smallness of $x^{σ}$ for $σ < 1$ .

Answer

On the slit strip ${Re (s) \geq 1 - c / (1 + ∣ Im (s) ∣)} ∖ (- \infty, 1]$ , the integrand $ζ (s)^{z} G (s) x^{s} / s$ is holomorphic: $ζ$ has no zeros there (so $ζ^{z}$ is well-defined off the cut), $G$ is holomorphic by hypothesis, and the only multivaluedness is the cut. By Cauchy's theorem the truncated vertical segment from $c - i T$ to $c + i T$ equals the integral over the deformed contour consisting of (i) two horizontal segments at $Im (s) = \pm T$ running left to the strip boundary, (ii) two near-vertical pieces descending to the cut along the strip edge, and (iii) the Hankel loop $H$ hugging $(- \infty, 1]$ near $s = 1$ .

The horizontal segments (i) contribute $O (x^{c} / T) = O (x lo g x / T)$ from the Perron tail, negligible for $T = (lo g x)^{A}$ . The strip-edge pieces (ii) carry $x^{σ}$ with $σ \leq 1 - c / (1 + T)$ , so $x^{σ} \leq x exp (- c lo g x / (1 + T))$ , decaying faster than any power of $lo g x$ once $T$ is a power of $lo g x$ ; the bound on $G$ keeps the integrand controlled. What survives is the Hankel loop (iii), giving the stated main term. The error is therefore governed by two competing inputs: the width $c$ of the zero-free strip (the analytic continuation of $ζ$ past $Re (s) = 1$ ) and the truncation height $T$ ; balancing them produces the secondary terms $λ_{j} (lo g x)^{- j}$ and the final $O (x (lo g x)^{Re (z) - N - 1})$ remainder. $□$

Advanced results Master

Full asymptotic expansion. The leading theorem is the first term of an expansion in descending powers of $lo g x$ . Writing $A (s) := H (s) G (s) / s = \sum_{j \geq 0} a_{j} (s - 1)^{j}$ for the Taylor expansion of the regular part at $s = 1$ , the Hankel-contour evaluation term by term produces $$ \sum_{n \le x} f(n) = x \sum_{j=0}^{N-1} \frac{a_j}{\Gamma(z - j)},(\log x)^{z - 1 - j} + O!\left(x (\log x)^{\operatorname{Re}(z) - N - 1}\right), $$ because the Hankel integral of $w^{- (z - j)} e^{w}$ is $1/Γ (z - j)$ . The coefficients $λ_{j} (z) = a_{j} /Γ (z - j)$ are entire in $z$ ; at $z = 1, 2, 3, \dots$ the poles of $1/Γ$ at non-positive integers truncate the expansion, recovering the polynomial-in- $lo g x$ main terms that the integer-power cases produce directly by residues ^{[Tenenbaum §II.5]}.

The Landau-Selberg-Sathe theorem in full. For the count $π_{k} (x) = # {n \leq x : ω (n) = k}$ of integers with exactly $k$ distinct prime factors, the uniform two-variable form holds for $k \leq (2 - δ) lo g lo g x$ : $$ \pi_k(x) = \frac{x}{\log x},\frac{(\log\log x)^{k-1}}{(k-1)!}\left(\mathcal{G}!\left(\frac{k-1}{\log\log x}\right) + O!\left(\frac{k}{(\log\log x)^2}\right)\right), $$ where $G (z) = G (1, z) /Γ (z + 1)$ is an entire function with $G (0) = 1$ . Sathe obtained this by elementary-analytic recursion in 1953; Selberg's 1954 note re-derived it in a few lines from the analytic factorisation $F (s, z) = ζ (s)^{z} G (s, z)$ and the contour method, which is the form that became the template ^{[Selberg 1954]}. The same machine yields the count $Ω (n) = k$ with $ζ (s)^{z}$ replaced by $(1 - z p^{- s})^{- 1}$ products convergent only for $∣ z ∣ < 2$ .

Delange's Tauberian theorem. Delange's 1954 formulation dispenses with the smoothness assumptions of classical Tauberian theorems (Ikehara, Wiener-Ikehara) and works directly from the analytic continuation of $F (s) = ζ (s)^{z} G (s)$ across $Re (s) = 1$ ^{[Delange 1954]}. It is the Tauberian engine behind the method: where Wiener-Ikehara extracts $x$ from a simple pole, Delange extracts $x (lo g x)^{z - 1}$ from a branch point of exponent $- z$ , and the proof is the contour deformation of the leading theorem made uniform. This is the precise sense in which the Selberg-Delange method is "Perron's formula with a branch point in place of a pole."

Sums of two squares and the Landau-Ramanujan law. The indicator of integers expressible as a sum of two squares is not multiplicative, but the associated multiplicative density $b (n) = \prod_{p \equiv 3 (4)} 1 [2 ∣ v_{p} (n)]$ has Dirichlet series $ζ (s)^{1/2} G (s)$ with $G$ analytic for $Re (s) > 1/2$ . The method gives $# {n \leq x : n = a^{2} + b^{2}} \sim K x / lo g x$ with the Landau-Ramanujan constant $K = \frac{1}{2} \prod_{p \equiv 3 (4)} (1 - p^{- 2})^{- 1/2}$ . The half-integer exponent $z = 1/2$ — and hence the $lo g x$ denominator and the $π$ in $1/Γ (1/2)$ — is the cleanest illustration that the method handles genuinely non-integer branch points where residue calculus cannot reach.

Erdős-Kac as a central limit theorem. The Landau-Selberg-Sathe count $π_{k} (x)$ , viewed as a distribution in $k$ , is asymptotically Poisson with parameter $lo g lo g x$ ; since a Poisson law with large parameter is approximately Gaussian, $ω (n)$ for $n \leq x$ satisfies a central limit theorem with mean and variance $lo g lo g x$ , which is the Erdős-Kac theorem. The Selberg-Delange method supplies the moment generating function $\sum_{n \leq x} z^{ω (n)} \sim \frac{G ( 1 , z )}{Γ ( z )} x (lo g x)^{z - 1}$ uniformly in a complex neighbourhood of $z = 1$ ; evaluating at $z = e^{i t / l o g l o g x}$ and applying the continuity theorem for characteristic functions yields the Gaussian limit.

Synthesis. The Selberg-Delange method is the foundational reason that mean values of multiplicative functions obey a single universal law $x (lo g x)^{z - 1} /Γ (z)$ , and the central insight is that the strength $z$ of the branch point of $F (s) = ζ (s)^{z} G (s)$ at $s = 1$ sets the power of $lo g x$ exactly as a pole sets a power of $x$ . This is exactly the analytic continuation of the residue calculus from integer to complex order: putting these together, the Hankel contour is the geometric image of $1/Γ (z)$ , so every mean-value constant is a Gamma value, and the method is dual to the Dirichlet-series packaging that produced $F = ζ^{z} G$ . The template generalises Perron's formula 21.11.04 from poles to branch points, the bridge is the deformation of the vertical contour onto the cut, and the same machine appears again in 06.01.16 through the analytic structure of $ζ$ whose single pole at $s = 1$ is the seed of every branch point the method exploits — from the divisor law ( $z = 2$ ) to the Landau-Ramanujan law ( $z = 1/2$ ) to the Erdős-Kac theorem (the $z$ -derivative at $z = 1$ ).

Full proof set Master

Proposition (Hankel evaluation of the model integral). For every $z \in C$ and the Hankel contour $H^{'}$ wrapping the negative real axis in the $w$ -plane, $$ \frac{1}{2\pi i}\int_{\mathcal{H}'} e^{w}, w^{-z}, dw = \frac{1}{\Gamma(z)}. $$

Proof. For $Re (z) < 1$ the integral converges absolutely and the standard Hankel-loop evaluation applies. Parametrise the three pieces of $H^{'}$ : the lower edge $w = u e^{- iπ}$ , $u : \infty \to r$ ; the circle $w = r e^{i θ}$ , $θ : - π \to π$ ; the upper edge $w = u e^{iπ}$ , $u : r \to \infty$ . On the edges $w^{- z} = u^{- z} e^{\mp iπ z}$ and $e^{w} = e^{- u}$ , while the circular part contributes $O (r^{1 - Re (z)}) \to 0$ as $r \to 0$ when $Re (z) < 1$ . The two edges give $$ \frac{1}{2\pi i}\left( e^{i\pi z} - e^{-i\pi z}\right)\int_0^\infty e^{-u} u^{-z}, du = \frac{2 i \sin(\pi z)}{2\pi i},\Gamma(1 - z) = \frac{\sin(\pi z)}{\pi},\Gamma(1-z). $$ By Euler's reflection formula 06.01.15, $Γ (z) Γ (1 - z) = π / sin (π z)$ , so $\frac{s i n ( π z )}{π} Γ (1 - z) = 1/Γ (z)$ . Both sides are entire in $z$ (the left by holomorphy of the loop integral, the right because $1/Γ$ is entire), so the identity extends from $Re (z) < 1$ to all $z \in C$ by the identity theorem. $□$

Proposition (branch-point factorisation of $ζ (s)^{z}$ ). Near $s = 1$ , $ζ (s)^{z} = (s - 1)^{- z} H (s)$ with $H$ holomorphic at $s = 1$ and $H (1) = 1$ , where $H (s) = ((s - 1) ζ (s))^{z}$ for the principal branch.

Proof. The zeta function has a simple pole at $s = 1$ with residue $1$ 06.01.16, so $(s - 1) ζ (s)$ extends to a function holomorphic and equal to $1$ at $s = 1$ , hence non-vanishing in a neighbourhood. On that neighbourhood $lo g ((s - 1) ζ (s))$ has a holomorphic principal branch vanishing at $s = 1$ , so $H (s) := exp (z lo g ((s - 1) ζ (s)))$ is holomorphic with $H (1) = e^{0} = 1$ . Then $ζ (s)^{z} = exp (z lo g ζ (s)) = exp (z lo g ((s - 1) ζ (s)) - z lo g (s - 1)) = H (s) (s - 1)^{- z}$ for the matching branch of $(s - 1)^{- z}$ . The factor $(s - 1)^{- z}$ carries all the multivaluedness; $H$ is single-valued. $□$

Proposition (consistency at integer $z$ ). For $z = m \in Z_{\geq 1}$ , the Hankel main term $G (1) x (lo g x)^{m - 1} /Γ (m)$ agrees with the residue obtained by treating $s = 1$ as a pole of order $m$ of $ζ (s)^{m} G (s) x^{s} / s$ .

Proof. For $z = m \in Z_{\geq 1}$ , $(s - 1)^{- z} = (s - 1)^{- m}$ is single-valued, so $ζ (s)^{m} G (s) x^{s} / s$ has a genuine pole of order $m$ at $s = 1$ and the Hankel loop collapses to a circle. The residue is the coefficient of $(s - 1)^{- 1}$ in $(s - 1)^{- m} H (s) G (s) x^{s} / s$ , namely $\frac{1}{( m - 1 )!} \frac{d ^{m - 1}}{d s ^{m - 1}} [H (s) G (s) x^{s} / s]_{s = 1}$ . The dominant contribution as $x \to \infty$ comes from differentiating $x^{s} = e^{s l o g x}$ , giving $(lo g x)^{m - 1}$ , with leading coefficient $H (1) G (1) /1 = G (1)$ . Thus the residue is $\frac{G ( 1 )}{( m - 1 )!} x (lo g x)^{m - 1} (1 + o (1)) = \frac{G ( 1 )}{Γ ( m )} x (lo g x)^{m - 1} (1 + o (1))$ , since $Γ (m) = (m - 1)!$ . This matches the Hankel formula, confirming that the branch-point method specialises to residue calculus at integer exponents. $□$

Connections Master

Perron's formula and Mellin inversion 21.11.04. The Selberg-Delange method begins with the truncated Perron formula and differs only in the final move: where the prime number theorem slides the contour past the simple pole of $- ζ^{'} / ζ$ and reads a residue, Selberg-Delange deforms onto a Hankel loop around a branch point of $ζ (s)^{z}$ and reads a Gamma value. Perron's formula is the contour engine; this unit is its branch-point upgrade.
Riemann zeta function 06.01.16. The entire method is built on the single simple pole of $ζ$ at $s = 1$ with residue $1$ : raising $ζ$ to a complex power $z$ converts that pole into the branch point of exponent $- z$ that every mean-value law exploits, and the zero-free region of $ζ$ is exactly the strip into which the contour is deformed. The analytic structure of $ζ$ established there is the raw material the method processes.
Gamma function 06.01.15. The constant $1/Γ (z)$ in every Selberg-Delange asymptotic is the Hankel-loop integral $\frac{1}{2 π i} \int_{H} e^{w} w^{- z} d w$ , and the reflection formula $Γ (z) Γ (1 - z) = π / sin (π z)$ is what evaluates that loop. The Gamma function is not incidental decoration: it is the analytic shadow of the branch point, and its reciprocal being entire is why the asymptotic constant is finite for every $z$ , including the integers where naive residue calculus would suggest a pole.

Historical & philosophical context Master

The prototype is Landau's 1900 determination of the count of integers that are products of exactly two primes, $π_{2} (x) \sim \frac{x l o g l o g x}{l o g x}$ ^{[Landau 1900]}, obtained by an intricate elementary-analytic argument. L. G. Sathe extended this to general $k$ in a long 1953 series of papers using elementary recursions, and Atle Selberg, in a short 1954 note in the Journal of the Indian Mathematical Society ^{[Selberg 1954]}, re-derived Sathe's results in a few pages by introducing the generating Dirichlet series $\sum_{n} z^{ω (n)} n^{- s} = ζ (s)^{z} G (s)$ and extracting the asymptotic from its branch point at $s = 1$ . In the same year Hubert Delange published a general Tauberian theorem ^{[Delange 1954]} isolating the analytic content: a Dirichlet series of the shape $ζ (s)^{z} G (s)$ has summatory function $\sim \frac{G ( 1 )}{Γ ( z )} x (lo g x)^{z - 1}$ . The pairing of the two 1954 papers gave the method its name.

The analytic device — deforming a contour onto a loop around a branch point and reading off a reciprocal-Gamma constant — descends from Hankel's 1864 loop integral for $1/Γ$ ^{[Hankel 1864]}, itself a complex-analytic reworking of Euler's integral. Tenenbaum's textbook treatment systematised the method with its full power-of- $lo g x$ expansion and uniformity in $z$ , making it the standard tool for mean values of multiplicative functions and for the analytic half of the Erdős-Kac theorem, whose probabilistic statement Erdős and Kac proved in 1940 by sieve and central-limit arguments before the Selberg-Delange machine supplied the moment generating function directly.

Bibliography Master

@article{Landau1900,
  author = {Landau, Edmund},
  title = {Sur quelques probl\`emes relatifs \`a la distribution des nombres premiers},
  journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
  volume = {28},
  year = {1900},
  pages = {25--38},
  note = {Asymptotic for integers that are products of two primes}
}

@article{Selberg1954,
  author = {Selberg, Atle},
  title = {Note on a paper by L. G. Sathe},
  journal = {Journal of the Indian Mathematical Society},
  volume = {18},
  year = {1954},
  pages = {83--87},
  note = {Analytic derivation of the Sathe-Selberg theorem via zeta(s)^z}
}

@article{Delange1954,
  author = {Delange, Hubert},
  title = {G\'en\'eralisation du th\'eor\`eme de Ikehara},
  journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure (3)},
  volume = {71},
  year = {1954},
  pages = {213--242},
  note = {General Tauberian theorem for Dirichlet series of type zeta(s)^z G(s)}
}

@article{Hankel1864,
  author = {Hankel, Hermann},
  title = {Die Euler'schen Integrale bei unbeschr\"ankter Variabilit\"at des Argumentes},
  journal = {Zeitschrift f\"ur Mathematik und Physik},
  volume = {9},
  year = {1864},
  pages = {1--21},
  note = {Loop-contour representation of the reciprocal Gamma function}
}

@book{Tenenbaum2015,
  author = {Tenenbaum, G\'erald},
  title = {Introduction to Analytic and Probabilistic Number Theory},
  edition = {3},
  publisher = {American Mathematical Society},
  series = {Graduate Studies in Mathematics},
  volume = {163},
  year = {2015},
  note = {\S II.5-6: the Selberg-Delange method and the Landau-Selberg-Sathe theorem}
}

@book{MontgomeryVaughan2007,
  author = {Montgomery, Hugh L. and Vaughan, Robert C.},
  title = {Multiplicative Number Theory I: Classical Theory},
  publisher = {Cambridge University Press},
  series = {Cambridge Studies in Advanced Mathematics},
  volume = {97},
  year = {2007},
  note = {\S II.5: mean values of multiplicative functions via zeta(s)^z G(s)}
}

Prerequisites

21.11.04
06.01.16
06.01.15

Tier anchors

beginner: Derbyshire 2003 *Prime Obsession* Ch. 19-21 (informal contour summation of arithmetic functions); Stewart *Calculus* early-transcendentals on power functions and logarithms
intermediate: Tenenbaum 2015 *Introduction to Analytic and Probabilistic Number Theory* (3rd ed., AMS GSM 163) §II.5-6 (the Selberg-Delange method, mean values of multiplicative functions, the Landau-Selberg-Sathe theorem); Montgomery-Vaughan 2007 *Multiplicative Number Theory I* (Cambridge Studies 97) §II.5 (mean values via zeta^z G(s))
master: Selberg 1954 *J. Indian Math. Soc.* 18, 83 (the note on the number of primes in short intervals and the Sathe-Selberg theorem); Delange 1954 *Ann. Sci. ENS* 71, 213 (the general Tauberian theorem for zeta^z G(s)); Tenenbaum 2015 *Introduction to Analytic and Probabilistic Number Theory* §II.5-6; Montgomery-Vaughan 2007 *Multiplicative Number Theory I* §II.5; Hankel 1864 *Z. Math. Phys.* 9, 1 (the loop-contour integral for 1/Gamma)

References

Selberg, A. — Note on a paper by L. G. Sathe · *Journal of the Indian Mathematical Society* 18 (1954), 83-87. The analytic derivation of the asymptotic for the count of integers with exactly k prime factors via the singular series zeta(s)^z.
Delange, H. — Generalisation du theoreme de Ikehara · *Annales scientifiques de l'Ecole Normale Superieure* (3) 71 (1954), 213-242. The general Tauberian theorem extracting the main term of the summatory function of a multiplicative function whose Dirichlet series is zeta(s)^z G(s).
Tenenbaum, G. — Introduction to Analytic and Probabilistic Number Theory · American Mathematical Society Graduate Studies in Mathematics 163, 3rd edition (2015). §II.5 (the Selberg-Delange method and the Hankel-contour main term), §II.6 (applications to the Landau-Selberg-Sathe theorem and integers free of large prime factors).
Montgomery, H. L. & Vaughan, R. C. — Multiplicative Number Theory I: Classical Theory · Cambridge Studies in Advanced Mathematics 97 (2007). §II.5 (mean values of multiplicative functions, the zeta(s)^z G(s) model, and the resulting x (log x)^{z-1}/Gamma(z) asymptotic).
Landau, E. — Sur quelques problemes relatifs a la distribution des nombres premiers · *Bulletin de la Societe Mathematique de France* 28 (1900), 25-38. The original asymptotic for the count of integers that are products of exactly two primes, the prototype of the Landau-Selberg-Sathe theorem.
Hankel, H. — Die Euler'schen Integrale bei unbeschrankter Variabilitat des Argumentes · *Zeitschrift fuer Mathematik und Physik* 9 (1864), 1-21. The loop-contour (Hankel) representation 1/Gamma(z) = (1/2 pi i) integral over a contour wrapping the negative real axis of e^t t^{-z} dt, the analytic core of the Selberg-Delange main term.

Estimated time

beginner: 18m
intermediate: 46m
master: 82m