Math 259: Introduction to Analytic Number Theory

More about the Gamma function

We collect some more facts about Γ(s) as a function of a complex variable that will figure in our treatment of ζ(s) and L(s, χ). All of these, and most of the Exercises, are standard textbook fare; one basic reference is Ch. XII (pp. 235–264) of [WW 1940]. One reason for not just citing Whittaker & Watson is that some of the results concerning Euler’s integrals B and Γ have close analogues in the Gauss and Jacobi sums associated to Dirichlet characters, and we shall need these analogues before long.

The product formula for Γ(s). Recall that Γ(s) has simple poles at s = 0, −1, −2, … and no zeros. We readily concoct a product that has the same behavior: let

g(s) := \frac{1}{s} ∏_{k=1}^{∞} e^{s/k} /(1 + \frac{s}{k} ) ,

the product converging uniformly in compact subsets of C − {0, −1, −2, …} because e^{x}/(1 + x) = 1 + O(x^{2}) for small x. Then Γ/g is an entire function with neither poles nor zeros, so it can be written as exp α(s) for some entire function α. We show that α(s) = −γs, where γ = 0.57721566490 … is Euler’s constant:

γ := lim_{N→∞}(− log N + ∑_{k=1}^{N} \frac{1}{k} ).

That is, we show:

Lemma. The Gamma function has the product formulas

Γ(s) = e^{−γs}g(s) = \frac{e^{−γs}}{s} ∏_{k=1}^{∞} e^{s/k} /(1 + \frac{s}{k} ) = \frac{1}{s} lim_{N→∞} (N^{s} ∏_{k=1}^{N} \frac{k}{s + k} ) . (1)

Proof : For s ≠ 0, −1, −2, …, the quotient g(s + 1)/g(s) is the limit as N→∞ of

\frac{s}{s + 1} ∏_{k=1}^{N} e^{1/k} \frac{1 + \frac{s}{k}}{1 + \frac{s+1}{k}} = \frac{s}{s + 1} (exp ∑_{k=1}^{N} \frac{1}{k} ) ∏_{k=1}^{N} \frac{k + s}{k + s + 1}

= s · \frac{N}{N + s + 1} · exp (− log N + ∑_{k=1}^{N} \frac{1}{k} ) .

Now the factor N/(N + s + 1) approaches 1, while − log N + ∑_{k=1}^{N} \frac{1}{k} →γ. Thus g(s+1) = se^{γ}g(s), and if we define Γ^{?}(s) := e^{−γs}g(s) then Γ^{?} satisfies the same functional equation Γ^{?}(s + 1) = sΓ^{?}(s) satisfied by Γ. We are claiming that in fact Γ^{?} = Γ.

Consider q := Γ/Γ^{?}, an entire function of period 1. Thus it is an analytic function of e^{2πis} ∈ C^{∗}. We wish to show that q = 1 identically. By the definition of g we have lim_{s→0} sg(s) = 1; hence

lim_{s→0} sΓ^{?}(s) = lim_{s→0} sg(s) = 1 = lim_{s→0} sΓ(s),

and q(0) = 1. We claim that there exists a constant C such that

|q(σ + it)| ≤ Ce^{π|t|/2}  (2)

for all real σ, t; since the coefficient π/2 in the exponent is less than 2π, it will follow that q is constant, and thus that Γ^{?} = Γ as claimed.

Since q is periodic, we need only prove (2) for s = σ + it with σ ∈ [1, 2]. For such s, we have |Γ(σ + it)| ≤ Γ(σ) by the integral formula and

|\frac{Γ^{?}(σ + it)}{Γ^{?}(σ)}| = ∏_{k=0}^{∞} \frac{σ + k}{|σ + k + it|} = exp −\frac{1}{2} ∑_{k=0}^{∞} log (1 + \frac{t^{2}}{(σ + k)^{2}} ) .

The summand is a decreasing function of k, so the sum is

≤ ∫_{0}^{∞} log(1 + (t/x)^{2}) dx = |t| ∫_{0}^{∞} log(1 + (1/x)^{2}) dx,

which on integration by parts becomes 2|t| ∫_{0}^{∞} dx/(x^{2} + 1) = π|t|. This proves (2) with C = sup_{1≤σ≤2} q(σ), and completes the proof of (1). □

Consequences of the product formula. Our most important application of the product formula for Γ(s) is the Stirling approximation^{1} to log Γ(s). Fix ϵ > 0 and let R_{ϵ} be the region

{s ∈ C^{∗} : |Im(log s)| < π − ϵ}.

Then R_{ϵ} is a simply-connected region containing none of the poles of Γ, so there is an analytic function log Γ on R_{ϵ}, real on R_{ϵ} ∩ R, and given by the above product formula:

log Γ(s) = lim_{N→∞}(s log N + log N! − ∑_{k=0}^{N} log(s + k)).  (3)

We prove:

Lemma. The approximation

log Γ(s) = (s − \frac{1}{2}) log s − s + \frac{1}{2} log(2π) + O_{ϵ}(|s|^{−1})  (4)

holds for all s in R_{ϵ}.

1Originally only for n! = Γ(n + 1), but we need it for complex s as well.

Proof : The estimate holds for small s, say |s| < 1, because O_{ϵ}(|s|^{−1}) well exceeds all the other terms. We thus assume |s| ≥ 1, and estimate the sum in (3) as we did for log x! in obtaining the original form of Stirling’s approximation. The sum differs from

∫_{− \frac{1}{2}}^{N+ \frac{1}{2}} log(s + x) dx = (N + \frac{1}{2} + s) log(N + \frac{1}{2} + s) − (s − \frac{1}{2}) log(s − \frac{1}{2}) − N − 1

= (N + \frac{1}{2} +s) log N +(N + \frac{1}{2} +s) log(1+ \frac{1}{N} (s+ \frac{1}{2}))−(s− \frac{1}{2}) log(s− \frac{1}{2})−N −1

by

\frac{1}{2} ∫_{− \frac{1}{2}}^{N+ \frac{1}{2}} \frac{∥x + \frac{1}{2}∥^{2}}{(s + x)^{2}} dx ≪_{ϵ} |s|^{−1}.

We already know that log N! = (N + 1/2) log N − N + A + O(N^{−1}) for some constant A. The estimate (4) follows upon taking N→∞, except for the value \frac{1}{2} log(2π) of the constant term. This constant can be obtained by letting s→∞ in the duplication formula Γ(2s) = π^{−1/2}2^{2s−1}Γ(s)Γ(s + \frac{1}{2}). □

One can go on to expand the O_{ϵ}(|s|^{−1}) error in an asymptotic series in inverse powers of s (see the Exercises), but (4) is already more than sufficient for our purposes, in that we do not need the identification of the constant term with \frac{1}{2} log 2π.

The logarithmic derivative of our product formula for Γ(s) is

\frac{Γ′(s)}{Γ(s)} = −γ − \frac{1}{s} + ∑_{k=1}^{∞} (\frac{1}{k} − \frac{1}{s + k} ) = lim_{N→∞} [log N − ∑_{k=0}^{N} \frac{1}{s + k} ] .

Either by differentiating^{2} (4) or by applying the same Euler-Maclaurin step to ∑_{0}^{N} 1/(s + k) we find that

\frac{Γ′(s)}{Γ(s)} = log s − \frac{1}{2s} + O_{ϵ}(|s|^{−2}).  (5)

Remark

The product formula for Γ(s) can also be obtained for real s by elementary means, starting from the characterization of Γ as the unique logarithmically convex function satisfying the recursion Γ(s + 1) = sΓ(s) and normalized by Γ(1) = 1 (the Bohr-Mollerup theorem, see for instance [Rudin 1976, p.193]). The theorem for complex s can then be obtained by analytic continuation. The method used here, though less elegant, generalizes to a construction of product formulas for a much more general class of functions, as we shall see next.

2While real asymptotic series cannot in general be differentiated (why?), complex ones can, thanks to Cauchy’s integral formula for the derivative. The logarithmic derivative of Γ(s) is often called ψ(s) in the literature, but alas we cannot use this notation because it conflicts with ψ(x) = ∑_{n<x} Λ(n)…

Exercises On the product formula:

1. Verify that the duplication formula for Γ(2s) yields the correct constant term in (4). Apply Euler-Maclaurin to the sum in (3) to show that the O_{ϵ}(|s|^{−1}) error can be expanded in an asymptotic series in inverse powers of s.

2. Use (1) to obtain a product formula for Γ(s)Γ(−s), and deduce that

Γ(s)Γ(1 − s) = π/ sin πs.  (6)

(This can also be obtained from Γ(s)Γ(1−s) = B(s, 1−s) by using the change of variable x = y/(y−1) in the Beta integral and evaluating the resulting expression by contour integration.) Use this together with the duplication formula and Riemann’s formula for ζ(1 − s) to obtain the equivalent asymmetrical form

ζ(1 − s) = π^{−s}2^{1−s}Γ(s) cos \frac{πs}{2}ζ(s)

of the functional equation for ζ(s). Note that the duplication formula, and its generalization

Γ(ns) = (2π)^{\frac{1−n}{2}} n^{ns− \frac{1}{2}} ∏_{k=0}^{n−1} Γ (s + \frac{k}{n}) ,

can also be obtained from (1).

3. Show that log Γ(s) has the Taylor expansion

log Γ(s) = −γ(s − 1) + ∑_{n=2}^{∞} \frac{(−1)^{n}}{n} ζ(n)(s − 1)^{n}

about s = 1. Recover from this the Laurent expansion

Γ(s) = \frac{1}{s} − γ + (γ^{2} + \frac{π^{2}}{6} )\frac{s}{2} + O(s^{2})

of Γ(s) about s = 0. Behavior of Γ(s) on vertical lines:

4. Deduce from (4) that for fixed σ ∈ R

Re(log Γ(σ + it)) = (σ − \frac{1}{2}) log |t| − \frac{π}{2} |t| + C_{σ} + O_{σ}(|t|^{−1})

as |t|→∞. Check that for σ = 0, 1/2 this agrees with the exact formulas

|Γ(it)|^{2} = \frac{π}{t sinh πt}, |Γ(1/2 + it)|^{2} = \frac{π}{cosh πt}

obtained from (6).

5. For a, b, c > 0, determine the Fourier transform of f(x) = exp(ax − be^{cx}), and check your answer by using contour integration to calculate the Fourier transform of f̂. Now apply Poisson summation, let a→0 and C = e^{c} > 1, and describe the behavior of ∑_{n=0}^{∞} z^{C^{n}} as z→1 from below. What does

∑_{n=0}^{∞}(−1)^{n}z^{2^{n}} = z − z^{2} + z^{4} − z^{8} + z^{16} − + ⋯

do as z→1? Use this to prove that Z ∩ ⋃_{m=0}^{∞}[2^{2^{2m}}, 2^{2^{2m+1}}) is an explicit example of a set of integers that does not have a logarithmic density.

An alternative proof of the functional equation for ζ(s):

6. Prove that

ζ(s) = \frac{1}{Γ(s)} ∫_{0}^{∞} u^{s−1} \frac{du}{e^{u} − 1}

for σ > 1, and that when s is not a positive integer an equivalent formula is

ζ(s) = −\frac{e^{−πis}}{2πi} Γ(1 − s) ∫_{C} u^{s−1} \frac{du}{e^{u} − 1}

where C is a contour coming from +∞, going counterclockwise around u = 0, and returning to +∞:

[figure: u; C]

Show that this gives the analytic continuation of ζ to a meromorphic function on C; shift the line of integration to the left to obtain the functional equation relating ζ(s) to ζ(1 − s) for σ < 0, and thus for all s by analytic continuation.

References

[WW 1940] Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis…^{3} (fourth edition). Cambridge University Press, 1940 (reprinted 1963). [HA 9.40 / QA295.W38]

[Rudin 1976] Rudin, W.: Principles of Mathematical Analysis (3rd edition). New York: McGraw-Hill, 1976.

3The full title is 26 words long, which was not out of line when the book first appeared in 1902. You can find the title in Hollis.