Convolution of Riemann zeta-values
Copyright policy )Convolution of Riemann zeta-values
Let \(\zeta_N(s)=\sum_{n\leq N}1/n^s\), the \(N\)-th partial sum of the Riemann zeta-function \(\zeta(s)\), let \(H_N=\zeta_N(1)\), and let \[ \zeta_2(s_1,s_2)=\sum_{m0\), the authors establish the asymptotic formula \[ m\zeta_N(m+1)-2\sum_{h\leq N}{H_{h-1}\over h^m}+o(1)=\sum_{j=1}^{m-2}\zeta_N(j+1)\zeta_N(m-j), \qquad N\to\infty, \] and thus \[ 2\zeta_2(1,m)=m\zeta(m+1)-\sum_{j=1}^{m-2}\zeta(j+1)\zeta(m-j). \] The identity here is equivalent to \(\zeta(k)=\sum_{2\leq j
- Nagoya University Japan
- Kindai University Japan
11M06, Riemann zeta-values, \(\zeta (s)\) and \(L(s, \chi)\), Multiple Dirichlet series and zeta functions and multizeta values, Euler-Zagier sum, 11M41, Other Dirichlet series and zeta functions, Mellin transform
11M06, Riemann zeta-values, \(\zeta (s)\) and \(L(s, \chi)\), Multiple Dirichlet series and zeta functions and multizeta values, Euler-Zagier sum, 11M41, Other Dirichlet series and zeta functions, Mellin transform
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