An expansion of the Riemann Zeta function on the critical line

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An expansion of the Riemann Zeta function on the critical line

Bernard Candelpergher

To cite this version:

Bernard Candelpergher. An expansion of the Riemann Zeta function on the critical line. 2021. �hal- 03271709�

An expansion of the Riemann Zeta function on the critical line B.Candelpergher

Universit´e Cˆote d’Azur, CNRS, LJAD (UMR 7351), Nice, France. candel@unice.fr

Abstract

We give an expansion of the Riemann Zeta function on the critical line as a converging series P

m≥0a_mq_m(¹₂ +it) in the space L²(R,_cosh(πt)^dt ), where the functions q_m are related to Meixner polynomials of the first kind and the coefficients am are linear combinations of the Euler constant γ and the values ζ(2), ζ(3), . . . , ζ(m+ 1).

1 Laguerre functions

The Laguerre polynomials x7→Lm(2x) are defined by the generating function (cf. [6]) 1

1−ae^−1−a2ax = 1 1 +a

+∞

X

m=0

L_m(2x)a^m where |a|<1.

They are given by L_m(2x) =Pm

k=0C_m^k(−2)^{k x}_k!^k. The Laguerre functions ϕ_m(x) =√

2 e^−xL_m(2x),

form an orthonormal basis of L²(]0,+∞[, dx), and we have the generating function

√2

1−ue^−x1+u1−u =

+∞

X

m=0

ϕ_m(x)u^m where |u|<1. (1)

With z = ^1+u_1−u, we get for Re(z)>0

e^−xz =√ 2π

+∞

X

m=0

ϕ_m(x)ψ_m(z), (2)

where ψ_m is the function defined in the half-plane {Re(z)>0} by ψ_m(z) = 1

√π(1 +z)

z−1 z+ 1

m

. By (2) we see that the function ψ_m is related toϕ_m by

ψ_m(z) = (e^−xz

√2π, ϕ_m) = Z +∞

0

e^−xz

√2π ϕ_m(x)dx.

Thus we have ψ_m(z) = ^√1

2π Lϕ_m(z), where L is the Laplace transform Lf(z) =

Z +∞

0

e^−xzf(x) dx.

This transformation maps the space L²(]0,+∞[, dx) to the Hardy spaceH²(P) of analytic functions g in the half-plane P ={Re(z >0}such that: there exists M_g >0 with

Z

R

|g(x+iy)|² dy≤M_g for all x >0.

By a theorem of Paley and Wiener each such g has non-tangential limits g(iy) = lim_x→0g(x+iy) at almost every point of the imaginary axis. The space H²(P) is an Hilbert space with the inner product

(g, h)_H² = Z

R

g(iy)h(iy)dy,

and the functions ψ_m (m ≥0), form an orthonormal basis ofH²(P). If f ∈L²(]0,+∞[, dx) then f =X

m≥0

(f, ϕ_m)ϕ_m ⇔ Lf(z) =√ 2πX

m≥0

(f, ϕ_m)ψ_m(z).

2 Mellin transform and Meixner polynomials

For a function f on ]0,+∞[, we define the Mellin transform of f by M(f)(s) =

Z +∞

0

x^s−1f(x)dx, which is supposed to be defined for s∈C such that 0< Re(s)<1.

We have MLf(s) = Γ(s)Mf(1−s) for 0< Re(s)< 1, and iff and g are in L²(]0,+∞[), we have the Parseval-Mellin formula (cf. [4]) :

1 2iπ

Z ¹₂+i∞

1 2−i∞

M(f)(z)M(g)(z) dz = Z +∞

0

f(x)g(x) dx.

The Mellin transform of ϕ_m is given by Z +∞

0

ϕ_m(x)x^s−1 dx=√ 2

m

X

k=0

C_m^k(−2)^k k!

Z +∞

0

e^−xx^s+k−1 dx=√

2 Γ(s)q_m(s), with

qm(s) =

m

X

k=0

C_m^k(−2)^k (s)_k

k! where (s)k=s(s+ 1)· · ·(s+k−1) (with (s0) = 1).

We have (cf. [2])q_m(s) =F(−m, s; 1; 2) whereF is the Gauss hypergeometric function (also denoted by 2F1). We have also qk(s) = _k!¹mk(−s,1,−1), wheremk is a Meixner polynomial of the first kind.

Since Mϕ_m(s) = √

2 Γ(s)q_m(s) and ψ_m = ^√1

2πLϕ_m, we get for 0< Re(s)<1 Mψ_m(s) = 1

√2πMLϕ_m(s) = 1

√2πΓ(s)Γ(1−s)Mϕ_m(1−s) =

√π

sin(πs)q_m(1−s).

By the definition of ψ_m we verify that ¹_xψ_m(¹_x) = (−1)^mψ_m(x), thus we have Mψ_m(1−s) = (−1)^mMψ_m(s),

this gives

q_m(1−s) = (−1)^mq_m(s).

By the Parseval-Mellin formula we get 1

2π Z +∞

−∞

M(ϕm)(1

2+it)M(ϕn)(1

2+it) dt = Z +∞

0

ϕm(x)ϕn(x) dx=δm,n

(with δ_m,n = 1 ifm =n and δ_m,n = 0 ifm 6=m). This gives δ_m,n =

Z +∞

−∞

Γ(1

2 +it)q_m(1

2+it)Γ(1

2+it)q_n(1

2 +it) dt π =

Z +∞

−∞

q_m(1

2 +it)q_n(1

2+it) dt cosh(πt) Thus the polynomials t7→q_m(¹₂+it) form an orthonormal basis ofL²(R,_cosh(πt)^dt ) with respect to the scalar product

(f|g) = Z +∞

−∞

f(t)g(t) dt cosh(πt)

This implies that all the zeros of the polynomials t7→q_m(¹₂ +it) are real.

We have q₀ = 1 and

q₁(1

2 +it) = −2it q₂(1

2 +it) = 1 2 −2t² q₃(1

2 +it) = −5 3it+4

3it³ q₄(1

2 +it) = 3 8 −7

3t²+ 2 3t⁴ . . .

By Mellin transform of (1), we see that the generating function of the polynomials q_m is

+∞

X

m=0

q_m(s)u^m = 1 1−u

1 +u 1−u

−s

for u∈]−1,1[. (3)

This gives, with y= ^1+u_1−u, the relation y^−s = 2√

π

+∞

X

m=0

ψm(y)qm(s) for y >0. (4)

Let s= ¹₂ +it with t ∈Rand y=e^−ξ, we get e^itξ = 2√

πe^−ξ/2

+∞

X

m=0

ψ_m(e^−ξ)q_m(1

2+it) for ξ ∈R.

The latter series converges in L²(R,_cosh(πt)^dt ) since P+∞

m=0|ψ_m(e^−ξ)|² <+∞.

If a function h∈L²(R,_cosh(πt)^dt ) has, in this space, an expansion kike h(t) =X

n≥0

a_nq_n(1 2 +it), then we have

(h|e^itξ) = 2√ πe^−ξ/2

+∞

X

m=0

a_mψ_m(e^−ξ), that is

F h(t) cosh(πt)

(ξ) = 2√ πe^−ξ/2

+∞

X

m=0

a_mψ_m(e^−ξ) (5)

where F is the Fourier transform defined by Fg(ξ) =R+∞

−∞ g(t)e^−itξdt.

3 An expansion of Zeta

3.1 A Fourier transform

In the critical strip 0< Re(s)<1, we have (cf. [7]) ζ(s) = 1

Γ(s) Z +∞

0

f(x)x^s−1dx= 1

Γ(s)Mf(s) where f(x) = 1

e^x−1 − 1 x (also we have ζ(s) =sR+∞

0 ([x]−x)x^−s−1dx, which gives |ζ(¹₂ +it)|=O(|t|) for t→ ±∞).

Since we have (cf.[2]) for x >0

L(f)(x) = log(x)−Ψ(1 +x) where Ψ = Γ⁰ Γ, then we get for 0< Re(s)<1

M(log(x)−Ψ(1 +x))(s) =ML(f)(s) = Γ(s)M(f)(1−s) = Γ(s)Γ(1−s)ζ(1−s),

thus π

sin(πs)ζ(1−s) =M(log(x)−Ψ(1 +x))(s).

By Mellin inversion, we obtain for x >0 log(x)−Ψ(1 +x) = 1

2iπ

Z c+i∞

c−i∞

π

sin(πs)ζ(1−s)x^−s ds for all 0< c <1.

With c= ¹₂, we have for x >0

log(x)−Ψ(1 +x) = 1 2

Z +∞

−∞

ζ(1

2+it) 1

√xe^itlog(x) dt cosh(πt).

Let x=e^−ξ with ξ ∈R, then we get the Fourier transform Fζ(¹₂ +it)

cosh(πt)

(ξ) = −2e^−ξ/2(ξ+ Ψ(1 +e^−ξ)) = 2e^−ξ/2Lf(e^−ξ). (6) Remark. The Fourier transform given by the relation (6), gives for g ∈L²(R), the relation

Z +∞

−∞

ζ(1

2 +it)Fg(t) dt

cosh(πt) =−2 Z +∞

−∞

g(ξ)e^−ξ/2(ξ+ Ψ(1 +e^−ξ))dξ.

For example, let g(t) = t^s−1e^−atχ[0,+∞[(t) with a > 0 and Re(s) > ¹₂. Then Fg(t) = _(a+it)^Γ(s)s and we have

−1 2

Z +∞

−∞

ζ(1

2+it) 1 (a+it)^s

dt

cosh(πt) = 1 Γ(s)

Z +∞

0

ξ^s−1e^−ξ(a+12)(ξ+ Ψ(1 +e^−ξ)) dξ.

Expanding the Ψ function as

Ψ(1 +e^−ξ) =−γ+

+∞

X

n=1

(−1)ⁿ⁺¹ζ(n+ 1)e^−nξ since 0< e^−ξ <1, we get for α=a+ ¹₂ > ¹₂

−1 2

Z +∞

−∞

ζ(1

2 +it) 1 (α− ¹₂ +it)^s

dt

cosh(πt) = s

α^s+1 −γ 1 α^s +

+∞

X

n=1

(−1)ⁿ⁺¹ζ(n+ 1) 1 (n+α)^s, Thus, for x >−¹₂ and Re(s)> ¹₂, we have a generalization of a formula of I.V.Blagouchine (cf.[1])

γ (x+ 1)^s +

+∞

X

n=2

(−1)ⁿ⁻¹ ζ(n)

(n+x)^s = s

(x+ 1)^s+1 +1 2

Z +∞

−∞

ζ(1

2+it) 1 (x+¹₂ +it)^s

dt

cosh(πt) (7)

3.2 Expansion of ζ (

+ it)

By (5) and (6), the coefficients a_m of the expansion ζ(¹₂ +it) = P

m≥0a_mq_m(¹₂ +it) in the space L²(R,_cosh(πt)^dt ) are given by

Lf(e^−ξ) =√ π

+∞

X

m=0

a_mψ_m(e^−ξ). (8)

For an explicit evaluation of a_m, let u= ^e_e^−ξ−ξ⁻¹+1 in the relation (8), then we get for −1< u <1 1

1−u

log(1 +u

1−u)−Ψ(1 + 1 +u 1−u)

=

+∞

X

m=0

b_mu^m where b_m = a_m

2 . (9)

Now, take the Taylor expansion of the left side of (8). For the logarithmic part, we have simply 1

1−ulog(1 +u

1−u) = 1 1−u

+∞

X

n=0

1−(−1)ⁿ n uⁿ=

+∞

X

n=1

Xⁿ

p=1

1−(−1)^p p

uⁿ,

For the part involving the function Ψ, we need the help of (cf.[2]) the integral formula Ψ(x+ 1) = 1

x + Ψ(x) = 1

x −γ+ Z +∞

0

e^−t−e^−xt 1−e^−t dt, this gives

− 1 1−uΨ

1 + 1 +u 1−u)

=− 1

1 +u + 1

1−uγ− 1 1−u

Z +∞

0

e^−t−e^−1+u1−ut 1−e^−t dt, and, with (1), we get

− 1 1−uΨ

1 + 1 +u 1−u

=− 1

1 +u+ 1 1−uγ−

+∞

X

m=1

Z +∞

0

e^−t(1−L_m(2t))

1−e^−t dt u^m. Since, for m integer ≥1, we have

− Z +∞

0

e^−t(1−L_m(2t)) 1−e^−t dt =

Z +∞

0

e^−t 1−e^−t(

m

X

k=1

C_m^k(−2)^kt^k k!)dt=

m

X

k=1

C_m^k(−2)^kζ(k+ 1).

then we have proved the following theorem.

Theorem. The expansion of t7→ζ(¹₂ +it) in the space L²(R,_cosh(πt)^dt ) is given by ζ(1

2+it) = 2X

m≥0

b_mq_m(1

2+it) (10)

with b₀ =−1 +γ, and for m≥1 b_m =

m

X

p=1

1−(−1)^p

p + (−1)^m+1+γ+

m

X

k=1

(−2)^kC_m^k ζ(k+ 1). (11) For example, we have b₁ = 3 +γ−2ζ(2), and

b₂ = 1 +γ−4ζ(2) + 4ζ(3) b₃ = 11

3 +γ−6ζ(2) + 12ζ(3)−8ζ(4) b₄ = 5

3 +γ−8ζ(2) + 24ζ(3)−32ζ(4) + 16ζ(5) Since we have, for m ≥1, the combinatorial identity

m

X

p=1

1−(−1)^p

p =−

m

X

k=1

C_m^k(−2)^k1 k,

then we get, for the coefficients of (11), the simple expression b_m =

m

X

k=0

C_m^k(−1)^kz_k with z₀ =γ−1 and z_k = 2^k ζ(k+ 1)−1− 1 k

if k ≥1. (12) Remark. We have for any integer k ≥0

Z +∞

0

1

e^x−1 − 1 x

e^−x2^kx^k k! =z_k, thus we get

b_m = Z +∞

0

1

e^x−1− 1 x

e^−xL_m(2x)dx= 1

√2(f, ϕ_m), as we expected, since by Mellin transform we have formally

ζ(1

2 +it) = 2X

m≥0

b_mq_m(1

2+it)⇔f =√ 2

+∞

X

m=0

b_mϕ_m

3.3 An integral formula

Since the binomial transform bv_m =Pm

k=0(−1)^kC_m^kv_k is involutive then we have by (12) z_m =

m

X

k=0

(−1)^kC_m^kb_k.

From (10), we have for m ≥0 b_m = 1

2 Z +∞

−∞

ζ(1

2 +it)q_m(1

2 −it) dt

cosh(πt), (13)

thus the binomial transform of (b_m) is given by the binomial transform of (q_m(s)). We have q_m(s) =

m

X

k=0

C_m^k(−2)^k (s)_k

k! ⇒2^m (s)_m m! =

m

X

k=0

(−1)^kC_m^kq_k(s), thus

m

X

k=0

(−1)^kC_m^kbk = 2^m Z +∞

−∞

ζ(1

2+it)(¹₂ −it)_m m!

dt cosh(πt). Finally we get the integral expression γ = 1 + ¹₂R+∞

−∞ ζ(¹₂ +it)_cosh(πt)^dt ,and ζ(m+ 1) = 1 + 1

m + 1 2π

Z +∞

−∞

ζ(1

2 +it)Γ(¹₂ +it)Γ(¹₂ −it+m)

Γ(m+ 1) dt for m≥1.

We see that the analytic functions f and g defined by f(s) =ζ(s+ 1)− 1

s for s6= 0 with f(0) =γ, and

g(s) = 1 + 1 2π

Z +∞

−∞

ζ(1

2+iu)Γ(¹₂ +iu)Γ(¹₂ −iu+s)

Γ(s+ 1) du for Re(s)>−1 2.

are such that f(m) = g(m) for all integersm ≥0. Then by the Carlson’s theorem we get f(s) = g(s) for Re(s)>−¹₂. This gives the integral formula

ζ(s+ 1)− 1

s = 1 + 1 2π

Z +∞

−∞

ζ(1

2 +iu)Γ(¹₂ +iu)Γ(¹₂ −iu+s)

Γ(s+ 1) du for Re(s)>−1

2, s6= 0. (14) Remark. Let the function defined for Re(s)>0 by ζ_a(s) =P+∞

n=1

(−1)ⁿ⁻¹

n^s = (1− ₂²s)ζ(s). We have ζa(s) = 1

Γ(s) Z +∞

0

f(x)x^s−1dx where fa(x) = 1 e^x+ 1.

By similar calculations as before we get the following expansion in the space L²(R,_cosh(πt)^dt ) (1−√

2 2^−it)ζ(1

2+it) = 2X

m≥0

c_mq_m(1

2+it) with c_m = (−1)^m−Log(2)−

m

X

k=1

C_m^k(−1)^k(2^k−1)ζ(k+1).

We have c₀ := 1−ln (2), and

c1 = −1−ln (2) +ζ(2)

c₂ = 1−ln(2) + 2ζ(2)−3ζ(3)

c₃ = −1−ln +3ζ(2)−9ζ(3) + 7ζ(4)

c₄ = 1−ln (2) + 4ζ(2)−18ζ(3) + 28ζ(4)−15ζ(5)

Acknowledgments.

4 Bibliography

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[3] G.Hetyei. Meixner polynomials of second kind and quantum algebras representing su(1,1).

Proceedings of the Royal Society A 466 (2010) (p.1409-1428)

[4] A.Ivic. The Riemann Zeta-function. Theory and Applications. Dover (2003)

[5] A. Kuznetsov. Expansion of the Riemann Ξ Function in Meixner-Pollaczec Polynomials.

Canad. Math. Bull. Vol. 51 (4), 2008 (p.561-569).

[6] N.N.Lebedev. Special functions and their applications. Dover (1972)

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Heath-Brown. Clarendon Press Oxford. (1988)