André Voros
email: andre.voros@cea.fr

Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis

We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz zeta function. As a concrete application, a superzeta function enters an integral representation for the Keiper-Li coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point entirely depends on the Riemann Hypothesis being true or not, and the outcome is a sharp exponential-asymptotic criterion for the Riemann Hypothesis that only refers to the large-order Keiper-Li coefficients. As a new result, that criterion, then Li's criterion, are transposed to a novel sequence of Riemann-zeta expansion coefficients based at the point 1/2 (vs 1 for Keiper-Li).

It is a great honor and pleasure to dedicate this work to Professor AOKI, who has had zeta functions among his numerous activities; we are specially grateful to him for setting up a major conference in Osaka in 2003 [START_REF]Zeta Functions, Topology and Quantum Physics[END_REF] that much stimulated our work afterwards. And the present work also touches another theme in which we have much longer fruitfully interacted with him (and collaborators, mainly at the RIMS), namely complex exponential asymptotics (here embodied in the method of steepest descents).

(Generalized) zeta functions

Discrete numerical sets {w k } that have "natural", "collective" definitions (e.g., roots of equations, spectra, . . . ), are often better accessible through their symmetric functions, which easily display richer and more explicit properties than the initial data. Specially fruitful symmetric functions turn out to be the zeta functions, defined (formally) as

Z(x) = k w -x k (standard zeta), Z(x, w) = k (w k + w) -x (generalized zeta), (1)
typically for complex arguments. As a rule, the exponent (here, x) is the key variable; the shift parameter (w) adds useful flexibility but retains an auxiliary status.

Based on [START_REF] Voros | Zeta functions over zeros of zeta functions[END_REF] and references therein, we will specifically review cases where {w k } is the set of nontrivial zeros {ρ} of Riemann's zeta function (or some related set, or a generalization) -we will name the resulting functions (1) "superzeta" functions for brevity.

Our starting set {w k } will however be the natural integers, with the zeta functions

ζ(x) = ∞ k=1 k -x (Riemann), ζ(x, w) = ∞ k=0 (k + w) -x (Hurwitz) (Re x > 1) (2)
(the standard notations, in which ζ(x) ≡ ζ(x, w = 1)), because a) ζ(x) gives rise to the first superzeta functions, those over the Riemann zeros; b) ζ(x, w) provides the basic template for the full set of explicit properties ultimately displayed by those less elementary superzeta functions.

Predicted pattern of general results for zeta functions

For our later sets {w k } (all countably infinite), typical results can be categorized as:

• A -Analytic structure in the whole complex x-plane (at fixed w): our zeta functions will be defined by [START_REF]Zeta Functions, Topology and Quantum Physics[END_REF] only over a half-plane in x, as in ( 2), but will then admit a meromorphic continuation to the whole x-plane, with exactly and fully computable poles and principal parts.

(E.g., Hurwitz's ζ(x, •) = 1/(x -1)+ [entire function].)
• F -Functional relation (often): this may link the analytic continuation of Z(x, w) to another function of Mellin-transform type (as in [START_REF] Guinand | A summation formula in the theory of prime numbers[END_REF] for the Hurwitz case); when it exists, F may also readily supply all the explicit features announced here.

• R

T -Special-value formulae for Z(x, w) at all integers x = ∓n: Rational Transcendental respectively (such will be the layout of the Tables), and having quite distinct origins: R for x = -n (in conjunction with A): our zeta functions relate to Mellin transforms, e.g., Z(x, w) = Γ(x) -1 ∞ 0 f (z, w) z x-1 dz (like [START_REF] Edwards | Riemann's Zeta Function[END_REF] for Hurwitz) where f (z, w) will have an explicit z → 0 power expansion: this feature makes the Mellin integral meromorphic for all x by continuation and also yields its explicit poles and principal parts (A), plus (upon the division by Γ(x)) explicit rational (R) values for Z(x, w) at all x = -n (0 and the negative integers). By "rational" we mean: polynomial in w and in the expansion coefficients of f at z = 0.

T for x = +n : zeta functions relate to zeta-regularized products

D(w) = k (w + w k ) by log D(w) def = -∂ x Z(x, w)| x=0 for Z(x, w) = k (w k + w) -x (3) =⇒ Z(n, w) = (-1) n-1 (n -1)! (log D) (n) (w) for n = 1, 2, . . . outside poles of Z, ( 4
)
vs FP x=1 Z(x, w) = (log D) ′ (w) (FP: finite-part extraction) FP x=n Z(x, w) = [a bulkier formula] for n ≥ 2 [31, § 2.6.1] on poles of Z (5)
(the very last case is unused here); thus, formulae (3)- [START_REF] Delsarte | Formules de Poisson avec reste[END_REF] express zeta values at all x = +n ( 0 and the positive integers), in terms of a function D(w) which is entire, has {-w k } as zeros, and will be transcendental (T) but hopefully known. E.g., in the Hurwitz case:

{w k } = {0, 1, 2, . . .} leads to D(w) = √ 2π/Γ(w). ( 6
)

Results for the Hurwitz zeta function ζ(x, w)

We recall classic results [START_REF]Higher Transcendental Functions (Bateman Manuscript Project)[END_REF], to be needed for the notations and for later reference.

• F-Functional relation (will imply A): we start from the (obvious) Mellin-transform representation of ζ(x, w), then convert it to a Hankel integral:

ζ(x, w) = 1 Γ(x) ∞ 0 e (1-w)z e z -1 z x-1 dz (Re x > 1, Re w > 0) (7) = Γ(1 -x) 2iπ C ′ e (1-w)z e z -1 (-z) x-1 dz (x ∈ N * , 1 ≥ Re w > 0) (8) = Γ(1 -x) 2iπ lim R→+∞ C R e (1-w)z e z -1 (-z) x-1 dz (x ∈ N * , 1 ≥ Re w > 0) (9)
(see Fig. 1); then this contour integral is readily evaluated by the residue calculus, giving [START_REF] Guinand | A summation formula in the theory of prime numbers[END_REF] for F (u, y) • R T -Special values: R resulting from [START_REF] Edwards | Riemann's Zeta Function[END_REF], followed by T from [START_REF] Deninger | Local L-factors of motives and regularized determinants[END_REF], are listed in Table 1.

ζ(x, w) = Γ(1 -x) (2π) 1-x i [e iπx/2 F (e 2iπw , 1 -x) -e -iπx/2 F (e -2iπw , 1 -x)] (F)
Moreover, [START_REF] Guinand | A summation formula in the theory of prime numbers[END_REF] reduces at w = 1 to Riemann's Functional Equation:

ζ(x) ≡ 2(2π) x-1 sin πx 2 Γ(1 -x) ζ(1 -x) ⇐⇒ Ξ( 1 2 + t) ≡ Ξ( 1 2 -t), (12)
in terms of a completed zeta function [START_REF] Hirano | Half zeta functions[END_REF], ζ(x) has trivial zeros: x = -2k, k = 1, 2, . . ., plus those of Ξ(x) : the Riemann zeros ρ; the latter are countably many, 2-by-2 symmetrical about t = 0, as

Ξ(x) = x(x -1)π -x/2 Γ(x/2) ζ(x) (our normalization: Ξ(0) = Ξ(1) = 1), ( 13
) x ζ(x, w) = ∞ k=0 (k + w) -x -n ≤ 0 -1 n + 1 B n+1 (w) 0 1 2 -w R 0 (x-derivative) ζ ′ (0, w) = log(Γ(w)/ √ 2π) T +1 (finite part) FP x=1 ζ(x, w) = -(log Γ) ′ (w) +n > 1 (-1) n (n -1)! (log Γ) (n) (w)
ρ = 1 2 ± iτ k (Re τ k > 0) (k = 1, 2, . . .), (14)
and lie within the critical strip {0 < Re ρ < 1}; moreover, the Riemann Hypothesis [START_REF] Riemann | Über die Anzahl der Primzahlen unter einer gegebenen Grösse[END_REF] (still an open conjecture) puts them all on the critical line:

Re ρ = 1 2 ⇐⇒ τ k is real for all Riemann zeros. [RH] (15)
2 Superzeta functions, in the Riemann zeros' case

Over the set of Riemann zeros {ρ} (always counted with multiplicities, if any), several generalized zeta functions are conceivable:

Z (s | t) = ρ ( 1 2 + t -ρ) -s , Re s > 1 (1 st kind) (16)
Z (σ | t) = ∞ k=1 (τ k 2 + t 2 ) -σ , Re σ > 1 2 (2 nd kind) (17)
Z(s | τ ) = ∞ k=1 (τ k + τ ) -s , Re s > 1 (3 rd kind) ( 18
)
where the parameter t has the same meaning as in [START_REF] Hirano | Half zeta functions[END_REF]; those functions are 2-by-2 inequivalent, except at a single parameter location where a confluence occurs:

(2 cos 1 2 πs) -1 Z (s | 0) ≡ Z ( 1 2 s | 0) ≡ Z(s | 0) ≡ ∞ k=1 τ -s k ; (19)
(19) can be taken as a standard (one-variable) zeta function over the Riemann zeros.

Z (s | t) = ρ ( 1 2 + t -ρ) -s Z(σ | t) = ∞ k=1 (τ 2 k + t 2 ) -σ Z(s | τ ) = ∞ k=1 (τ k + τ ) -s 1860 Riemann (unpubl.): ρ ρ -1 T 1917 Mellin t = ± 1 2 AR Mellin τ = ± 1 2 i A 1949 1966 1970 1971 ← --------- Guinand AF Delsarte A Chakravarty F " AR        t = 0 (confluent case) ---------→ 1985 1988 Matsuoka Lehmer ρ ρ -n t = 1 2 s = n T 1988 Kurokawa t = 1 2 A 1989 Matiyasevich t = 1 2 σ = n T 1992 1994 Keiper Zhang-Williams ρ ρ -n T 1992 1994 Deninger Schröter-Soulé ∂ sZ (0 | t) A T

Table 2:

The pre-2000 literature on "superzeta" functions of all 3 kinds (to our knowledge), sorted by kind (columns). (A-T : classes of results, cf. Sect. 1.1; n means any positive integer; the references are listed in full at the end.)

Those zeta functions over the Riemann zeros (or "superzeta" functions, for brevity) got considered quite sporadically until the turn of the century: we found but a dozen studies, all partially focused and incomplete even as a whole (Table 2). From 2000 onwards we have gradually filled gaps in the description [START_REF] Voros | Zeta functions for the Riemann zeros[END_REF][29] [START_REF] Voros | Zeta functions over zeros of general zeta and L-functions[END_REF], ending up with a more global perspective, also set in new coherent notations, as a book [START_REF] Voros | Zeta functions over zeros of zeta functions[END_REF]. The present text then basically condenses the main results of [START_REF] Voros | Zeta functions over zeros of zeta functions[END_REF], where the details skipped here can be found. 16) where its series converges, i.e., in the half-plane {Re s > 1}.

The superzeta function of the 1

st kind Z (s | t) 2.1.1 Basic functional relation / continuation formula Z (s | t) is only defined by (
Now, in terms of the partner zeta function over the trivial zeros of ζ(x),

Z (s | t) def = ∞ k=1 ( 1 2 + t + 2k) -s ≡ 2 -s ζ(s, 5 4 + 1 2 t), (20)
and of another Mellin transform,

J (s | t) = ∞ 0 ζ ′ ζ ( 1 2 + t + y) y -s dy (Re s < 1), (21)
Z (s | t) admits an analytic continuation formula which reads as

Z (s | t) = -Z (s | t) + (t -1 2 ) -s + sin πs π J (s | t) (Re s < 1), ( 22
)
valid in a cut t-plane: 1 2 + t must avoid all negatively oriented half-lines drawn from the Riemann zeros and the pole (whose cut t ∈ (-∞, + 1 2 ] imposes special treatments for both remarkable values t = 0 and 1 2 ). The argument of proof (parallel to ( 7)-( 9), and sketched by Fig. 2) shows [START_REF] Matsuoka | A note on the relation between generalized Euler constants and the zeros of the Riemann zeta function[END_REF] to be the counterpart of the functional relation (10) from the Hurwitz case, i.e., [START_REF] Matsuoka | A note on the relation between generalized Euler constants and the zeros of the Riemann zeta function[END_REF] gives the property F for Z . Moreover, by the same logic as in Sect. 1.1.R, the y → 0 Taylor expansion of the integrand in [START_REF] Yu | A relationship between certain sums over trivial and nontrivial zeros of the Riemann zeta-function[END_REF] induces the continuation of J (s | t) (and of the F-identity [START_REF] Matsuoka | A note on the relation between generalized Euler constants and the zeros of the Riemann zeta function[END_REF]) to the whole s-plane, and it fully specifies the poles and residues of J . As a result, the product [sin πsJ (s | t)] in ( 22) is regular everywhere (and computable at all zeros of sin πs as well, in preparation to Sect. 2.1.2). Its regularity alone implies that Z (s | t) has the same singular structure as -Z (s | t) : i.e., only the simple pole s = 1, of residue -1 2 (property A).

Im y Im (x+y ) Re y x+y Re ( ) R x - -6 -2 0 1 0 C' C -4

Special values of Z (s | t)

Once we can compute [sin πsJ (s | t)] at all integers s, the corresponding values of Z (s | t) plus ∂ s Z (0 | t) readily follow from the F-identity (22) (as extended to all s). Then, in the resulting Table 3 of special values, the superzeta function Z proves fully on par with the much more elementary Hurwitz zeta function of (1) ! Since J (s | t) is regular for s < 1, its transcendental values get killed in [START_REF] Matsuoka | A note on the relation between generalized Euler constants and the zeros of the Riemann zeta function[END_REF] by the sin πs factor at all s = 0, -1, -2, . . ., making those Z (-n | t) values rational (R).

The remaining, transcendental (T), values Z (n | t) (n = 1, 2, . . .; more detail about n = 1 in [31, § 7.4]) also directly stem from a generating function that goes back to a symmetric Hadamard product for Ξ(x) [7, § 1.10], The special values Z (n | t) also display some imprints of the fundamental symmetry of the Riemann zeros (ρ ←→ (1-ρ), t ←→ -t) (vs nothing we know of for non-integer s):

Ξ(x) = ρ (1 -x/ρ) =⇒ d dy log Ξ( 1 2 + t + y) ≡ ∞ n=1 Z (n | t) (-y) n-1 . ( 23
) s Z (s | t) = ρ ( 1 2 + t -ρ) -s ζ(s, w) = ∞ k=0 (k + w) -s -n < 0 2 n n + 1 B n+1 ( 1 4 + 1 2 t) + (t + 1 2 ) n + (t -1 2 ) n -1 n + 1 B n+1 (w) 0 1 2 (t + 7 2 ) R 1 2 -w 0 (s-derivative) -1 2 (log 2π) t + 1 4 log 8π -log Ξ( 1 2 + t) T -1 2 log 2π + log Γ(w) +1 (finite part) 1 2 log 2π + (log Ξ) ′ ( 1 2 + t) -(log Γ) ′ (w) +1 (log Ξ) ′ ( 1 2 + t) ∞ +n > 1 (-1) n-1 (n -1)! (log Ξ) (n) ( 1 2 + t) (-1) n (n -1)! (log Γ) (n) (w)
-obvious: Z (n | -t) ≡ (-1) n Z (n | t) for n = 1, 2, . . ., implying Z (n | 0) ≡ 0 for n odd (vs e iπs/2 Z (s | t) -e -iπs/2 Z (s | -t) ≡ 2i(sin πs) Z(s | it) ≡ 0 for non-integer s, cf. (30)); -less obvious: Z (-n | -t) ≡ (-1) n Z (-n | t) + 1 n + 1 B n+1 ( 1 2 -t) for n = 0,
(skeleton of proof: expand identity ( 1 2 -ρ) -n ≡ ( 1 2 + t -ρ) -n [1 -t/( 1 2 + t -ρ)]
-n in powers of t, then sum over {ρ}, and finally use Z (n | 0) ≡ 0 for n odd).

The next two Tables exemplify Table 3 for Z0 (s)

def = Z (s | 0), resp. Z * (s) def = Z (s | 1 2 ). s Z0 (s) ≡ ρ (ρ -1 2 ) -s [t = 0] even 2 -n+1 (1 -1 8 E n ) -n ≤ 0 odd -1 2 (1-2 -n ) B n+1 n+1 0 7/4 R 0 (s-derivative) Z ′ 0 (0) = log [2 11/4 π 1/2 Γ( 1 4 ) -1 |ζ( 1 2 )| -1 ] T +1 (finite part) FP s=1 Z0 (s) = 1 2 log 2π odd 0 +n ≥ 1 even 2 n+1 -1 2 [(2 n -1) ζ(n) + 2 n β(n)] - (log |ζ|) (n) ( 1 2 ) (n -1)! ζ(n) ≡ (2π) n |B n | 2 n! Table 4: ( E n B n : Euler Bernoulli numbers; β(x) def = ∞ k=0 (-1) k (2k + 1) -x : Dirichlet β-function.) s Z * (s) ≡ ρ ρ -s [t = 1 2 ] -n < 0 1 -(2 n -1) B n+1 n+1 0 2 R 0 (s-derivative) Z ′ * (0) = 1 2 log 2 T +1 (finite part) FP s=1 Z * (s) = 1 -1 2 log 2 + 1 2 γ +1 1 -1 2 log 4π + 1 2 γ +n > 1 1 -(1-2 -n ) ζ(n) + g c n (n -1)! ≡ 1 -(-1) n 2 -n ζ(n) - (log |ζ|) (n) (0) (n -1)!
)] ≡ - ∞ n=1 (-1) n n! g c n y n ; ζ(n) ≡ (2π) n |B n | 2 n! for n even.)
2.2 The superzeta function of the 2 nd kind Z (σ | t)

Defined by [START_REF] Lagarias | Li coefficients for automorphic L-functions[END_REF] for Re σ > 1 2 , Z (σ | t) appears (for t = 0) functionally independent from the function of the 1 st kind Z (s | t); contrary to the latter, it manifestly embodies the full symmetry of the Riemann zeros, through the identity Z (σ | t) ≡ Z (σ | -t).

On the other hand, Z (σ | t) for t = 0 displays no tractable functional relation F. Still, by expansion in powers of t around the known confluent case [START_REF] Li | The positivity of a sequence of numbers and the Riemann Hypothesis[END_REF] [27, § 6.1][31, § 8.3], Z (σ | t) is proved meromorphic in σ, now with double poles σ = 1 2 , -1 2 , -3 2 , . . ., and all principal parts plus rational values Z (-m | t) (m ∈ N) explicitly computable (A+R). E.g., the leading pole σ = 1 2 has the principal part (t-independent)

Z ( 1 2 + ε | t) = 1 8π ε -2 - log 2π 4π ε -1 + [regular term], ε → 0. (A) (24) 2.2.1 Special values of Z (σ | t) (continued)
The remaining, transcendental (T), values Z (m | t) (m = 1, 2, . . .) stem from a variant of ( 23)

[7, § 1.10], Ξ(x) = Im ρ>0 1 - x(1 -x) ρ(1 -ρ) ⇒ d dw log Ξ( 1 2 + (t 2 + w) 1/2 ) ≡ ∞ m=1 Z (m | t) (-w) m-1 ; (25) only the derivative ∂ σ Z (σ | t)| σ=0 is harder to obtain (see [27, § 4][31, § 8.4]
), and that completes Table 6.

The change of variables t + y = (t 2 + w) 1/2 in the generating function [START_REF] Hj | Über die Nullstellen der Zetafunktion[END_REF] (at fixed t, say t ≥ 0) also has to yield [START_REF] Schröter | On a result of Deninger concerning Riemann's zeta function[END_REF], hence the Z (m | t) (m = 1, 2, . . .) must relate to the Z (n | t) (1 ≤ n ≤ m). To get a fully explicit connection, our shortcut (new) is to use the residue calculus twice, inside small positive contours encircling y = 0, resp. w = 0 :

(-1) n-1 2πi d log Ξ( 1 2 + t + y) y n = Z (n | t) for n = 1, 2, . . . , 0 for n ∈ Z otherwise; (26) Z (m | t) = (-1) m-1 2πi d log Ξ( 1 2 + (t 2 + w) 1/2 ) w m = (-1) m-1 2πi d log Ξ( 1 2 + t + y) (2ty + y 2 ) m = (-1) m-1 2πi ∞ ñ=-m (-1) m+ñ 2m+ ñ-1 m -1 (2t) -2m-ñ y ñ d log Ξ( 1 2 +t+y) (t = 0) ≡ m n=1 2m-n-1 m -1 (2t) -2m+n Z (n | t) for m = 1, 2, .
. . , and t = 0 [START_REF] Voros | Zeta functions for the Riemann zeros[END_REF] by [START_REF] Sekatskii | Generalized Bombieri-Lagarias' theorem and generalized Li's criterion[END_REF] (vs [START_REF] Voros | Zeta functions over zeros of general zeta and L-functions[END_REF] below for t = 0); e.g., the first of the identities [START_REF] Voros | Zeta functions for the Riemann zeros[END_REF] is All in all and notwithstanding its lack of a tractable functional relation F, the superzeta function of the 2 nd kind matches its partner of the 1 st kind (or the Hurwitz zeta function) for its stock of explicit fixed-σ properties.

Z (1 | t) ≡ Z (1 | t) 2t . σ Z (σ | t) = ∞ k=1 (τ k 2 + t 2 ) -σ -m < 0 (t 2 -1 4 ) m -2 -2m-3 m j=0 m j (-1) j E 2j (2t) 2(m-j) 0 7/8 R 0 (σ-derivative) ∂ σ Z (0 | t) = 1 4 log 8π -log Ξ( 1 2 + t) T +m ≥ 1 (-1) m-1 (m -1)! d m d(t 2 ) m (log Ξ)( 1 2 + t)
Next, we exemplify those formulae at the special parameter values t = 0 and 1 2 . For Z0 (σ) = Z (σ | t = 0) (the confluent case), Table 4 can be reused since [START_REF] Li | The positivity of a sequence of numbers and the Riemann Hypothesis[END_REF] entails

Z0 (m) ≡ 1 2 (-1) m Z0 (2m) (m ∈ Z), d dσ Z0 (0) = d ds Z0 (0). ( 28
)
Whereas for Z * (σ) = Z (σ | t = 1 2 ), Table 6 plus ( 27) specialize to:

σ Z * (σ) = ∞ k=1 (τ k 2 + 1 4 ) -σ [t = 1 2 ] -m < 0 -2 -2m-3 m j=0 m j (-1) j E 2j 0 7/8 R 0 (σ-derivative) Z ′ * (0) = 1 4 log 8π T +m ≥ 1 m n=1 2m-n-1 m -1 Z * (n)
Z(s | τ ) ≡ 1 2i sin πs [e iπs/2 Z (s | -iτ ) -e -iπs/2 Z (s | +iτ)]. (30)
The singular structure of Z is thus computable just as before; on the other hand, only one special value remains explicit: the finite part of Z(s | τ ) at s = 0, which reads 7 8 + log 2π 2π τ .

Extensions

Those currently include, but should not be limited to, (zeta functions over the) zeros of:

• Dedekind zeta functions (for algebraic number fields) [START_REF] Kurokawa | Special values of Selberg zeta functions[END_REF][12] [START_REF] Hashimoto | Euler's constants for the Selberg and the Dedekind zeta functions[END_REF][13][28]

• Dirichlet L-functions (for real primitive Dirichlet characters) [START_REF] Voros | Zeta functions over zeros of general zeta and L-functions[END_REF] • Selberg zeta functions (for cocompact subgroups of SL(2,R)) -with more work ([31, App. B] and refs. therein).

In every such extension, Z * (1) = ρ 1/ρ is closely related to the associated generalized Euler constant (= Euler-Kronecker invariant), a major invariant of the zeta function [START_REF] Hashimoto | Euler's constants for the Selberg and the Dedekind zeta functions[END_REF] [START_REF] Ihara | On the Euler-Kronecker constants of global fields and primes with small norms, in: Algebraic Geometry and Number Theory[END_REF].

3 Complex-asymptotic view of Riemann Hypothesis

The Keiper-Li coefficients λ n

The sequence of real numbers

λ n = ρ [1 -(1 -1/ρ) n ] (n = 1, 2, . . .), (31)
(in Li's notation [START_REF] Li | The positivity of a sequence of numbers and the Riemann Hypothesis[END_REF], amounting to n times Keiper's λ n [START_REF] Keiper | Power series expansions of Riemann's ξ function[END_REF]), of generating function

log Ξ x = 1 1 -z ≡ ∞ n=1 λ n n z n , (32)
has served to recast the Riemann Hypothesis (RH) (cf. ( 15)), as RH ⇐⇒ λ n > 0 for all n (Li's criterion) [START_REF] Li | The positivity of a sequence of numbers and the Riemann Hypothesis[END_REF][2]. Here we review the (large-n) asymptotics of the λ n instead [START_REF] Keiper | Power series expansions of Riemann's ξ function[END_REF] [START_REF] Maślanka | Li's criterion for the Riemann hypothesis -numerical approach[END_REF][30] [START_REF] Lagarias | Li coefficients for automorphic L-functions[END_REF], and our resulting asymptotic criterion for RH that involves just the tail of the sequence {λ n } [30][31, Chap. 11].

Exponential-asymptotic large-n analysis of λ n

The Hadamard product formula in [START_REF] Schröter | On a result of Deninger concerning Riemann's zeta function[END_REF]

implies log Ξ 1 1 -z = ∞ k=1 log 1 + z (τ k 2 + 1 4 )(1 -z) 2 = - ∞ m=1 Z * (m) m (-z) m (1 -z) 2m (33)
(the Z * (m) are in Table 7); then, expanding (1 -z) -2m by the generalized binomial formula, reordering in powers of z and substituting into (32), we get [30][31, § 8.6.2]

λ n = -n n m=1 (-1) m m m + n -1 2m -1 Z * (m), n = 1, 2, . . . . (34)
This yields a representation by an integral over the superzeta function Z * (σ) ≡ Z (σ | 1 2 ) (of the 2 nd kind),

λ n = (-1) n n i π C Γ(σ + n)Γ(σ -n) Γ(2σ + 1) Z * (σ) dσ, (35)
as proved by reduction to (34) using the residue calculus, see Fig. 3. Our point is now that for n → +∞ (and after using the Stirling formula in the numerator), the integral form ( 35) is asymptotically computable thanks to the method of steepest descents [START_REF] Erdélyi | Asymptotic Expansions[END_REF]. With the integrand in (35) behaving like n 2σ-1 for n → +∞, that means to push the integration path toward decreasing Re σ, up to saddle-points of the integrand where dominant contributions to the integral will localize. Here, saddle-points σ * of two species compete in the half-plane {Re σ > 1 2 } for dominance (= largest Re σ * ), see Fig. 4 [30]: 2 + iτ k on the critical line (and is irrelevant, beyond reach), σ k ′ (n) comes from a putative ("?") zero 1 2 + iτ k ′ off the critical line (and then, gives a dominant contribution), finally σ r (n) is the real saddle-point tending to the pole σ = 1/2 from above (and gives the dominant contribution if and only if RH holds).

0 C Re σ 1/2 1 2 n -1 Im σ
C σ Re Im σ C' ? 0 -1 1 1/2 n σ r σ k' σ k
• using Z * (σ) = k (τ k 2 + 1 4 ) -σ to integrate (35) term by term: then the k-th integrand has a saddle-point σ k (n) ∼ 1 2 n i/τ k , which is relevant iff arg τ k > 0, as this is the necessary and sufficient condition for that saddle-point to ultimately reside in the half-plane {Re σ > 1 2 } (that is, for n | Im 1/τ k | -1 );

• the pair of poles σ = + 1 2 of Z * as in [START_REF] Riemann | Über die Anzahl der Primzahlen unter einer gegebenen Grösse[END_REF] and σ = 1 of Γ(σ -n) together force a real local minimum in the modulus of the integrand, unconditionally: σ r (n) ∼ 1 2 + 1 log n , this is another saddle-point, and the dominant one iff all arg τ k ≡ 0 (i.e., RH true).

Thus, the competition between both types of saddle-point contributions to (35) results in this asymptotic criterion for RH [30]: as n → +∞, -if RH is false, λ n has an exponentially growing oscillatory behavior:

λ n ∼ - {arg τ k >0} τ k + i/2 τ k -i/2 n + c. c. (mod o(e εn ) ∀ε > 0) (36)
-if RH is true, λ n has a tempered growth to +∞ :

λ n ∼ 1 2 n (log n -1 + γ -log 2π) (mod o(n)). (37)
However, if a numerical crossover from the latter (RH) to the former (non-RH) behavior is sought as a signal of a violation of RH, that has to await a huge value n ≈ 10 18 at least, corresponding to the present height up to which Re ρ = 1 2 has been verified [START_REF] Lagarias | Li coefficients for automorphic L-functions[END_REF].

Extensions

For zeros of a more general zeta function as in Sect. 2.4, and such that

Z * ( 1 2 + ε) = R -2 ε -2 + R -1 ε -1 + O(1) ε→0 (with R -2 > 0), (38)
(generalizing [START_REF] Riemann | Über die Anzahl der Primzahlen unter einer gegebenen Grösse[END_REF] from the case of the Riemann zeros), then the Generalized Riemann Hypothesis is equivalent to

λ n ∼ (-1) n 2n Res σ=1/2 Γ(σ + n)Γ(σ -n) Γ(2σ + 1) Z * (σ) (39) ∼ 2πn [2R -2 (log n -1 + γ) + R -1 ] (mod o(n)) (40)
for its corresponding Keiper-Li sequence {λ n } as n → ∞ [START_REF] Lagarias | Li coefficients for automorphic L-functions[END_REF]; the constants R -2 , R -1 are also those governing the asymptotic counting function of the zeros' ordinates, as

N(T ) ∼ 2T [2R -2 (log T -1) + R -1 ] (T → +∞). ( 41
)
3.4 (New) Shifted asymptotic and Li criteria for RH (returning to the Riemann-zeta case alone, for the sake of definiteness). The Keiper-Li (KL) generating series [START_REF] Zhang | Some results on the generalized Stieltjes constants[END_REF] combines two features: a high sensitivity to the Riemann Hypothesis through its radius of convergence (|z| = 1 if and only if RH holds vs strictly less otherwise), obtained by the use of a conformal mapping z → x that sends the disk {|z| < 1} to the half-plane {Re x > 1 2 }; and an expansion basepoint (the image of z = 0) set to x = 1 (hence the parameter t

≡ x -1 2 is 1 2 in the superzeta function entering § 3.2: Z * (σ) def = Z (σ | 1 2 )
). The two features always appeared coupled, as the conformal mapping was invariably prescribed to be x = 1/(1 -z), as in [START_REF] Zhang | Some results on the generalized Stieltjes constants[END_REF].

Yet, we claim that imposing only x = 1 as basepoint is an unnecessary limitation. To wit, we present an alternative KL-like framework based at the other remarkable point x = 1 2 (the center of symmetry), i.e., t = 0. In the last minute, we saw that the idea of a general basepoint had been developed elsewhere, only recently and in a different manner that just excludes our preferred symmetrical point x = 1 2 [START_REF] Sekatskii | Generalized Bombieri-Lagarias' theorem and generalized Li's criterion[END_REF]. For expanding Ξ(x) about x = 1 2 , due to (12) the good variable is w = t 2 , in which the above half-plane {Re x > 1 2 } maps to the cut w-plane C \ R -; this cut can then be truncated to any half-line (-∞, -w ′ ] with 0 < w ′ < inf ρ {| Im ρ| 2 } (that only deletes a zero-free segment), and here we choose w ′ = 1 4 just to make the formulae simpler and closest to the KL case. There now remains to map to the cut w-plane C \ (-∞, - 1 4 ] from a unit disk {|y| < 1}, which is achieved by w = y/(1 -y) 2 . Under the resulting y → x correspondence, we will need the preimages of a Riemann zero ρ = 1 2 + iτ k , as (43) thus, we created a KL-like sequence of "central" coefficients λ 0 n . Comparing to (32)-(33), we see that the relationships (34) persist with the λ n substituted by the λ 0 n and the superzeta function Z * by Z0 . (Equivalently, by [START_REF] Voros | Zeta functions over zeros of general zeta and L-functions[END_REF] and Table 3

y k,± = 1 - 1 2τ k 2 ± i τ k 1 - 1 4τ k 2

Figure 1 :

 1 Figure 1: Integration paths in (8)-(9) leading to the functional relation (10) for ζ(x, w).

  year

Figure 2 :

 2 Figure 2: Integration paths applied to the Mellin transform (21) similarly to Fig. 1, leading to the analytic continuation formula (22) for Z (s | t). (x ≡ 1 2 + t; • : zeros of ζ(x + y), including mock "Riemann zeros" with imaginary parts contracted to fit into the frame.)

1 , 2 ,

 12 . . . (as computed from Table 3-R, with B n+1 (•) : Bernoulli polynomial); -even less obvious: 0 ≡ ∞ k=n k-1 n-1 t k-n Z (k | t) for each odd n = 1, 3, . . .

Figure 3 :

 3 Figure 3: Contour of integration for (35). ( × × × : poles of the integrand.)

Figure 4 :

 4 Figure4: Integration-path deformation for a large-n evaluation of the integral (35). • : typical saddle-points (none on scale); σ k (n) comes from a Riemann zero1 2 + iτ k on the critical line (and is irrelevant, beyond reach), σ k ′ (n) comes from a putative ("?") zero1 2 + iτ k ′ off the critical line (and then, gives a dominant contribution), finally σ r (n) is the real saddle-point tending to the pole σ = 1/2 from above (and gives the dominant contribution if and only if RH holds).

1 / 2 ≡

 12 e ±iθ k with θ k def from the generating function (25) at t = 0 and integrate in w, to getlog Ξ 0 ( 1 2 + w 1/2 ) ≡m (1 -y) 2m ;

Table 1 :

 1 Special-value formulae for the Hurwitz zeta function,

	Rational Transcendental	at all integer

Table 3 :

 3 Central column: the special values for the superzeta function of the 1 st kind Z (s | t),

	Rational

Transcendental at all integer s = ∓n. Right column, for comparison: the same for the Hurwitz zeta function, rewritten from Table

1

. (B n (•) : Bernoulli polynomials; Ξ(•) : the completed Riemann zeta function

[START_REF] Ihara | On the Euler-Kronecker constants of global fields and primes with small norms, in: Algebraic Geometry and Number Theory[END_REF]

.)

Table 5 :

 5 (γ : Euler's constant; g c n : cumulants of the Stieltjes constants [14, § 5][2][27, § 3.3][4], of generating function log [y ζ(1 + y

Table 6 :

 6 The special values for the superzeta function of the 2

nd kind Z(σ | t),

Rational

Transcendental at all integer σ = ∓m. (E n : Euler numbers; Ξ(•) : the completed Riemann zeta function

[START_REF] Ihara | On the Euler-Kronecker constants of global fields and primes with small norms, in: Algebraic Geometry and Number Theory[END_REF]

.)

Table 7 :

 7 As Table 6, but at t = 1 The superzeta function of the 3 rd kind Z(s | τ ) Defined by (18) for Re s > 1, Z(s | τ ) is less regular than the other two kinds [12, § 3][27, § 6.2], nevertheless it reduces to functions of the 1 st kind: indeed, the straightforward identity Z (s | t) ≡ e iπs/2 Z(s | it) + e -iπs/2 Z(s | -it)

	(29)

2 . (E n : Euler numbers; values Z * (n) : see Table 5.) 2.3

  at t = 0, Being based upon (34), our asymptotic analysis of {λ n } sketched in § 3.2 [30][START_REF] Voros | Zeta functions over zeros of zeta functions[END_REF] Chap. 11] then carries over likewise to {λ 0 n }, and it now yields this shifted asymptotic criterion for RH:

	λ 0 n = -	n 2	n m=1	m+n-1 2m -1	1 m	Z0 (2m) = n	n m=1	m+n-1 2m -1	1 (2m)!	(log Ξ) (2m) ( 1 2 ), (44)
	and the last line in Table 4 displays Z0 (2m) in most reduced form.) E.g.,
	λ 0 1 = Z0 (1) = 1 2 (log Ξ) ′′ ( 1 2 ) ≈ 0.0231050, λ 0 2 = 4 Z0 (1) -Z0 (2) = 2(log Ξ) ′′ ( 1 2 ) + 1 12 (log Ξ) (4) ( 1 2 ) ≈ 0.0923828, . . .	(45) (46)
			-if RH is false, λ 0 n ∼ -{arg τ k >0}	y -n k,-+ c. c.,	(47)

an exponentially growing oscillatory behavior (like (36

)) since |y -1 k,-| > 1 if arg τ k > 0; -if RH is true, λ 0 n ∼ 1 2 n (log n -1 + γ -log 2π),

(48)

the latter being unchanged from (37), as according to § 3.3 it arises from the principal part

[START_REF] Riemann | Über die Anzahl der Primzahlen unter einer gegebenen Grösse[END_REF]

of Z (σ | t) at σ = 1 2 , and (

24

) is t-independent.

Acknowledgements

We are most grateful to the conference organizers and to the Research Institute for Mathematical Sciences of Kyoto University for their kind invitation and generous support.

Finally, Li's criterion [START_REF] Li | The positivity of a sequence of numbers and the Riemann Hypothesis[END_REF] shifts as well: by the Hadamard product formula in [START_REF] Schröter | On a result of Deninger concerning Riemann's zeta function[END_REF],

Now if RH is true, then for all k, τ k is purely real and so is θ k in (42), in which case y k,± = e ±iθ k makes (52) manifestly positive for all n. Whereas if RH is false, then as n → +∞, the growing oscillatory behavior (47) will force some λ 0 n into the negative range. So, the Li criterion transfers to this "central" sequence:

Remark: Other expressions for the λ 0 n , derivable from (43) by using the residue calculus both ways (similarly to the proof of ( 27)), are