The fractional parts of the bernoulli numbers
نویسندگان
چکیده
منابع مشابه
Fractional Parts of Bernoulli Numbers
The sequence {B2}, {B4}, {B6}, ... is dense in the unit interval [0, 1], but it is not uniformly distributed [4]. Certain rational numbers appear with positive probability: 1/6 is most likely with probability 0.151..., 29/30 is next with probability 0.064... [5]. In fact, the limiting distribution F is piecewise linear with countably many jump discontinuities: F increases only when jumping (see...
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The fractional parts of the Bernoulli numbers are dense in the interval (0, 1). For every positive integer k, the set of all m for which B 2. has the same fractional part as B 21 has positive asymptotic density .
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In this paper, we calculate the values of the integrals ∫ 1 0 { 1 x}dx, ∫ ∫ 0≤x,y≤1 { 1 x+y}dxdy, ∫ ∫ ∫ 0≤x,y,z≤1 { 1 x+y+z}dxdydz and ∫ 1 0 { 1 x}{ 1 1−x}dx, where m and n are positive integers and {u} is the fractional part of u, and express their values in terms of Euler’s constant and Riemann-Zeta function. We also obtain a set of identities involving the Bernoulli and Harmonic numbers.
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The three fundamental properties of the Bernoulli numbers, namely, the theorem of von Staudt-Clausen, von Staudt’s second theorem, and Kummer’s original congruence, are generalized to new numbers that we call generalized Bernoulli-Hurwitz numbers. These are coefficients of power series expansion of a higher genus algebraic function with respect to suitable variable. Our generalization strongly ...
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ژورنال
عنوان ژورنال: Illinois Journal of Mathematics
سال: 1980
ISSN: 0019-2082
DOI: 10.1215/ijm/1256047799