کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
6415524 1630667 2015 11 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Does the Riemann zeta function satisfy a differential equation?
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
Does the Riemann zeta function satisfy a differential equation?
چکیده انگلیسی

In Hilbert's 1900 address at the International Congress of Mathematicians, it was stated that the Riemann zeta function is the solution of no algebraic ordinary differential equation on its region of analyticity. It is natural, then, to inquire as to whether ζ(z) satisfies any non-algebraic differential equation. In the present paper, an elementary proof that ζ(z) formally satisfies an infinite order linear differential equation with analytic coefficients, T[ζ−1]=1/(z−1), is given. We also show that this infinite order differential operator T may be inverted, and through inversion of T we obtain a series representation for ζ(z) which coincides exactly with the Euler-MacLauren summation formula for ζ(z). Relations to certain known results and specific values of ζ(z) are discussed.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Number Theory - Volume 147, February 2015, Pages 778-788
نویسندگان
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