Проблема Бибербаха

Репутация которого запятнана связями с нацистами) уходит корнями в значительно более раннее утверждение, высказанное в 1851 г. Бернхардом Рима- ном. Теорема Римана — одно из первых утверждений, связывающих анализ и теорию функций с топологией — наукой, представляющей собой обобщение определенных аспектов геометрии. Чтобы понять теорему Римана, нужно рассмотреть на плоскости произвольную односвязную область, т.е. область, устроенную сколь угодно сложно, но не имеющую «дыр». Риман, апеллируя к физическим соображениям, доказал, что для любой такой области обязана существовать «хорошая» функция, отображающая круг единичного радиуса на эту область, и притом так, чтобы каждой точке области отвечала одна и только одна точка внутри круга (строгое доказательство его теоремы было получено в начале нашего века).

Функции, отображающие одну область плоскости в другую, удобнее всего представлять с помощью комплексных чисел. В математическом отношении с такими функциями обычно легче иметь дело, если их можно записать в виде степенного ряда — суммы бесконечного числа членов. Например, функция ez, где z — комплексное число, может быть представлена в виде бесконечного ряда 1 + г + z2/2! + г3/3! + .

Предположение Бибербаха состояло в том, что есть определенная связь между условиями на данную функцию, фигурирующими в гипотезе Римана, и числовыми коэффициентами степенного ряда, представляющего эту функцию. Бнбербах рассмотрел первоначально одну специальную функцию комплексной переменной, исследованную математиком Паулем Кёбе. Это функция г/( 1 — z)2, где z также комплексное число. При отображении, осуществляемом функцией Кёбе, образы всех точек, лежащих внутри единичного круга на плоскости, заполняют всю плоскость, за исключением лишь луча на горизонтальной оси координат, начинающегося в точке с координатой — 1/4 и уходящего влево на бесконечность. При этом внутри круга нет двух точек, которые переводились бы функцией Кёбе в одну и ту же точку!

Если разложить функцию Кёбе в степенной ряд, выполнив формальное алгебраическое деление числителя на знаменатель, то получится ряд z + 2г2 + Зг3 + . У Бибербаха возникла смелая догадка: если функция задает взаимно-однозначное соответствие между точками единичного круга и точками некоторой односвязной плоской области, то коэффициенты степенного ряда, представляющего функцию,



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