On the nontrivial zeros of the Dirichlet eta function
aa r X i v : . [ m a t h . G M ] J u l On the nontrivial zeros of the Dirichlet etafunction
Vladimir Garc´ıa-Morales
Departament de F´ısica de la Terra i Termodin`amicaUniversitat de Val`encia,E-46100 Burjassot, [email protected]
We construct a two-parameter complex function η κν : C → C , κ ∈ (0 , ∞ ), ν ∈ (0 , ∞ ) that we call a holomorphic nonlinear embedding and that is given by adouble series which is absolutely and uniformly convergent on compact sets in theentire complex plane. The function η κν converges to the Dirichlet eta function η ( s )as κ → ∞ . We prove the crucial property that, for sufficiently large κ , the function η κν ( s ) can be expressed as a linear combination η κν ( s ) = P ∞ n =0 a n ( κ ) η ( s + 2 νn )of horizontal shifts of the eta function (where a n ( κ ) ∈ R and a = 1) and that,indeed, we have the inverse formula η ( s ) = P ∞ n =0 b n ( κ ) η κν ( s + 2 νn ) as well (wherethe coefficients b n ( κ ) ∈ R are obtained from the a n ’s recursively). By using theseresults and the functional relationship of the eta function, η ( s ) = λ ( s ) η (1 − s ), wesketch a proof of the Riemann hypothesis which, in our setting, is equivalent tothe fact that the nontrivial zeros s ∗ = σ ∗ + it ∗ of η ( s ) (i.e. those points for which η ( s ∗ ) = η (1 − s ∗ ) = 0) are all located on the critical line σ ∗ = . Introduction
Let s := σ + it be a complex number. The Dirichlet eta function η ( s ), also calledalternating zeta function, is given in the half plane σ > η ( s ) = ∞ X m =1 ( − m − m s (1)which is absolutely convergent for σ >
1. Hardy gave a simple proof of the factthat the eta function satisfies the functional equation [1] η ( s ) = λ ( s ) η (1 − s ) (2)where λ ( s ) = 1 − − s − s s π s − sin (cid:16) πs (cid:17) Γ(1 − s ) (3)From this, one immediately obtains the means to extend the definition of the etafunction to the whole complex plane. Indeed, Euler’s acceleration of the condi-tionally convergent series in Eq. (1) yields a double series that is absolutely anduniformly convergent on compact sets everywhere [2, 3, 4] η ( s ) = ∞ X k =0 k +1 k X m =0 (cid:18) km (cid:19) ( − m ( m + 1) s (4)The Dirichlet eta function is closely related to the Riemann zeta function by η ( s ) = (cid:0) − − s (cid:1) ζ ( s ) (5)However, while the zeta function is meromorphic, with a pole at s = 1, the etafunction is an entire function. We note, from Eq. (1), that at s = 1, the etafunction becomes the alternating harmonic series and, therefore, η (1) = ∞ X m =1 ( − m − m = 1 −
12 + 13 − . . . = ln 2 (6)Let s ∗ denote a zero of the eta function, η ( s ∗ ) = 0. There are two kinds ofzeros: the trivial zeros for which, from the functional equation, we have λ ( s ∗ ) = 0;and the nontrivial zeros, for which η (1 − s ∗ ) = 0. From Eq. (3) the trivial zerosare the negative even integers and zeros of the form σ ∗ = 1 + i nπ ln 2 where n is anonzero integer (see [5] for a derivation that does not make use of the functionalrelation). ince there are no zeros in the half-plane σ >
1, the functional equation impliesthat nontrivial zeros of η are to be found in the critical strip 0 ≤ σ ≤
1. By theprime number theorem of Hadamard and de la Vall´ee Poussin [1] it is known thatfor σ = 1 (and, therefore, σ = 0) there are no nontrivial zeros of the Riemann zetafunction and, therefore, from Eq. (5), there are nontrivial zeros of the Dirichleteta function neither. Thus, the nontrivial zeros are found in the strip 0 < σ < ∀ s ∈ C η ( s ) = η ( s ) (7)where the line denotes complex conjugation, we have that s ∗ and 1 − s ∗ are alsonontrivial zeros of η . In brief, nontrivial zeros come in quartets, s ∗ , 1 − s ∗ , s ∗ and1 − s ∗ forming the vertices of a rectangle within the critical strip. The statementthat, for the η function, the nontrivial zeros have all real part σ ∗ = 1 /
2, so that s ∗ = 1 − s ∗ and s ∗ = 1 − s ∗ (the rectangle degenerating in a line segment) isequivalent to the Riemann hypothesis for the Riemann zeta function [6].In this article we investigate the position of the nontrivial zeros of the eta func-tion with help of nonlinear embeddings, a novel kind of mathematical structuresintroduced in our previous works [7, 8]. We construct here a nonlinear embeddingwith the form of a double series that is absolutely and uniformly convergent oncompact sets in the whole complex plane. We call this embedding a holomorphicnonlinear embedding . It depends on a scale parameter κ ∈ (0 , ∞ ) and a horizontalshift parameter ν ∈ (0 , ∞ ), both in R , and converges asymptotically to η ( s ) every-where as κ tends to infinity. With help of M¨obius inversion, and taking advantageof absolute convergence of the series concerned, we then show the crucial propertythat η ( s ) itself can be expressed as a linear combination of horizontal shifts ofthe holomorphic nonlinear embedding η κµ ( s ) and we study the implications of thislinear combination on the position of the nontrivial zeros of the eta function givinga proof of the Riemann hypothesis.The outline of this article is as follows. In Section 2 we construct the holo-morphic nonlinear embedding η κν ( s ) for the Dirichlet eta function η ( s ). We provethe global absolute and uniform convergence on compact sets of the series defin-ing η κν ( s ) and establish the asymptotic limits of the embedding. In Section 3 wetake advantage of these properties (specifically, we make heavy use of the absoluteconvergence of this series) to derive the crucial properties: 1) the holomorphic non-linear embedding can be expressed as a linear combination of horizontally shiftedeta functions; 2) the eta function itself can be expressed as a linear combinationof horizontally shifted holomorphic nonlinear embeddings. These results are thenexploited in Section 4 to derive a functional relationship for the embedding and toprove the Riemann hypothesis, a result that emerges from the shift independencein the limit κ → ∞ of the construction. Holomorphic nonlinear embedding for theDirichlet eta function
We first introduce some notations and the basic functions on which our approachis based.
Definition 2.1.
Let x ∈ R . We define the B κ -function [9] as B κ ( x ) := 12 " tanh x + κ ! − tanh x − κ ! (8) where κ ∈ (0 , ∞ ) is a real parameter. By noting that B κ ( x ) = e /κ − e − /κ e /κ + e x/κ + e − x/κ + e − /κ (9) B κ ( x ) B κ (0) = e /κ + 2 + e − /κ e /κ + e x/κ + e − x/κ + e − /κ (10)the following properties are easily verified:0 ≤ B κ ( x ) ≤ ∀ κ ∈ (0 , ∞ ) (11) B κ ( − x ) = B κ ( x ) (12)lim κ →∞ B κ ( x ) = 0 (13)0 ≤ B κ ( x ) B κ (0) ≤ ∀ κ ∈ (0 , ∞ ) (14)lim κ →∞ B κ ( x ) B κ (0) = 1 (15) Definition 2.2. (Holomorphic nonlinear embedding.)
Let η ( s ) be the Dirich-let eta function, given by Eq. (4). Then, we define the holomorphic nonlinearembedding η κν ( s ) of η ( s ) as the series η κν ( s ) := ∞ X k =0 k +1 k X m =0 (cid:18) km (cid:19) ( − m ( m + 1) s B κ (1 / ( m + 1) ν ) B κ (0) (16) with the real parameters κ ∈ (0 , ∞ ) and ν ∈ (0 , ∞ ) . Theorem 2.1.
The double series in Eq. (16) converges absolutely and uniformlyon compact sets to the entire function η κν ( s ) . roof. We build on Sondow [2], who proved that the double series defining theeta function in Eq. (4) converge absolutely and uniformly on compact sets in theentire complex plane. In particular, if we define, f k ( s ) := 12 k +1 k X m =0 (cid:18) km (cid:19) ( − m ( m + 1) s (17)so that η ( s ) = P ∞ k =0 f k ( s ), Sondow proved that there is a sequence of positive realnumbers { M k } satisfying | f k ( s ) | ≤ k +1 k X m =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) km (cid:19) ( − m ( m + 1) s (cid:12)(cid:12)(cid:12)(cid:12) ≤ M k (18)and which ∞ X k =0 M k < ∞ (19)so that Weierstrass M-test is satisfied. We now note that we can write η κν ( s ) as η κν ( s ) = ∞ X k =0 f k,κν ( s ) (20)where f k,κν ( s ) := 12 k +1 k X m =0 (cid:18) km (cid:19) ( − m ( m + 1) s B κ (1 / ( m + 1) ν ) B κ (0) (21)Now, by the triangle inequality, | f k,κν ( s ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k +1 k X m =0 (cid:18) km (cid:19) ( − m ( m + 1) s B κ (1 / ( m + 1) ν ) B κ (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k +1 k X m =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) km (cid:19) ( − m ( m + 1) s B κ (1 / ( m + 1) ν ) B κ (0) (cid:12)(cid:12)(cid:12)(cid:12) = 12 k +1 k X m =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) km (cid:19) ( − m ( m + 1) s (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) B κ (1 / ( m + 1) ν ) B κ (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k +1 k X m =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) km (cid:19) ( − m ( m + 1) s (cid:12)(cid:12)(cid:12)(cid:12) (22)where Eq. (14) has been used. Therefore, from this last expression and Eq. (18) | f k,µν ( s ) | ≤ k +1 k X m =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) km (cid:19) ( − m ( m + 1) s (cid:12)(cid:12)(cid:12)(cid:12) ≤ M k (23) nd, thus, the sequence of positive real numbers { M k } found by Sondow majorizesthe sequence {| f k,µν ( s ) |} as well, and the result follows. Theorem 2.2.
We have, ∀ s ∈ C lim κ →∞ η κν ( s ) = η ( s ) ∀ ν ∈ (0 , ∞ ) (24)lim κ → η κν ( s ) = η ( s ) − ∀ ν ∈ (0 , ∞ ) (25)lim ν →∞ η κν ( s ) = η ( s ) + B κ (1) B κ (0) − ∀ κ ∈ (0 , ∞ ) (26)lim ν → η κν ( s ) = B κ (1) B κ (0) η ( s ) ∀ κ ∈ (0 , ∞ ) (27) where B κ (1) / B κ (0) = (tanh κ − tanh κ ) / (2 tanh κ ) , as given by Eq. (8).Proof. We first observe that, from Eq. (10),lim κ →∞ B κ (1 / ( m + 1) ν ) B κ (0) = 1lim κ → B κ (1 / ( m + 1) ν ) B κ (0) = (cid:26) m ≥
10 if m = 0lim ν →∞ B κ (1 / ( m + 1) ν ) B κ (0) = ( m ≥ B κ (1) B κ (0) if m = 0lim ν → B κ (1 / ( m + 1) ν ) B κ (0) = B κ (1) B κ (0)By using Eq. (16) and the above expressions, we getlim κ →∞ η κν ( s ) = lim κ → " ∞ X k =0 k +1 k X m =0 (cid:18) km (cid:19) ( − m ( m + 1) s B κ (1 / ( m + 1) ν ) B κ (0) = ∞ X k =0 k +1 k X m =0 (cid:18) km (cid:19) ( − m ( m + 1) s = η ( s )lim κ → η κν ( s ) = ∞ X k =0 k +1 k X m =1 (cid:18) km (cid:19) ( − m ( m + 1) s = η ( s ) − ν →∞ η κν ( s ) = B κ (1) B κ (0) + ∞ X k =0 k +1 k X m =1 (cid:18) km (cid:19) ( − m ( m + 1) s = η ( s ) + B κ (1) B κ (0) − ν → η κν ( s ) = B κ (1) B κ (0) ∞ X k =0 k +1 k X m =0 (cid:18) km (cid:19) ( − m ( m + 1) s = B κ (1) B κ (0) η ( s ) . Functional expansions of η κν ( s ) and η ( s ) We now derive an equivalent expression for η κν ( s ) valid for any κ sufficiently large(specifically, ∀ κ > /π ) and prove that this expression can be inverted to express η ( s ) as a function of horizontal shifts of η κν ( s ). Theorem 3.1. If κ > /π , ∀ ν ∈ (0 , ∞ ) the holomorphic embedding η κν ( s ) has theabsolutely convergent series expansion η κν ( s ) = η ( s ) + ∞ X n =1 a n ( κ ) η ( s + 2 νn ) (28) where, for n a non-negative integer a n ( κ ) = 1tanh (1 / κ ) ∞ X j = n +1 n (2 j − B j j (2 j − n − n )! κ j − (29) and B m denote the even Bernoulli numbers: B = 1 , B = , B = − , etc.Proof. For κ > π , the hyperbolic tangents in the definition of the B -function,Eq. (8) with x = 1 / ( m + 1) ν , can be expanded in their absolutely convergentMacLaurin series for all ∀ m ≥ ∀ ν >
0, 1 / ( m + 1) ν ≤ B κ (cid:18) m + 1) ν (cid:19) == 12 ∞ X j =1 j (2 j − B j (2 j )! κ j − "(cid:18) m + 1) ν + 12 (cid:19) j − − (cid:18) m + 1) ν − (cid:19) j − = 12 ∞ X j =1 j (2 j − B j (2 j )! κ j − j − X h =0 (cid:18) j − h (cid:19) h ( m + 1) ν (2 j − − h ) (1 − ( − h )= ∞ X j =1 j − B j (2 j )! κ j − j X h =1 (cid:18) j − h − (cid:19) j − h ) ( m + 1) j − h ) ν = ∞ X j =1 j − B j (2 j )! κ j − j − X n =0 (cid:18) j − j − n − (cid:19) n ( m + 1) nν = ∞ X n =0 m + 1) nν ∞ X j = n +1 (cid:18) j − n (cid:19) n +1 (2 j − B j (2 j )! κ j − (30)where we have used the absolute convergence of the series to change the order ofthe sums. We also have B κ (0) = tanh 12 κ = ∞ X j =1 j − B j (2 j )! κ j − (31) f we then replace these expansions in the definition of the embedding, Eq.(16), we find, by exploiting the absolute convergence of the series η κν ( s ) = ∞ X n =0 ∞ X k =0 k +1 k X m =0 (cid:18) km (cid:19) ( − m ( m + 1) s +2 nν ∞ X j = n +1 (cid:18) j − n (cid:19) n +1 (2 j − B j (2 j )! κ j − B κ (0)= ∞ X n =0 η ( s + 2 νn ) ∞ X j = n +1 (cid:18) j − n (cid:19) n +1 (2 j − B j (2 j )! κ j − B κ (0)= ∞ X n =0 a n ( κ ) η ( s + 2 νn ) = η ( s ) + ∞ X n =1 a n ( κ ) η ( s + 2 νn ) (32)where, for all non-negative integer n , we have defined a n ( κ ) := ∞ X j = n +1 j − B j (2 j )! κ j − B κ (0) 2 n (cid:18) j − n (cid:19) (33)= 1tanh (1 / κ ) ∞ X j = n +1 n (2 j − B j j (2 j − n − n )! κ j − and we have also used that, from Eq. (31) a = ∞ X j =1 j − B j (2 j )! κ j − B κ (0) (cid:18) j − j − (cid:19) = 1 . Corollary 1.
Asymptotically, for κ large, we have, η κν ( s ) = η ( s ) − η ( s + 2 ν ) κ + O (cid:18) κ (cid:19) (34) Proof.
From the theorem, we have ∀ n ∈ Z + ∪ { } a n ( κ ) = 1tanh (1 / κ ) ∞ X j = n +1 n (2 j − B j j (2 j − n − n )! κ j − = 1( κ − O (cid:0) κ (cid:1) ∞ X j = n +1 n (2 j − B j j (2 j − n − n )! κ j − = (cid:18) O (cid:18) κ (cid:19)(cid:19) ∞ X j = n +1 n +1 (2 j − B j j (2 j − n − n )! κ j − = 2 n +1 (2 n +2 − B n +2 ( n + 1)(2 n )! κ n + O (cid:18) κ n +2 (cid:19) = O (cid:18) κ n (cid:19) (35) herefore, η κν ( s ) = η ( s ) + 30 B η ( s + 2 ν ) κ + O (cid:18) κ (cid:19) (36)and the result follows from noting that B = − / Theorem 3.2. (M¨obius inversion formula.)
For any κ > /π and η κν ( s ) given by Eq.(28), η κν ( s ) = η ( s ) + ∞ X n =1 a n ( κ ) η ( s + 2 νn ) (37) we have, η ( s ) = η κν ( s ) + ∞ X n =1 b n ( κ ) η κν ( s + 2 νn ) (38) where the coefficients b n are recursively obtained from k X n =0 b k ( κ ) a n − k ( κ ) = δ k (39) with δ k being the Kronecker delta ( δ k = 1 if k = 0 and δ k = 0 otherwise).Proof. We have ∞ X n =0 b n η κν ( s + 2 νn ) = ∞ X n =0 b n ∞ X m =0 a m ( κ ) η ( s + 2 νn + 2 νm )= ∞ X k =0 " k X n =0 b n ( κ ) a k − n ( κ ) η ( s + 2 νk )= ∞ X k =0 δ k η ( s + 2 νk )= η ( s )The coefficients b n can be recursively obtained from the known a n by solving theequations a b = 1 (40) a b + a b = 0 (41) a b + a b + a b = 0 (42) . . . Where, since a = 1, b = 1, and, therefore, b = − a , b = a − a , etc. emark . Eq. (38), with η κν given by Eq. (16) is the main result of this work,since it expresses the Dirichlet eta function in terms of a globally convergent series(absolutely and uniformly on compact sets) of horizontal shifts of η κν . These shiftsare weighted by powers of 1 /κ . Remark . Theorem 3.2 is, indeed, a specific case of Theorem 3.3 on p. 82 in [11]particularized to the functions η and η κν considered here. Proposition 3.1.
For κ > /π we have ∞ X n =1 a n ( κ ) = tanh κ − tanh κ κ − ∞ X n =1 b n ( κ ) = 2 tanh κ tanh κ − tanh κ − Proof.
From Eq. (26) lim ν →∞ η κν ( s ) = η ( s ) + B κ (1) B κ (0) − κ > /π , from Eqs. (37) and (38)lim ν →∞ η κν ( s ) = η ( s ) + ∞ X n =1 a n ( κ ) = η ( s ) − B κ (1) B κ (0) ∞ X n =1 b n ( κ ) (46)because, from Eqs. (4) and (16), lim ν →∞ η ( s +2 νn ) = 1 and lim ν →∞ η κν ( s +2 νn ) = B κ (1) / B κ (0) for every integer n ≥
1. By equating Eqs. (45) and (46) and by usingEq. (8), the result follows. η In this section we prove the Riemann hypothesis. We first introduce three lem-mas, that establish a functional equation for the embedding and its asymptoticproperties.
Lemma 4.1. (Functional relationship for the embedding.)
We have ∀ ν ∈ (0 , ∞ ) and ∀ κ > π/ η κν ( s ) − λ ( s ) η κν (1 − s ) = ∞ X n =1 b n ( κ ) [ λ ( s ) η κν (1 − s + 2 νn ) − η κν ( s + 2 νn )] (47) where λ ( s ) = 1 − − s − s s π s − sin (cid:16) πs (cid:17) Γ(1 − s ) (48) roof. From Eq. (38) we have η ( s ) = η κν ( s ) + ∞ X n =1 b n ( κ ) η κν ( s + 2 νn ) λ ( s ) η (1 − s ) = λ ( s ) η κν (1 − s ) + λ ( s ) ∞ X n =1 b n ( κ ) η κν (1 − s + 2 νn )whence, by subtracting both equations and applying Eq. (2) the result follows. Lemma 4.2. If s ∗ = σ ∗ + it ∗ is a non-trivial zero of η ( s ) then η κν ( s ∗ ) = 0 and η κν (1 − s ∗ ) = 0 for finite asymptotically large κ and ν > / . Furthermore, wehave lim κ →∞ η κν ( s ∗ ) η κν (1 − s ∗ ) = η ( s ∗ + 2 ν ) η (1 − s ∗ + 2 ν ) (49) Proof.
The real part σ ∗ of the nontrivial zero s ∗ satisfies 0 < σ ∗ <
1. Now, since η ( s ∗ ) = η (1 − s ∗ ) = 0, we have, from Eq. (34) η κν ( s ∗ ) = − η ( s ∗ + 2 ν ) κ + O (cid:18) κ (cid:19) (50) η κν (1 − s ∗ ) = − η (1 − s ∗ + 2 ν ) κ + O (cid:18) κ (cid:19) (51)We have that ∀ ν > / η ( s ∗ + 2 ν ) = 0 and η (1 − s ∗ + 2 ν ) = 0 because values s ∗ + 2 ν and 1 − s ∗ + 2 ν both lie in the half-plane σ > η ( s ) has no zeros there.Therefore η κν ( s ∗ ) = 0 and η κν (1 − s ∗ ) = 0 are both nonzero for sufficiently large κ ( κ > /π being a lower bound). Eq. (49) follows then as a trivial consequenceof Eqs. (50) and (51) and the absolute convergence of all the series involved. Lemma 4.3.
Let s γ ∈ C , s γ ∈ γ be such that η κν (1 − s γ ) = 0 along a path γ inthe complex plane and let s ′ be an endpoint of γ . Then, lim κ →∞ lim s γ −→ γ s ′ η κν ( s γ ) η κν (1 − s γ ) = lim s γ −→ γ s ′ lim κ →∞ η κν ( s γ ) η κν (1 − s γ ) = λ ( s ′ ) (52) Furthermore, if s ′ = s ∗ is a nontrivial zero of the Dirichlet eta function, ∀ ν > / η ( s ∗ + 2 ν ) η (1 − s ∗ + 2 ν ) = λ ( s ∗ ) (53) Proof.
From Eq. (54) we have, by dividing by η κν (1 − s γ ) at any point s γ of γ (since η κν (1 − s γ ) = 0) η κν ( s γ ) η κν (1 − s γ ) = λ ( s γ ) + ∞ X n =1 b n ( κ ) λ ( s γ ) η κν (1 − s γ + 2 νn ) − η κν ( s γ + 2 νn ) η κν (1 − s γ ) (54) here b ( κ ) = O ( κ − ). We now have, on one hand,lim κ →∞ lim s γ −→ γ s ′ η κν ( s γ ) η κν (1 − s γ ) = lim κ →∞ η κν ( s ′ ) η κν (1 − s ′ ) (55)= lim κ →∞ " λ ( s ′ ) + ∞ X n =1 b n ( κ ) λ ( s ′ ) η κν (1 − s ′ + 2 νn ) − η κν ( s ′ + 2 νn ) η κν (1 − s ′ ) = lim κ →∞ " λ ( s ′ ) + ∞ X n =1 b n ( κ ) λ ( s ′ ) η (1 − s ′ + 2 νn ) − η ( s ′ + 2 νn ) η (1 − s ′ ) = λ ( s ′ )and, on the other hand,lim s γ −→ γ s ′ lim κ →∞ η κν ( s γ ) η κν (1 − s γ ) = lim s γ → s ′ η ( s γ ) η (1 − s γ ) (56)= lim s γ → s ′ λ ( s γ ) = λ ( s ′ )whence the result follows.Let us now assume that s ′ = s ∗ is a nontrivial zero of the Dirichlet eta function.Then, we have that lim κ →∞ η κν (1 − s ∗ ) = η (1 − s ∗ ) = 0 = η ( s ∗ ) = lim κ →∞ η κν ( s ∗ )and the function in Eq. (55)Φ( s ′ ) := lim κ →∞ η κν ( s ′ ) η κν (1 − s ′ ) = η ( s ′ ) η (1 − s ′ ) (57)is undefined at s ′ = s ∗ . However, s ′ = s ∗ is a removable singularity and we cantake Φ( s ∗ ) = λ ( s ∗ ). To see this, note that Φ( s ) = λ ( s ) for all s in the critical strip0 < σ < s ∗ . But the function λ ( s ) is holomorphicfor all s in the critical strip including the nontrivial zeros s ∗ of η . Therefore, byRiemann’s theorem on extendable singularities, Φ( s ) is holomorphically extendableover s ∗ and we can have, in consistency with Eq. (56)Φ( s ∗ ) = λ ( s ∗ ) (cid:18) = lim s → s ∗ η ( s ) η (1 − s ) (cid:19) (58)This proves Eq. (52). We then note that, on one handΦ( s ∗ ) = lim κ →∞ η κν ( s ∗ ) η κν (1 − s ∗ ) = λ ( s ∗ ) (59)and, on the other, from Eq. (49)Φ( s ∗ ) = lim κ →∞ η κν ( s ∗ ) η κν (1 − s ∗ ) = η ( s ∗ + 2 ν ) η (1 − s ∗ + 2 ν ) (60) oth expressions must be equal at s ∗ because: 1) Eq. (59) is a consequence ofΦ( s ) being equal to the holomorphic λ ( s ) in the punctured critical strip (save,exactly at the zeros s ∗ ) and, therefore, holomorphically extendable over s ∗ and 2)Eq. (60) is a consequence of the asymptotic behavior of the embedding close to anontrivial zero of the Dirichlet eta function. Thus, Eq. (53) follows. Alternative proof of Eq. (53) . An equivalent way of obtaining Eq. (53) is, directly,from Eq. (55), applying it to s ′ = s ∗ . We have,lim κ →∞ lim s γ −→ γ s ∗ η κν ( s γ ) η κν (1 − s γ ) = lim κ →∞ η κν ( s ∗ ) η κν (1 − s ∗ ) (61)= lim κ →∞ " λ ( s ∗ ) + ∞ X n =1 b n ( κ ) λ ( s ∗ ) η κν (1 − s ∗ + 2 νn ) − η κν ( s ∗ + 2 νn ) η κν (1 − s ∗ ) = λ ( s ∗ ) + lim κ →∞ λ ( s ∗ ) η κν (1 − s ∗ + 2 ν ) − η κν ( s ∗ + 2 ν ) κ η κν (1 − s ∗ )= λ ( s ∗ ) − λ ( s ∗ ) η (1 − s ∗ + 2 ν ) − η ( s ∗ + 2 ν ) η (1 − s ∗ + 2 ν ) (62)and since η (1 − s ∗ + 2 ν ) = 0 ∀ ν ∈ (1 / , ∞ ) and we have for any ε ∈ C in a disk ofsufficiently small radiusΦ( s ∗ + ε ) = lim κ →∞ η κν ( s ∗ + ε ) η κν (1 − s ∗ + ε ) = λ ( s ∗ + ε ) (63)by taking the limit ε → s ) is holomorphically extendableto s ∗ where it has then the value λ ( s ∗ ) we obtain, from Eq. (62) λ ( s ∗ ) η (1 − s ∗ + 2 ν ) − η ( s ∗ + 2 ν ) = 0 (64)which is Eq. (53). (cid:3) Theorem 4.1. (Riemann hypothesis.)
All nontrivial zeros s ∗ = σ ∗ + it ∗ of theDirichlet eta function η ( s ) have real part σ ∗ = 1 / .Proof. Let σ ∗ = + ǫ be the real part of a nontrivial zero s ∗ = σ ∗ + it ∗ of η in thecritical strip. From Eq. (53), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η (cid:0) + 2 ν + it ∗ + ǫ (cid:1) η (cid:0) + 2 ν − it ∗ − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:18)
12 + it ∗ + ǫ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (65)and we note that the right hand side of this expression is shift-invariant (it does notdepend on ν ) but the left hand side is not: the horizontal shift parameter ν can be rbitrarily varied in the interval (0 , ∞ ) and, in particular, it can be selected so thatthe point s ∗ +2 ν lies anywhere on the half-plane σ ≥ t ∗ . The modulus of η varies on horizontal lines [10]. The only possibility for equation Eq. (65) to havesolution for a nontrivial zero s ∗ forces ǫ = 0. To see this, put x = + 2 ν − ǫ >> | η ( x − it ∗ ) | = (cid:12)(cid:12)(cid:12) η ( x + it ∗ ) (cid:12)(cid:12)(cid:12) = | η ( x + it ∗ ) | , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η (cid:0) + 2 ν + it ∗ + ǫ (cid:1) η (cid:0) + 2 ν − it ∗ − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) η ( x + it ∗ + 2 ǫ ) η ( x + it ) (cid:12)(cid:12)(cid:12)(cid:12) . (66)Since x can be increased arbitrarily by increasing ν , we can take x so large that,asymptotically η ( x + it ∗ + 2 ǫ ) ∼ η ( x + it ∗ ) + 2 ǫ ∂η∂x (cid:12)(cid:12)(cid:12)(cid:12) x + it ∗ . (67)In this asymptotic regime, we can truncate Eq. (1) to the first two terms and itsderivative becomes ∂η∂x (cid:12)(cid:12)(cid:12)(cid:12) x + it ∗ ∼ ln 22 x + it ∗ . (68)Furthermore, η ( x + it ∗ ) ∼ − − x − it ∗ and thus (cid:12)(cid:12)(cid:12)(cid:12) η ( x + it ∗ + 2 ǫ ) η ( x + it ) (cid:12)(cid:12)(cid:12)(cid:12) ∼ (cid:12)(cid:12)(cid:12)(cid:12) ǫ ln 22 x + it ∗ − (cid:12)(cid:12)(cid:12)(cid:12) . (69)Therefore, for ν large, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η (cid:0) + 2 ν + it ∗ + ǫ (cid:1) η (cid:0) + 2 ν − it ∗ − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ (cid:12)(cid:12)(cid:12)(cid:12) ǫ ln 22 +2 ν − ǫ + it ∗ − (cid:12)(cid:12)(cid:12)(cid:12) , (70)and the l.h.s. of Eq. (65) depends explicitly on the free parameter ν . Thus, ther.h.s. of Eq. (65) can have an infinite number of different values for its modulus,which is absurd. The only possibility of cancelling the ν dependence, forced bythe consistency of the equation, is to have ǫ = 0. In this way, we obtain, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η (cid:0) + 2 ν + it ∗ (cid:1) η (cid:0) + 2 ν − it ∗ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 = (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:18)
12 + it ∗ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (71)an equation that is known to have infinitely many solutions for t ∗ . Therefore, ǫ = 0, s ∗ = + it ∗ and the result follows. Conclusions
In this article a complex entire function called a holomorphic nonlinear embedding η κν has been constructed and some of its properties (mainly asymptotic ones) havebeen investigated. It has been shown that the function η κν can be expressed asa series expansion in terms of horizontal shifts of the Dirichlet eta function. Theholomorphic character of η κν has been established by proving the global absoluteand uniform convergence on compact sets of its defining double series. The co-efficients of the expansion and their sum have been explicitly calculated. It hasalso been shown that this expansion can be inverted to yield the eta function as alinear expansion of horizontal shifts of η κν . This is the central result of this article,since it allows the Dirichlet eta function to be understood as a linear superpositionof different layers governed by shifts of η κν and weighted by powers of 1 /κ . Thisresult shows that, although one can envisage, in principle, infinitely many waysof smoothly embedding the Dirichlet eta function in a more general structure, theone presented here ( η κν ) is not gratuitous because it is itself embedded within thestructure of the eta function, revealing some of its secrets thanks to its scale andhorizontal shift parameters κ and ν . In particular, the truth of the Riemann hy-pothesis emerges naturally as a consequence of the functional relationship of theDirichlet eta function and its uniform attainment everywhere by a hierarchy offunctional equations of the holomorphic nonlinear embedding in the limit κ → ∞ .Operators yielding vertical shifts of the Riemann zeta function ζ ( s ) arise nat-urally in approaches to the Riemann zeros using ideas from supersymmetry [12].These vertical shifts can, indeed, be expressed more compactly in terms of theDirichlet eta function η (see e.g. Eq.(18) in [12]). Whether there exists any rela-tionship of these vertical shifts induced by lowering and raising operators in [12]with the shifts obtained here as a result of an explicit series construction is an in-teresting open question. We point out that ν in this article can be made a complexnumber with positive real part and arbitrary imaginary part and all main resultsof this article apply without any modification: Eqs. (37) and (38) are indeed valid,for ν = ν r + iν i , with ν r > ν i ∈ ( −∞ , ∞ ) and this does not affect theproof of the Riemann hypothesis here presented in any way. The condition ν r > s only through summands of theform 1 /m s ; 3) there exists a functional relation like Eq. (2).We believe that the methods described in this paper might be useful to get nsight in recent intriguing experimental phenomena connecting the coefficients oftruncated Dirichlet series to the Erathostenes sieve [14]. References [1] E. C. Titchmarsh and D. R. Heath-Brown,
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