Jacob's ladders, Bessel's functions and the asymptotic solutions of a new class of nonlinear integral equations
aa r X i v : . [ m a t h . C A ] N ov JACOB’S LADDERS, BESSEL’S FUNCTIONS AND THEASYMPTOTIC SOLUTIONS OF A NEW CLASS OF NONLINEARINTEGRAL EQUATIONS
JAN MOSER
Abstract.
It is shown in this paper that there is a connection between theRiemann zeta-function ζ (cid:0) + it (cid:1) and the Bessel’s functions. In this direction,a new class of the nonlinear integral equations is introduced. The first result Z ( t ) = e iϑ ( t ) ζ Å
12 + it ã that is generated by the Riemann zeta-function, where ϑ ( t ) = − t π + Im ln Γ Å
14 + i t ã = t t π − t − π O Å t ã , namely, the properties connected with the interaction of the function ζ (cid:0) + it (cid:1) with the Bessel’s functions J ν ( x ) = ∞ X r =0 ( − r r !Γ( ν + r + 1) (cid:16) x (cid:17) ν +2 r where x > , ν > − Z ( t ) = d ϕ ( t )d t , ϕ ( t ) = 12 ϕ ( t )where(1.1) ˜ Z ( t ) = Z ( t )2Φ ′ ϕ [ ϕ ( t )] = Z ( t ) (cid:8) O (cid:0) ln ln t ln t (cid:1)(cid:9) ln t , (see [1], (3.9); [3], (1.3); [7], (1.1), (3.1), (3.2)), and ϕ ( t ) is the Jacob’s ladder, i.e.a solution to the nonlinear integral equation (see [1]) Z µ [ x ( T )]0 Z ( t ) e − x ( T ) t d t = Z T Z ( t )d t. Key words and phrases.
Riemann zeta-function. an Moser Jacob’s ladder . . .1.2. The system of the Bessel’s functions { J ν ( µ ( ν ) n x ) } ∞ n =1 , x ∈ [0 , , J ν ( µ ( ν ) n ) = 0is the orthogonal system on the segment [0 ,
1] with the weight x , i.e. the followingformulae hold true Z J ν ( µ ( ν ) m x ) J ν ( µ ( ν ) n x ) x d x = 0 , m = n, Z î J ν ( µ ( ν ) n x ) ó x d x = 12 J ν +1 ( µ ( νn ) . (1.2)It is shown in this paper that the ˜ Z -transformation of the Bessel’s functions gen-erates a new orthogonal system of functions that is connected with (cid:12)(cid:12) ζ (cid:0) + it (cid:1)(cid:12)(cid:12) .Namely, the following theorem holds true. Theorem 1.
Let x = t − T, t ∈ [ T, T + 1] and ϕ { [˚ T , ˚ ˘ T + 1] } = [ T, T + 1] , T ≥ T [ ϕ ] . Then the system of functions J ν [ µ ( ν ) n ( ϕ ( t ) − T )] , t ∈ [˚ T , ˚ ˘ T + 1] , n = 1 , , . . . is orthogonal on [˚ T , ˚ ˘ T + 1] with the weight ( ϕ ( t ) − T ) ˜ Z ( t ) , i.e. the following system of the new-type integrals Z ˚ ˜ T +1˚ T J ν [ µ ( ν ) m ( ϕ ( t ) − T )] J ν [ µ ( ν ) n ( ϕ ( t ) − T )] ·· ( ϕ ( t ) − T ) ˜ Z ( t )d t = 0 , m = n, Z ˚ ˜ T +1˚ T { J ν [ µ ( ν ) n ( ϕ ( t ) − T )] } ( ϕ ( t ) − T ) ˜ Z ( t )d t = 12 [ J ν +1 ( µ ( ν ) n )] (1.3) is obtained, where ϕ ( t ) − T ∈ [0 , , and (1.4) ρ { [0 , T , ˚ ˘ T + 1] } ∼ T, T → ∞ , (where ρ stands for the distance of corresponding segments).Remark . Theorem 1 gives the contact point between the functions ζ (cid:0) + it (cid:1) , ϕ ( t )and the Bessel’s functions J ν ( x ).This paper is the continuation of the series [1]-[18]. Page 2 of 6 an Moser Jacob’s ladder . . .2.
The second result: new class of nonlinear integral equations t − ϕ ( t ) ∼ (1 − c ) π ( t ) ⇒ ˚ T ∼ T, T → ∞ where c is the Euler’s constant and π ( t ) is the prime-counting function. Then thesecond formula in (1.3) via the mean-value theorem (comp. (1.1) leads to Corollary 1. Z ϕ − ( T +1) ϕ − ( T ) { J ν [ µ ( ν ) n ( ϕ ( t ) − T )] } ( ϕ ( t ) − T ) (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) d t ∼∼
12 [ J ν +1 ( µ ( ν ) n )] ln T, T → ∞ , n = 1 , , . . . . (2.2) Remark . Let the primary oscillations (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) , t ∈ [ ϕ − ( T ) , ϕ − ( T + 1)]interact with the complicated modulated oscillations(2.3) | J ν [ µ ( ν ) n ( ϕ ( t ) − T )] | » ϕ ( t ) − T .
Then the integral (2.2) expresses the energy of the resulting oscillations. Let usnote that the oscillations (2.3) comers to the point t with the big retardation (see(2.1)) t − { ϕ ( t ) − T } = t − ϕ ( t ) + T ∼ (1 − c ) π ( t ) + T, T → ∞ . Theorem 2.
Every Jacob’s ladder ϕ ( t ) = ϕ ( t ) where ϕ ( t ) is the (exact) solutionto the nonlinear integral equation Z µ [ x ( T )]0 Z ( t ) e − x ( T ) t d t = Z T Z ( t )d t is the asymptotic solution of the new-type nonlinear integral equation Z x − ( T +1) x − ( T ) { J ν [ µ ( ν ) n ( x ( t ) − T )] } ( x ( t ) − T ) (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) d t == 12 [ J ν +1 ( µ ( ν ) n )] ln T, T → ∞ , n = 1 , , . . . . (2.4) At the same time the Jacob’s ladder ϕ ( t ) is the asymptotic solution to the followingnonlinear integral equations (comp. (2.2) with [18] , (1.5), (2.2), (3.2), (3.3), (3.5),(3.6)) Z x − ( T +2) x − ( T ) [ P α,βn ( x ( t ) − T − ( T + 2 − x ( t )) α ( x ( t ) − T ) β (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) d t == 2 α + β +1 n + α + β + 1 Γ( n + α + 1)Γ( n + β + 1) n !Γ( n + α + β + 1) ln T, n = 1 , , . . . , (2.5)(2.6) Z x − ( T +2) x − ( T ) [ P n ( x ( t ) − T − (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) d t = 22 n + 1 ln T, n = 1 , , . . . , Page 3 of 6 an Moser
Jacob’s ladder . . .(2.7) Z x − ( T +2) x − ( T ) [ T n ( x ( t ) − T − (cid:12)(cid:12) ζ (cid:0) + it (cid:1)(cid:12)(cid:12) p − ( x ( t ) − T − d t = π T, n = 1 , , . . . , (2.8) Z x − ( T +2) x − ( T ) (cid:12)(cid:12) ζ (cid:0) + it (cid:1)(cid:12)(cid:12) p − ( x ( t ) − T − d t = π ln T, Z x − ( T +2) x − ( T ) [ U n ( x ( t ) − T − » − ( x ( t ) − T − (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) d t == π T, n = 1 , , . . . , (2.9)(2.10) Z x − ( T +2) x − ( T ) » − ( x ( t ) − T − (cid:12)(cid:12)(cid:12)(cid:12) ζ Å
12 + it ã (cid:12)(cid:12)(cid:12)(cid:12) d t = π T, for every fixed T ≥ T [ ϕ ] where P α,βn ( t ) , P n ( t ) , T n ( t ) , U n ( t ) denote the polyno-mials of Jacobi, Legendre and Chebyshev of the first and second kind, respectivelly.Remark . There are the fixed-point methods and other methods of the functionalanalysis used to study the nonlinear equations. What can be obtained by usingthese methods in the case of the nonlinear integral equations (2.4)-(2.10)?3.
Proof of Theorem 1 f ( x ) , x ∈ [ ϕ ( T ) , ϕ ( T + U )]we have(3.1) Z T + UT f [ ϕ ( t )] ˜ Z ( t )d t = Z ϕ ( T + U ) ϕ ( T ) f ( x )d x, U ∈ Å , T ln T ò where t − ϕ ( t ) ∼ (1 − c ) π ( t ) ,c is the Euler’s constant and π ( t ) is the prime-counting function. In the case (comp.Theorem 1) T = ϕ (˚ T ) , T + U = ϕ ( ˚ ˙ T + U ) we obtain from (3.1)(3.2) Z ˚ ¯ T + U ˚ T f [ ϕ ( t )] ˜ Z ( t )d t = Z T + UT f ( x )d x. f ( t ) = J ν [ µ ( ν ) m ( t − T )] J ν [ µ ( ν ) n ( t − T )]( t − T ) , U = 1 Page 4 of 6 an Moser Jacob’s ladder . . .we have by (3.2) and (1.2) the following ˜ Z -transformation Z ˚ ˜ T +1˚ T J ν [ µ ( ν ) m ( ϕ ( t ) − T )] J ν [ µ ( ν ) n ( ϕ ( t ) − T )] ·· ( ϕ ( t ) − T ) ˜ Z ( t )d t == Z T +1 T J ν [ µ ( ν ) m ( t − T )] J ν [ µ ( ν ) n ( t − T )]( t − T ) ˜ Z t d t == Z J ν ( µ ( ν ) m x ) J ν ( µ ( ν ) n x ) x d x = 0 , m = n, where t = x + T , i.e. the first formula in (1.3) holds true. Similarly, we obtain thesecond formula in (1.3).3.3. Next, for ξ ∈ (˚ T , ˚ ˘ T + 1) we have (see (2.1) and [18], (4.4))(3.3) ln ξ = ln ˚ T + O Å T ã = ln T + O Å T ã . The property (3.3) was used in (2.1), . . . .I would like to thank Michal Demetrian for helping me with the electronic versionof this work.
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Jacob’s ladder . . . [14] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function Z ( t ) with the function˜ Z ( t ) on the distance ∼ (1 − c ) π ( t ) for a collection of disconneted sets‘, (2010), arXiv:1006.5158[15] J. Moser, ‘Jacob’s ladders and the ˜ Z -transformation of the orthogonal system of trigono-metric functions‘, (2010), arXiv: 1007.0108.[16] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function Z ( t ) with the function˜ Z ( t ) on the distance ∼ (1 − c ) π ( t ) for the collections of disconnected sets‘, (2010), arXiv:1007.5147.[17] J. Moser, ‘Jacob’s ladders and some new consequences from A. Sleberg’s formula‘, (2010),arXiv: 1010.0868.[18] J. Moser, ‘Jacob’s ladders and new orthogonal systems generated by Jacobi polynomials‘,(2010), arXiv: 1010.3540. Department of Mathematical Analysis and Numerical Mathematics, Comenius Uni-versity, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
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