STRONG MULTIPLICITY ONE FOR SELBERG’S CLASS

M. Ram Murty^{1} and V. Kumar Murty^{2}

Abstract. Let F and G be two elements in the Selberg class of Dirichlet series. Write F (s) = ∑_{n=1}^{∞} a_{n}(F)/n^{s} and G(s) = ∑_{n=1}^{∞} a_{n}(G)/n^{s}. If for all but finitely many primes p, a_{p}(F) = a_{p}(G) and a_{p^{2}}(F) = a_{p^{2}}(G) then F = G. We also show the following. Let Z_{F} (T) be the multiset of zeroes of F (s) in the region Re(s) ≥ 1/2, |Im(s)| ≤ T. Suppose that as T →∞,

|Z_{F} (T)ΔZ_{G}(T)| = o(T),

where Δ denotes the symmetric difference. Then F = G.

Multiplicité un forte pour la classe de Selberg

Resumé. Soient F et G deux fonctions dans la classe de Selberg. Ecrivons F (s) = ∑_{n=1}^{∞} a_{n}(F)/n^{s} et G(s) = ∑_{n=1}^{∞} a_{n}(G)/n^{s}. Si pour presque tous les nombres premiers p, a_{p}(F) = a_{p}(G) et a_{p^{2}}(F) = a_{p^{2}}(G), alors F = G. Nous démontrons aussi le résultat suivant. Soit Z_{F} (T) l’ensemble des zéros de F (s) dans la région Re(s) ≥ 1/2 et |Im(s)| ≤ T, comptés avec multiplicité. Nous supposons que lorsque T →∞, on a

|Z_{F} (T)ΔZ_{G}(T)| = o(T)

où Δ désigne la différence symétrique. Alors on a F = G.

Version française abrégée La classe de Selberg est l’ensemble S des fonctions F (s) avec les propriétés suivantes:

(i) (série de Dirichlet) pour Re(s) > 1, on a l’égalité

F (s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

où a_{1} = 1. (Nous écrivons a_{n}(F) = a_{n} pour les coefficients de l’écriture de F en série de Dirichlet.);

(ii) (prolongement analytique) il existe un entier m ≥ 0, tel que (s − 1)^{m}F (s) ait un

prolongement en une fonction entière d’ordre fini; (iii) (équation fonctionelle) il existe des nombres Q > 0, α_{i} > 0, et r_{i} ∈ C avec Re(r_{i}) ≥ 0

tels que la fonction

Φ(s) = Q^{s} ∏_{i=1}^{d} Γ(α_{i}s + r_{i})F (s)

vérifie l’equation fonctionelle

Φ(s) = w\overline{Φ(1 − \overline{s})}

1 Research partially supported by NSERC and FCAR grants.

2 Research partially supported by NSERC.

où w est une nombre complexe avec |w| = 1. (iv) (produit eulérien) la fonction F (s) s’ecrit

F (s) = ∏_{p} F_{p}(s)

où

F_{p}(s) = exp (∑_{k=1}^{∞} \frac{b_{p^{k}}}{p^{ks}} )

où b_{p^{k}} = O(p^{kθ}) pour un certain θ < 1/2 et p désigne une nombre premier. (Nous ecrivons b_{p}(F) = b_{p}.)

(v) (l’hypothèse de Ramanujan) pour tout ϵ > 0 fixé, on a l’égalité

a_{n} = O(n^{ϵ}).

Une fonction F ∈ S est dite primitive s’il est impossible d’écrire F = F_{1}F_{2} avec F_{1}, F_{2} ∈ S, F_{1} ≠ 1, F_{2} ≠ 1. Selberg a conjecturé

(a) pour toute fonction primitive F, on a l’égalité

∑_{p≤x} \frac{|a_{p}(F)|^{2}}{p} = log log x + O(1);

(b) si F et G sont des fonctions primitives distinctes, on a l’egalité

∑_{p≤x} \frac{a_{p}(F)\overline{a_{p}(G)}}{p} = O(1)

quand x→∞.

Nous savons que chaque élément de S a une factorisation en produit de fonctions primitives. Ces conjectures impliquent que cette factorisation est unique. Elles impliquent aussi que, si pour presque tout p, on a l’égalité a_{p}(F) = a_{p}(G), alors on a F = G. (Une propriété est vraie, par définition, pour presque tout p, si le nombre d’exceptions en p est fini.) Théorème 1. Supposons que les égalités a_{p}(F) = a_{p}(G), a_{p^{2}}(F) = a_{p^{2}}(G) soient vraies pour presque tout p. Alors on a F = G. Corollaire. Si l’égalité F_{p}(s) = G_{p}(s) est vraie pour presque tout p, alors on a F = G. Théorème 2. Soit F ∈ S, et soit Z_{F} (T) l’ensemble des zéros de F (s), comptés avec multiplicité, dans la région Re(s) ≥ 1/2, |Im(s)| ≤ T. Pour F, G ∈ S supposons que la différence symétrique vérifie

|Z_{F} (T)ΔZ_{G}(T)| = o(T)

quand T →∞. Alors on a F = G.

La preuve du théorème 1 dépend du lemme suivant:

Lemme. Soit f(s) une fonction méromorphe sur C, sans pôle dans le demi-plan fermé Re(s) ≥ 1/2, et ne s’annulant pas dans ce demi plan. Supposons que log f(s) est une série de Dirichlet et qu’il existe A, α_{i}, γ_{i} > 0, β_{i}, δ_{i} ∈ C avec Re(β_{i}), Re(δ_{i}) ≥ 0 tels que la fonction

H(s) = A^{s} \frac{∏_{i=1}^{d_{1}} Γ(α_{i}s + β_{i})}{∏_{i=1}^{d_{2}} Γ(γ_{i}s + δ)}f(s)

verifie l’equation fonctionelle

H(s) = w\overline{H(1 − \overline{s})}

pour un certain nombre complexe w de module 1. Alors f(s) est une constante.

La preuve du théorème 2 commence par une généralisation d’une formule classique de Landau:

−b_{n}(F) log n = 2π lim_{T →∞} \frac{1}{T} ∑_{0<Im(ρ)<T} n^{ρ}

oú la somme est faite sur les zéros ρ de F (s).

The Selberg class S consists of functions F (s) of a complex variable s satisfying the following properties:

(i) (Dirichlet series) for Re(s) > 1,

F (s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

where a_{1} = 1. (We will write a_{n}(F) = a_{n} for the coefficients of the Dirichlet series of F.);

(ii) (Analytic continuation) for some integer m ≥ 0, (s − 1)^{m}F (s) extends to an entire

function of finite order; (iii) (Functional equation) there are numbers Q > 0, α_{i} > 0 and r_{i} ∈ C with Re(r_{i}) ≥ 0

so that

Φ(s) = Q^{s} ∏_{i=1}^{d} Γ(α_{i}s + r_{i})F (s)

satisfies

Φ(s) = w\overline{Φ}(1 − s)

where w is a complex number with |w| = 1 and \overline{Φ}(s) = \overline{Φ(\overline{s})}; (iv) (Euler product)

F (s) = ∏_{p} F_{p}(s)

where

F_{p}(s) = exp (∑_{k=1}^{∞} \frac{b_{p^{k}}}{p^{ks}} )

where b_{p^{k}} = O(p^{kθ}) for some θ < 1/2 and p denotes a prime number here and throughout the paper. (We shall write b_{p}(F) = b_{p}.);

(v) (Ramanujan hypothesis) For any fixed ϵ > 0, a_{n} = O(n^{ϵ}).

A function F ∈ S is called primitive if it cannot be written as a non-trivial product of two elements in S. It can be shown that every element in S can be written as a product of primitive elements. (See [1] and [2].) Selberg [4, (1.12) and (1.13)] conjectures:

(a) for any primitive function F,

∑_{p≤x} \frac{|a_{p}(F)|^{2}}{p} = log log x + O(1);

(b) for two distinct primitive functions F and G,

∑_{p≤x} \frac{a_{p}(F)\overline{a_{p}(G)}}{p} = O(1).

These conjectures imply that the factorization into primitive elements is unique. It is easy to see that the conjectures also imply that if a_{p}(F) = a_{p}(G) for all but finitely many primes p, then F = G. This looks difficult to prove unconditionally. We will show:

Theorem 1. Suppose a_{p}(F) = a_{p}(G), a_{p^{2}}(F) = a_{p^{2}}(G) for all but finitely many primes p. Then F = G.

Corollary . If F_{p}(s) = G_{p}(s) for all but finitely many primes p, then F = G.

The corollary can be viewed as an analogue of the strong multiplicity one theorem for automorphic L-functions on GL_{n}.

One can also prove analogously:

Theorem 2. For F ∈ S, denote by Z_{F} (T) the multiset of zeroes of F (s) in the region Re(s) ≥ 1/2, |Im(s)| ≤ T. For F, G ∈ S, suppose the symmetric difference satisfies

|Z_{F} (T)ΔZ_{G}(T)| = o(T)

as T →∞. Then F = G.

We begin by noting that in view of the functional equation and the Phragmén-Lindelöf theorem, any entire element of S is of order 1. (In general, if F (s) has a pole of order m at s = 1, then (s − 1)^{m}F (s) is necessarily of order 1.) The proof of Theorem 1 depends on the following lemma.

Lemma. Let f(s) be a a meromorphic function on C, analytic for Re(s) ≥ 1/2 and non-vanishing there. Suppose that log f(s) is a Dirichlet series and that f(s) satisfies the functional equation

H(s) = w\overline{H}(1 − s)

where w is some complex number of absolute value 1, and

H(s) = A^{s} \frac{∏_{i=1}^{d_{1}} Γ(α_{i}s + β_{i})}{∏_{i=1}^{d_{2}} Γ(γ_{i}s + δ_{i})}f(s)

with certain A, α_{i}, γ_{i} > 0 and Re(β_{i}), Re(δ_{i}) ≥ 0. Then f(s) is constant.

Proof. Recall that [5, p. 313] for any Dirichlet series F (s)

lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} F (σ + it)y^{σ+it}dt = \{ \begin{matrix}a_{n}(F) & if n = y \\ 0 & otherwise \end{matrix}

where σ > σ_{a}(F) the abscissa of absolute convergence of F. In other words,

\frac{1}{2πi} ∫_{(σ)} F (s)y^{s}ds = ∞

if y = n and a_{n}(F) ≠ 0. From the functional equation and the fact that the factor corresponding to the gamma functions introduce poles only in Re(s) ≤ 0, we find that H(s) is analytic in Re(s) ≥ 1/2 and does not vanish in this half-plane. By the functional equation, the same is true in Re(s) ≤ 1/2. Thus H(s) is entire. Moreover, Stirling’s formula and the functional equation imply that it is of order 1. By Hadamard’s theorem, it follows that H(s) = e^{as+b} for some constants a and b. Hence,

\frac{f′}{f} (s) = a − log A + ∑_{i=1}^{d_{2}} \frac{Γ′}{Γ} (γ_{i}s + δ_{i})γ_{i} − ∑_{i=1}^{d_{1}} \frac{Γ′}{Γ} (α_{i}s + β_{i})α_{i}

is a Dirichlet series. The derivative of this is again a Dirichlet series. Since

\frac{d}{ds} (\frac{Γ′}{Γ} (s)) = − ∑_{m=0}^{∞} \frac{1}{(s + m)^{2}}

and since

\frac{1}{2πi} ∫_{(σ)} \frac{y^{s}}{(αs + β)^{2}} ds = \{ \begin{matrix}α^{−2}y^{−β/α} log y & if y > 1 \\ 0 & if y ≤ 1 \end{matrix}

for σ sufficiently large, we deduce by an easy computation that

\frac{1}{2πi} ∫_{(σ)} \frac{d}{ds} (\frac{f′}{f} (s)) y^{s}dy = O(1)

for any y ≥ 1. Hence, every coefficient of d/ds(f′/f(s)) is zero. As f′/f is a Dirichlet series, this implies that f′/f = 0. Hence f is constant. This completes the proof of the lemma.

We can now prove Theorem 1. Set

f(s) = ∏_{p}(F_{p}(s)/G_{p}(s)).

Since log F_{p}(s) is an absolutely convergent Dirichlet series for Re(s) > θ, we deduce that F_{p}(s) is absolutely convergent for Re(s) > θ and is non-vanishing there. In particular, this holds for Re(s) ≥ 1/2. We can factor

(1 + \frac{a_{p}(F)}{p^{s}} + \frac{a_{p^{2}}(F)}{p^{2s}})^{−1}

from the numerator and denominator of F_{p}(s)/G_{p}(s) and write

f(s) = ∏_{p} f_{p}(s)

where each f_{p}(s) converges absolutely for Re(s) > θ (for some θ < 1/2, and does not vanish there. f(s) satisfies the conditions of the lemma. Hence f(s) is constant which must be 1 because that is the constant term of the Dirichlet series. Therefore F = G. This completes the proof of Theorem 1.

Set b_{x}(F) = 0 unless x is a prime power. To prove Theorem 2, we begin by noting the formula

−b_{n}(F) log n = 2π lim_{T →∞} \frac{1}{T} ∑_{0<Im(ρ)<T} n^{ρ}

where the sum is over the zeroes ρ of F (s). This is the analogue of a classical formula of Landau for the Riemann zeta function. It is easily derived from the integral identity used in the beginning of the proof of the lemma and by the method of [6, pp.61-62] in conjunction with the lemmas of [3, p.540-544]. Indeed, let q be a positive number, less than the ordinate of any zero of F in the region Re(s) > 0 and Im(s) > 0. Moreover, we suppose that q > max_{1≤i≤d} |Im(r_{i})/α_{i}| where r_{i} and α_{i} are as in (iii) in the definition of the Selberg class. Then,

| ∫_{2+iq}^{2+iT} \frac{F′}{F} (s)x^{s}ds + i(T − q)b_{x}(F) log x| ≤ 2x^{2} ∑_{{n=1 \atop n≠x}}^{∞} \frac{|b_{n}(F)| log n}{n^{2}| log(x/n)|}.

By Cauchy’s theorem,

∫_{2+iq}^{2+iT} \frac{F′}{F} (s)x^{s}ds − 2πi ∑_{q<Im(ρ)<T} x^{ρ} = ∫_{2+iq}^{a+iq} + ∫_{a+iq}^{a+iT} + ∫_{a+iT}^{2+iT} = I_{1} + I_{2} + I_{3} (say)

where a < 0 and T is not the ordinate of any zero. It is also seen that by the analogue of lemma 2 of [3,p.540],

\frac{F′}{F} (s) = O(log |s|)

for σ ≤ −1, t ≥ q. Thus, the integral I_{2} tends to zero as a→ − ∞ and so

∫_{2+iq}^{2+iT} \frac{F′}{F} (s)x^{s}ds − 2πi ∑_{q<Im(ρ)<T} x^{ρ} = ∫_{2+iq}^{−∞+iq} + ∫_{−∞+iT}^{2+iT} .

Again, we find

∫_{−∞+iT}^{−1+iT} \frac{F′}{F} (s)x^{s}ds = ∫_{−∞}^{−1} O(log(|σ| + T))x^{σ}dσ = O(log T)

which applies for T = q also. Now by the analogue of lemma 5 of [3,p.542] we find

∫_{−1+iT}^{2+iT} \frac{F′}{F} (s)x^{s}ds = ∑_{|T −γ|<1} ∫_{−1+iT}^{2+iT} \frac{x^{s}}{s − ρ}ds + O(log T).

We handle the integral in the sum as in [6, p. 62]: write

∫_{−1+iT}^{2+iT} = ∫_{−1+iT}^{−1+i(T +2)} + ∫_{−1+i(T +2)}^{2+i(T +2)} + ∫_{2+i(T +2)}^{2+iT}

and notice that each term is O(1) and there are O(log T) terms so we deduce that the sum on the right is O(log T). Since there are only a finite number of zeroes ρ with 0 < Im(ρ) ≤ q, we deduce that

\frac{−b_{x}(F) log x}{2π} = lim_{T →∞} \frac{1}{T} ∑_{0<Im(ρ)<T} x^{ρ}.

To prove the theorem, we have by hypothesis,

\frac{−b_{x}(F) log x}{2π} = lim_{T →∞} \frac{1}{T} ∑_{{0<Im(ρ)<T \atop F (ρ)=0}} x^{ρ}

= lim_{T →∞} \frac{1}{T} ( ∑_{{0<Im(ρ)<T \atop G(ρ)=0}} x^{ρ} + o(T))

= \frac{−b_{x}(G) log x}{2π}

so that F = G. This completes the proof.

It is clear that the proof actually establishes a stronger result. Let I_{T} be an interval contained in [0, T] and let Z_{F} (I_{T}) denote the multiset of zeroes of F whose imaginary parts are contained in I_{T}. Suppose that |I_{T} | = f(T) log T with f(T)→∞ as T →∞. If |Z_{F} (I_{T})ΔZ_{G}(I_{T})| = o(|I_{T} |) as T →∞, then F = G.

REFERENCES

[1] B. Conrey and A. Ghosh, On the Selberg class of Dirichlet series: Small weights, preprint.

[2] M. Ram Murty, Selberg’s conjectures and Artin L-functions, Bulletin of the Amer. Math. Soc., (1994) to appear.

[3] V. Kumar Murty, Explicit formulae and the Lang-Trotter conjecture, Rocky Mountain Journal of Mathematics, 15 (1985) 535-551.

[4] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Collected Papers, Vol. 2, pp. 47-63, Springer-Verlag, 1991.

[5] E.C. Titchmarsh, The Theory of Functions, Oxford University Press, Second Edition, 1939.

[6] E.C. Titchmarsh, The zeta function of Riemann, Cambridge Tracts in Mathematics, No. 26, Cambridge University Press, 1930.

Address of authors: Department of Mathematics, McGill University, Montreal, H3A 2K6, Canada murty@math.mcgill.ca Department of Mathematics, University of Toronto, Toronto, M5S 1A4, Canada murty@math.toronto.edu