## Selected of Norman LevinsonThe deep and original ideas of Norman Levinson have had a lasting impact on fields as diverse as differential & integral equations, harmonic, complex & stochas tic analysis, and analytic number theory during more than half a century. Yet, the extent of his contributions has not always been fully recognized in the mathematics community. For example, the horseshoe mapping constructed by Stephen Smale in 1960 played a central role in the development of the modern theory of dynami cal systems and chaos. The horseshoe map was directly stimulated by Levinson's research on forced periodic oscillations of the Van der Pol oscillator, and specifi cally by his seminal work initiated by Cartwright and Littlewood. In other topics, Levinson provided the foundation for a rigorous theory of singularly perturbed dif ferential equations. He also made fundamental contributions to inverse scattering theory by showing the connection between scattering data and spectral data, thus relating the famous Gel'fand-Levitan method to the inverse scattering problem for the Schrodinger equation. He was the first to analyze and make explicit use of wave functions, now widely known as the Jost functions. Near the end of his life, Levinson returned to research in analytic number theory and made profound progress on the resolution of the Riemann Hypothesis. Levinson's papers are typically tightly crafted and masterpieces of brevity and clarity. It is our hope that the publication of these selected papers will bring his mathematical ideas to the attention of the larger mathematical community." |

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### Contents

I | 3 |

II | 32 |

III | 38 |

IV | 53 |

V | 61 |

VI | 66 |

VII | 84 |

VIII | 90 |

XL | 256 |

XLI | 258 |

XLII | 265 |

XLIII | 267 |

XLIV | 268 |

XLV | 289 |

XLVI | 305 |

XLVII | 306 |

IX | 108 |

X | 126 |

XI | 136 |

XIII | 145 |

XIV | 150 |

XV | 153 |

XVI | 160 |

XVII | 163 |

XVIII | 181 |

XIX | 191 |

XX | 203 |

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XXIII | 223 |

XXIV | 224 |

XXV | 225 |

XXVI | 226 |

XXVII | 228 |

XXVIII | 234 |

XXIX | 236 |

XXX | 237 |

XXXI | 238 |

XXXII | 241 |

XXXIII | 243 |

XXXIV | 244 |

XXXV | 246 |

XXXVI | 247 |

XXXVII | 248 |

XXXVIII | 251 |

XXXIX | 253 |

XLVIII | 312 |

XLIX | 315 |

L | 323 |

LI | 327 |

LII | 330 |

LIII | 384 |

LIV | 388 |

LV | 389 |

LVI | 392 |

LVII | 396 |

LVIII | 414 |

LIX | 418 |

LX | 423 |

LXI | 426 |

LXII | 441 |

LXIII | 448 |

LXIV | 451 |

LXV | 454 |

LXVI | 463 |

LXVII | 476 |

LXVIII | 477 |

LXIX | 485 |

LXX | 487 |

LXXI | 493 |

LXXII | 517 |

LXXIII | 536 |

LXXIV | 547 |

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### Common terms and phrases

Acad Amer Anal analytic function apply approximate functional equation assume Asymptotic bang-bang bang-bang control bounded Carleman completes the proof condition constant continuous converges critical line defined denote Differential Equations Duke Math entire function equivalent to zero exists exponential type finite number formula Fourier series Fourier transform functional equation fur Math given gives half-plane Hence hypothesis implies inequality integral interval Lemma linear log log Massachusetts Institute Mathematical mollifier Moreover Norman Levinson number of zeros optimal control piecewise analytic Polya polynomial positive prime number theorem problem Proc Proof of Theorem proving Theorem replaced Reprinted result Riemann Hypothesis Riemann Zeta Function Riemann zeta-function right side satisfies Selberg sequence shows Singular solution Stirling's formula subanalytic Tauberian theorem theory uniformly values vanishes vector Wiener Zeta Function