Geodesic
Zeros of entire functions and related systems of infinitely many nonlinearly coupled evolution equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 193-213 Cet article a éte moissonné depuis la source Math-Net.Ru

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Recent findings concerning the zeros of generic polynomials are extended to entire functions featuring infinitely many distinct zeros, and related systems of infinitely many nonlinearly coupled evolution ODEs and PDEs are identified, the solutions of which display interesting properties.
Keywords: zero of an entire function, system of infinitely many evolution ODEs, system of infinitely many evolution PDEs, Riemann zeta function, Riemann hypothesis. 
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F. Calogero. Zeros of entire functions and related systems of infinitely many nonlinearly coupled evolution equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 193-213. http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a1/

[1] F. Calogero, “New solvable variants of the goldfish many-body problem”, Stud. Appl. Math., 137:1 (2016), 123–139 | DOI | MR | Zbl

[2] O. Bihun, F. Calogero, “A new solvable many-body problem of goldfish type”, J. Nonlinear Math. Phys., 23:1 (2016), 28–46 ; “Novel solvable many-body problems”, 23:2 (2016), 190–212 ; “Generations of monic polynomials such that the coefficients of each polynomials of the next generation coincide with the zeros of polynomial of the current generation, and new solvable many-body problems”, Lett. Math. Phys., 106:7 (2016), 1011–1031 ; “Generations of solvable discrete-time dynamical systems”, J. Math. Phys., 58:5 (2017), 052701, 21 pp. ; F. Calogero, “A solvable $N$-body problem of goldfish type featuring $N^2$ arbitrary coupling constants”, J. Nonlinear Math. Phys., 23:2 (2016), 300–305 ; “Three new classes of solvable $N$-body problems of goldfish type with many arbitrary coupling constants”, Symmetry, 8:7 (2016), 53, 16 pp. ; “Novel isochronous $N$-body problems featuring $N$ arbitrary rational coupling constants”, J. Math. Phys., 57:7 (2016), 072901, 11 pp. ; “Yet another class of new solvable $N$-body problems of goldfish type”, Qual. Theory Dyn. Syst., 16:3 (2017), 561–577 ; “New solvable dynamical systems”, J. Nonlinear Math. Phys., 23:4 (2016), 486–493 ; “Integrable Hamiltonian $N$-body problems in the plane featuring $N$ arbitrary functions”, 24:1 (2017), 1–6 ; “New C-integrable and S-integrable systems of nonlinear partial differential equations”, 142–148 ; M. Bruschi, F. Calogero, “A convenient expression of the time-derivative $z_n^{(k)}(t)$, of arbitrary order $k$, of the zero $z_n(t)$ of a time-dependent polynomial $p_N(z;t)$ of arbitrary degree $N$ in $z$, and solvable dynamical systems”, J. Nonlinear Math. Phys., 23:4 (2016), 474–485 | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | DOI | MR | DOI | MR | DOI | MR | DOI | MR

[3] F. Calogero, Zeros of Polynomials and Solvable Nonlinear Evolution Equations, Cambridge Univ. Press, Cambridge, 2018, in press

[4] F. Calogero, “Nonlinear differential algorithm to compute all the zeros of a generic polynomial”, J. Math. Phys., 57:8 (2016), 083508, 4 pp., arXiv: 1607.05081 | DOI | MR | Zbl

[5] F. Calogero, “Novel differential algorithm to evaluate all the zeros of any generic polynomial”, J. Nonlinear Math. Phys., 24:4 (2017), 469–472 | DOI | MR

[6] G. Devenport, Multiplikativnaya teoriya chisel, Nauka, M., 1971 | MR | MR | Zbl

[7] F. Calogero, “The neatest many-body problem amenable to exact treatments (a “goldfish”?)”, Phys. D, 152–153 (2001), 78–84 | DOI | MR | Zbl

[8] F. Calogero, “Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations, and related “solvable” many body problems”, Nuovo Cimento B, 43:2 (1978), 177–241 ; Classical Many-Body Problems Amenable to Exact Treatments, Lecture Notes in Physics. New Series m: Monographs, 66, Springer, Berlin, 2001 | DOI | MR | DOI | MR

[9] F. Calogero, Isochronous Systems, Oxford Univ. Press, Oxford, 2012 | MR

[10] F. Calogero, D. Gómez-Ullate, “Asymptotically isochronous systems”, J. Nonlinear Math. Phys., 15:4 (2008), 410–426 | DOI | MR | Zbl

[11] D. Gómez-Ullate, M. Sommacal, “Periods of the goldfish many-body problem”, J. Nonlinear Math. Phys., 12, supp. 1 (2005), 351–362 | DOI | MR

[12] E. Martínez Alonso, A. B. Shabat, “Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type”, Phys. Lett. A, 300:1 (2002), 58–64 ; В. Э. Адлер, А. Б. Шабат, “Модельное уравнение теории солитонов”, ТМФ, 153:1 (2007), 29–45 ; А. Б. Шабат, “Симметрические многочлены и законы сохранения”, Владикавк. матем. журн., 14:4 (2012), 83–94 | DOI | MR | Zbl | DOI | DOI | MR | Zbl | Zbl

[13] G. Gallavotti, C. Marchioro, “On the calculation of an integral”, J. Math. Anal. Appl., 44:3 (1973), 661–675 | DOI | MR | Zbl

[14] E. Bombieri, Problems of the Millennium: The Riemann Hypothesis, Clay Mathematics Institute, Cambridge, MA, 2000

[15] H. M. Edwards, Riemann's Zeta Function, Pure and Applied Mathematics, 58, Acad. Press, New York, 1974 | MR