Jacob's ladder as generator of new class of iterated L 2 -orthogonal systems and their dependence on the Riemann's function
aa r X i v : . [ m a t h . C A ] D ec JACOB’S LADDER AS GENERATOR OF NEW CLASS OFITERATED L -ORTHOGONAL SYSTEMS AND THEIRDEPENDENCE ON THE RIEMANN’S FUNCTION ζ (cid:0) + it (cid:1) JAN MOSER
Abstract.
In this paper new classes of L -orthogonal functions are con-structed as iterated L -orthogonal systems. In order to do this we use thetheory of the Riemann’s zeta-function as well as our theory of Jacob’s ladders.The main result is new one in the theory of the Riemann’s zeta-function andsimultaneously in the theory of L -orthogonal systems.DEDICATED TO THE MEMORY OF FOURIER’S EGYPTIAN ANABASE Main result ϕ ( t ),(b) the function ˜ Z ( t ) = d ϕ ( t )d t = 1 ω ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) ,ω ( t ) = ß O Å ln ln t ln t ã™ ln t, t → ∞ , (1.1)(c) the (direct) iterations of the Jacob’s ladder ϕ ( t ) = t, ϕ ( t ) = ϕ ( t ) , ϕ ( t ) = ϕ ( ϕ ( t )) , . . . , ϕ k ( t ) = ϕ ( ϕ k − ( t ))for every fixed k ∈ N ,(d) the reverse iterations (by means of ϕ ( t ))[ T , ˙ T + U ] , [ T , ˙ T + U ] , . . . , [ k T , k ˙ T + U ] ,U = o Å T ln T ã , T → ∞ , of the basic segment [ T, T + U ] = [ T , ˙ T + U ] , where(1.2) [ T , ˙ T + U ] ≺ [ T , ˙ T + U ] ≺ · · · ≺ [ k T , k ˙ T + U ] , that we have introduced into the theory of the Riemann’s zeta-function inour papers [1] – [4]. Key words and phrases.
Riemann zeta-function.
Theorem 1 . For every fixed L -orthogonal system(1.3) { f n ( t ) } ∞ n =0 , t ∈ [ a, a + 2 l ] , a ∈ R , l ∈ R + and for every fixed k ∈ N there is the set of k new iterated L -orthogonal systems(1.4) { f pn ( t ) } ∞ n =0 , t ∈ [ a, a + 2 l ] , p = 1 , , . . . , k, where f pn ( t ) = f n Ö ϕ p Ö p ˙ T + 2 l − p T l ( t − a ) + p T è − T + a è ×× p − Y r =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ Z Ö ϕ r Ö p ˙ T + 2 l − p T l ( t − a ) + p T èè (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (1.5)for all sufficiently big T > k new iteratedorthogonal systems P pn ( t ) = P n Ö ϕ p Ö p ˘ T + 2 − p T l ( t + 1) + p T è − T − è ×× p − Y r =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ Z Ö ϕ r Ö p ˘ T + 2 − p T l ( t + 1) + p T èè (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , for all sufficiently big T to the classical Legendre’s orthogonal system { P n ( t ) } ∞ n =0 , t ∈ [ − , a = − , l = 1 . For example P n ( t ) = P n Ö ϕ Ö ˘ T + 2 − T l ( t + 1) + T è − T − è ×× (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ Z Ö ϕ Ö ˘ T + 2 − T l ( t + 1) + T èè (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , t ∈ [ − , . Corollary 1 . For L -orthogonal system (1.3) there is the set of k new iterated L -orthogonal systems f n Ö ϕ p Ö p ˙ T + 2 l − p T l ( t − a ) + p T è − T + a è ∞ n =0 ,t ∈ [ a, a + 2 l ] , p = 1 , . . . , k, ACOB’S LADDER AS GENERATOR OF NEW CLASS OF ITERATED L -ORTHOGONAL SYSTEMS AND THEIR DEPENDENCE ON THE RIEMANN’S FUNCTION ζ (cid:0) + it (cid:1) with weights p − Y r =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ Z Ö ϕ r Ö p ˘ T + 2 − p T l ( t + 1) + p T èè (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and this last is (see (1.1)) ∼ p T p − Y r =0 ˜ Z Ö ϕ r Ö p ˙ T + 2 l − p T l ( t − a ) + p T èè , T → ∞ . Remark . Theorem 1 represents completely new result in the theory of the Rie-mann’s zeta-function and simultaneously in the the theory of L -orthogonal sys-tems. Remark . The last row for all sufficiently big
T > k -tuples of iterated L -orthogonal systems for every fixed system (1.3). Remark . Dependence of iterated L -orthogonal systems (1.4) on the Riemann’szeta-function ζ (cid:0) + it (cid:1) is evident one, see (1.1), (1.5). Remark . This paper is the continuation of 54 papers concerning Jacob’s ladders.These can be found in arXiv [math.CA] starting with the paper [1].2.
Jacob’s ladders Z µ [ x ( T )]0 Z ( t ) e − x ( T ) t d t = Z T Z ( t )d t we have introduced in the paper [1], where Z ( t ) = e iϑ ( t ) ζ Å
12 + it ã ,ϑ ( t ) = − t π + Im ß ln Γ Å
14 + i t ã™ , (2.2)and the class of functions { µ } specified as µ ∈ C ∞ ([ y , + ∞ ))monotonically increasing and unbounded from above and obeying the inequality(2.3) µ ( y ) ≥ y ln y. Lemma 1 . For any µ ∈ { µ } there is exactly one solution to the integral equation(2.1): ϕ ( T ) = ϕ ( T, µ ) , T ∈ [ T , + ∞ ) , T = T [ ϕ ] > ,ϕ ( T ) T →∞ −−−−→ ∞ . JAN MOSER
Let the symbol { ϕ } denote the system of these solutions. The function ϕ ( T )is related to the zeroes of the Riemann’s zeta-function on the critical line by thefollowing way. Let t = γ be such a zero of ζ Å
12 + it ã and of the order n ( γ ), where n ( γ ) = O (ln γ ) , γ → ∞ , of course. Then the following holds true. Remark . The points [ γ, ϕ ( γ )] , γ > T (and only these points) are inflection points with the horizontal tangent. In moredetail, the following system of equations holds true: ϕ ′ ( γ ) = ϕ ′′ ( γ ) = · · · = ϕ (2 n ) ( γ ) = 0 , ϕ (2 n +1) ( γ ) = 0 , n = n ( γ ) . Definition 1 . With respect the above mentioned property, an element ϕ ∈ { ϕ } iscalled Jacob’s ladder leading to [+ ∞ , + ∞ ]. The rungs of that ladder are segmentsof the curve ϕ lying in the neighbourhoods of the points[ γ, ϕ ( γ )] , γ > T [ ϕ ] . Remark . We call ϕ ( T ) as Jacob’s ladder in analogy with Jacob’s dream in Chu-mash, Bereishis, 28:12. Remark . Finally, the composite function g [ ϕ ( T )] is also called Jacob’s ladder forany function g that is increasing, C ∞ on [ y , + ∞ ) and unbounded from above. Forexample, the function(2.4) ϕ ( T ) = 12 ϕ ( T )as composition of g ( y ) = y , y ≥ y , y = ϕ ( T ), T ≥ T [ ϕ ] = T [ ϕ ], g ′ y = > Basic property of Jacob’s ladders: existence of almost exactexpressions for the classical Hardy-Littlewood integral (1918) Z T (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) d t can be expressed as follows:(3.2) Z T (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) d t = T ln T + (2 c − − ln 2 π ) T + R ( T ) , with, for example, Ingham’s error term(3.3) R ( T ) = O ( T ln T ) = O ( T + δ ) , δ > , T → ∞ for arbitrarily small δ .Next, by Good’s Ω-theorem (1977) we have that(3.4) R ( T ) = Ω( T ) , T → ∞ . ACOB’S LADDER AS GENERATOR OF NEW CLASS OF ITERATED L -ORTHOGONAL SYSTEMS AND THEIR DEPENDENCE ON THE RIEMANN’S FUNCTION ζ (cid:0) + it (cid:1) Remark . Let(3.5) R a ( T ) = O ( T + a ) , a ∈ [ δ,
14 + δ ] , T → ∞ . Then, by (3.4), it is true that for every valid estimate of type (3.5) one obtains:(3.6) lim sup T →∞ | R a ( T ) | = + ∞ . In other words, every expression of the type (3.2) possesses an unbounded errorterm at infinity.3.2. Under the circumstances (3.2) and (3.6) we have proved in [1] that the Hardy-Littlewood integral (3.1) has an infinite set of other completely new and almostexact representations expressed by the following:
Property 1 . Z T (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) d t = ϕ ( T ) ln { ϕ ( T ) } ++ ( c − ln 2 π ) ϕ ( T ) + c + O Å ln TT ã , T → ∞ , (3.7)(comp. (2.4)) with the following(3.8) lim T →∞ ˜ R ( T ) = lim T →∞ O Å ln TT ã = 0 , where c is the Euler’s constant and c is the constant from the Titchmarsh-Kober-Atkinson formula. Remark . Comparison of (3.6) and (3.8) completely characterizes the level ofexactness of our representation (3.7) of the Hardy-Littlewood integral (3.1).4.
Asymptotic relation between Jacob’s ladder and theprime-counting function
Lemma 2 . (4.1) T − ϕ ( T ) ∼ (1 − c ) π ( T ); π ( T ) ∼ T ln T , T → ∞ . Remark . As a consequence, the Jacob’s ladder can be viewed as complementaryfunction to the function (1 − c ) π ( T )in the sense that(4.2) ϕ ( T ) + (1 − c ) π ( T ) ∼ T, T → ∞ . y = ϕ ( t ) : ϕ ( t ) = t, ϕ ( t ) = ϕ ( t ) , ϕ ( t ) = ϕ ( ϕ ( t )) , . . . ,ϕ k ( t ) = ϕ ( ϕ k − ( t )) , . . . , t ∈ [ T, T + U ] , T ≥ T [ ϕ ] , (4.3)of course (see (4.1))(4.4) T > ϕ ( T ) , and the symbol ϕ k ( t ) represents the k -th iteration of the Jacob’s ladder. JAN MOSER
Remark . Let us remind that the functions ϕ k ( t ) , k = 2 , , . . . are increasing since ϕ ( t ) is increasing.4.3. In the case t ϕ k ( t ) , t ∈ [ T, T + U ]it follows from Lemma 2 that:(4.5) ϕ k ( t ) − ϕ k +11 ( t ) ∼ (1 − c ) ϕ k ( t )ln ϕ k ( t ) , k = 0 , , . . . , n, t → ∞ , where n ∈ N is arbitrary and fixed one. Now formulae (4.5) imply the followingproperties of the set { ϕ k ( t ) } n +1 k =0 : Lemma 3 . For(4.6) t ∈ [ T, T + U ] , U = o Å T ln T ã , T → ∞ the following statements hold true:(4.7) t ∼ ϕ ( t ) ∼ ϕ ( t ) ∼ · · · ∼ ϕ n +11 ( t ) , (4.8) t > ϕ ( t ) > ϕ ( t ) > · · · > ϕ n +11 ( t ) , (4.9) ϕ k ( T ) > (1 − ǫ ) T, k = 0 , , . . . , n + 1 , ǫ > , ǫ small and fixed , (4.10) ϕ k ( T + U ) − ϕ k ( T ) < n + 5 T ln T , k = 1 , . . . , n + 1 , (4.11) ϕ k ( T ) − ϕ k +11 ( T + U ) > . × T ln T , k = 0 , , . . . , n. D ( T, U, n ) = n +1 [ k =0 [ ϕ k ( T ) , ϕ k ( T + U )] . Remark . We list here the properties of the set (4.12):(a) It is disconnected set (see (4.11)) for every admissible U , (see (4.6));(b) Components of the set D are distributed as follows: (see (4.11))[ ϕ n +11 ( T ) , ϕ n +11 ( T + U )] ≺ [ ϕ n ( T ) , ϕ n ( T + U )] ≺ · · · ≺≺ [ ϕ ( T ) , ϕ ( T + U )] ≺ [ ϕ ( T ) , ϕ ( T + U )] = [ T, T + U ] . (4.13) Remark . Asymptotic behaviour of the set D is as follows: at T → ∞ its compo-nents receding unboundedly each from other (see (4.11)) and all together recede toinfinity. Hence at large T the set (4.12) behaves like one-dimensional Friedmann-Hubble expanding universe. ACOB’S LADDER AS GENERATOR OF NEW CLASS OF ITERATED L -ORTHOGONAL SYSTEMS AND THEIR DEPENDENCE ON THE RIEMANN’S FUNCTION ζ (cid:0) + it (cid:1) On the function ˜ Z ( t )Let us recall the following formula we have proved in [1]:(5.1) Z ( t ) = Φ ′ ϕ [ ϕ ( t )] d ϕ ( t )d t , t ∈ [ T, T + U ] , U = o Å T ln T ã , where Φ ′ ϕ [ ϕ ] = 2 ϕ Z µ [ ϕ ]0 te − ϕ t Z ( t )d t ++ Z { µ [ ϕ ] } e − ϕ µ [ ϕ ] d µ [ ϕ ]d ϕ , (5.2)(see [1], (3.5), (3.9)). Now we put (see (2.4) and [2], (9.1))(5.3) ˜ Z ( t ) = d ϕ ( t )d t , t ≥ T [ ϕ ] . In the next step we present just the result (see [2], Lemma 1, (7.7) – (7.9), (9.2)):
Lemma 4 . If µ a [ ϕ ] = aϕ ln ϕ, a ∈ [7 , ,t ∈ [ T, T + U ] , U = o Å T ln T ã , (5.4)then(5.5) Φ ′ ϕ [ ϕ ( t )] = 12 ß O Å ln ln t ln t ã™ ln t, i.e. (see (5.1), (5.3))(5.6) ˜ Z ( t ) = d ϕ ( t )d t = 1 ω ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) , where(5.7) ω ( t ) = 2Φ ′ ϕ [ ϕ ( t )] = ß O Å ln ln t ln t ã™ ln t, t → ∞ . Remark . The segmant [7 ,
8] is sufficient one for our purposes since the continuumset µ a [ ϕ ] corresponds to this one (comp. (2.3), (5.4)).6. Reverse iterations { k T } k k =0 by the formula(6.2) ϕ ( k T ) = k − T , k = 1 , , . . . , k , T = T, T ≥ T [ ϕ ](where k ∈ N is an arbitrary and fixed numer) since the function ϕ ( T ) , T → ∞ increases to + ∞ . Further we have(6.3) ϕ ( k T ) = k − T ⇒ . . . ⇒ ϕ k ( k T ) = T, k = 1 , . . . , k . JAN MOSER
Since(6.4) ϕ ( T ) = T ⇒ T = ϕ − ( T ) , and then we may use the inverse function ϕ − ( T ) to generate a reverse iterations.Namely we have: ϕ ( T ) = T ⇒ T = ϕ − ( T ) = ϕ − ( ϕ − ( T )) = ϕ − ( T ) , . . . , k T = ϕ − k ( T ) , k = 1 , . . . , k , (6.5)where the last row gives the k -th reverse iteration of the point T = T . Of course,we have ϕ k ( k T ) = ϕ k ( ϕ − k ( T )) = T. Lemma 5 . If(6.6) U = o Å T ln T ã , T → ∞ then(6.7) ϕ { [ k T , k ˙ T + U ] } = [ k − T , k − ˙ T + U ] , [ T , ˙ T + U ] = [ T, T + U ] , (6.8) | [ k T , k ˙ T + U ] | = k ˙ T + U − k T = o Å T ln T ã , (6.9) | [ k − ˙ T + U , k T ] | = k T − k − ˙ T + U ∼ (1 − c ) T ln T , (6.10) [
T, T + U ] ≺ [ T , ˙ T + U ] ≺ · · · ≺ [ k T , k ˙ T + U ] , k = 1 , . . . , k , (comp. Lemma 3 and (4.13)).From (6.6) – (6.10) we obtain the following property of the Jacob’s ladders. Property 2 . For every segment[
T, T + U ] , U = o Å T ln T ã , T → ∞ there is the following class of disconnected sets (comp. (4.12))(6.11) ∆( T, U, k ) = k [ r =0 [ r T , r ˙ T + U ] , ≤ k ≤ k , generated by the Jacob’s ladder ϕ ( T ). Remark . Asymptotic behaviour of the set ∆ is the same as behavior of theset (4.12), i.e. at T → ∞ its components receding unboundedly each from otherand all together recede to infinity. Hence at large T the set (6.11) behaves likeone-dimensional Friedmann-Hubble expanding universe. ACOB’S LADDER AS GENERATOR OF NEW CLASS OF ITERATED L -ORTHOGONAL SYSTEMS AND THEIR DEPENDENCE ON THE RIEMANN’S FUNCTION ζ (cid:0) + it (cid:1) Lemma 6 . If(6.12) t ∈ [ ϕ − k ( T ) , ϕ − k ( T + U )] , k = 1 , . . . , k , then (see (6.5))(6.13) ϕ r ( t ) ∈ [ ϕ r − k ( T ) , ϕ r − k ( T + U )] , r = 0 , , . . . , k, i.e. ϕ ( t ) = t ∈ [ ϕ − k ( T ) , ϕ − k ( T + U )] = [ k T , k ˙ T + U ] ,ϕ ( t ) ∈ [ ϕ − k +11 ( T ) , ϕ − k +11 ( T + U )] = [ k − T , k − ˙ T + U ] , ... ϕ k − ( t ) ∈ [ ϕ − ( T ) , ϕ − ( T + U )] = [ T , ˙ T + U ] ,ϕ k ( t ) ∈ [ ϕ ( T ) , ϕ ( T + U )] = [ T , ˙ T + U ] = [ T, T + U ] . (6.14) 7. Main lemma and proof of Theorem 1
Lemma 7 . If(7.1) U = o Å T ln T ã , T → ∞ , then for every Lebesgue-integrable function g ( t ) , t ∈ [ T, T + U ]the following holds true:(7.2) Z T + UT g ( t )d t = Z k ˙ T + U k T g [ ϕ k ( t )] k − Y r =0 ˜ Z [ ϕ r ( t )]d t, k = 1 , . . . , k . Remark . We have obtained the case k = 1:(7.3) Z T + UT g ( t )d t = Z ˙ T + U T g [ ϕ ( t )] ˜ Z ( t )d t in our paper [2], (9.5).7.2. Now we proceed to the proof of our Theorem 1. Proof of Theorem 1.
Since the system (1.3) is fixed one then the corresponding l is also fixed and consequently the condition (7.1) l = o Å T ln T ã , T → ∞ is fulfilled for all sufficiently big positive T . Now we have m = n : 0 = Z a +2 la f m ( t ) f n ( t )d t = Z T +2 lT f m ( τ − T + a ) f n ( τ − T + a )d τ =and next, by our Lemma 7 for sufficiently big T >
0, one obtains= Z p ˙ T +2 l p T f m [ ϕ p ( ρ ) − T + a ] f n [ ϕ p ( ρ ) − T + a ] p − Y r =0 ˜ Z [ ϕ r ( ρ )]d ρ =and next, by simple substitution ρ = p ˙ T + 2 l − p T l ( t − a ) + p T , t ∈ [ a, a + 2 l ] , ρ ∈ [ p T , p ˙ T + 2 l ]we obtain p ˙ T + 2 l − p T l Z a +2 la f m [ ϕ p ( p ˙ T + 2 l − p T l ( t − a ) + p T ) − T + a ] ×× f n [ ϕ p ( p ˙ T + 2 l − p T l ( t − a ) + p T ) − T + a ] ×× k − Y r =0 ˜ Z [ ϕ p ( p ˙ T + 2 l − p T l ( t − a ) + p T )]d t =(7.4)and, in the next step, we finish with (see (1.5))(7.5) = p ˙ T + 2 l − p T l Z a +2 la f pm ( t ) f pn ( t )d t ⇒ Z a +2 la f pm ( t ) f pn ( t )d t = 0 . Remark . If we use the last formula in (7.5) as the origin of a new process (ananalogue of this in the subsection 7.2), then we obtain k new iterated L -orthogonalsystems { f p ,p n ( t ) } ∞ n =0 , t ∈ [ a, a + 2 l ] , p , p = 1 , . . . , k ACOB’S LADDER AS GENERATOR OF NEW CLASS OF ITERATED L -ORTHOGONAL SYSTEMS AND THEIR DEPENDENCE ON THE RIEMANN’S FUNCTION ζ (cid:0) + it (cid:1) where f p ,p n ( t ) == f n [ ϕ p ( p ˙ T + 2 l − p T l ( ϕ p ( p ˙ T + 2 l − p T l ( t − a ) + p T ) − T ) + p T ) − T + a ] ×× p − Y r =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ Z [ ϕ r ( p ˙ T + 2 l − p T l ( ϕ p ( p ˙ T + 2 l − p T l ( t − a ) + p T ) − T ) + p T )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ×× p − Y r =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ Z [ ϕ r ( p ˙ T + 2 l − p T l ( t − a ) + p T )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and so on up to k l new iterated L -orthogonal systems { f p ,p ,...,p l n ( t ) } , t ∈ [ a, a + 2 l ] , p , . . . , p l = 1 , . . . , k for every fixed l ∈ N .7.4. Let us notice that the transformation ( ?? )(7.6) w = w ( t ) = ϕ p ( p ˙ T + 2 l − p T l ( t − a )) − T + a, t ∈ [ a, a + 2 l ]has the following properties:(a) by the subsection 6.1: w ( a ) = ϕ p ( p T ) − T + a = T − T + a = a ; p T = ϕ − p ( T ) ,w ( a + 2 l ) = ϕ p ( p ˙ T + 2 l ) = a + 2 l, (b) since the function ϕ p ( u ) is increasing one and u = p ˙ T + 2 l − p T l ( t − a ) + p T , u ∈ [ a, a + 2 l ]is evident then the composed function w ( t ) , t ∈ [ a, a + 2 l ]is increasing that is(7.7) w ( t ) ∈ [ a, a + 2 l ] . Remark . Consequently, it follows from (a) and (b) that by the one-to-one cor-respondence (7.6) we have defined new automorphism on [ a, a + 2 l ]. i.e. the k , k ,. . . , k l of new automorphisms for every fixed sufficiently big positive T .I would like to thank Michal Demetrian for his moral support of my study ofJacob’s ladders. References [1] J. Moser, ‘Jacob’s ladders and almost exact asymptotic representation of the Hardy-Littlewood integral‘, Math. Notes 88, (2010), 414-422, arXiv: 0901.3937.[2] J. Moser, ‘Jacob’s ladders, structure of the Hardy-Littlewood integral and some new class ofnonlinear integral equations‘, Proc. Steklov Inst. 276 (2011), 208-221, arXiv: 1103.0359.[3] J. Moser, ‘Jacob’s ladders, their iterations and new class of integrals connected with parts ofthe Hardy-Littlewood integral of the function | ζ (cid:0) + it (cid:1) | ’, arXiv: 1209.4719, (2012).[4] J. Moser, ‘Jacob’s ladders, reverse iterations and new infinite set of L -orthogonal systemsgenerated by the Riemann zeta-function, arXiv: 1402.2098. Department of Mathematical Analysis and Numerical Mathematics, Comenius Uni-versity, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
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