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- W2024584585 abstract "Abstract. The paper deals with the values at the negative integers ofa certain Dirichlet series related to the Riemann zeta function and withthe expression of these values in terms of Bernoulli numbers. 1. IntroductionWe consider the function(1) h ( s ) =X 1n =1 H n n s ; where(2) H n = 1+12+13+ ¢¢¢ +1 n; are the Harmonic numbers and Re s > 1. The function h ( s ) was studied bymany authors, starting with Euler, who evaluated this series in a closed formwhen s = k is a positive integer. An elementary derivation of Euler’s formulacan be found, for instance, in [3]. For general s this function was investigatedby Apostol-Vu [1] and Matsuoka [5], who provided an analytic extension to allcomplex numbers and discussed its values and poles at the negative integers.In this note we shall flnd a relation between the values h (1 i n ) and thenumbers A n , n = 1 ; 2 ;:::; deflned as the convolution(3) A n =X k + j = n B ( k ) k ! kB ( j ) j ! ; k = 1 ; 2 ;::: ; j = 0 ; 1 ;::: ; where B ( n ) = B" @default.
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- W2024584585 date "2008-05-31" @default.
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- W2024584585 title "THE VALUES OF AN EULER SUM AT THE NEGATIVE INTEGERS AND A RELATION TO A CERTAIN CONVOLUTION OF BERNOULLI NUMBERS" @default.
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- W2024584585 doi "https://doi.org/10.4134/bkms.2008.45.2.277" @default.
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