Ramanujan summation of divergent series
Sometimes it is desirable to associate values to divergent series, i.e., ``sum'' them symbolically by following certain rules. For example, a divergent series can be ``evaluated'' by using Abel summation, Borel summation, Cesàro summation, or Cauchy summation. The present book discusses the theory of Ramanujan summation, which we will denote here by \((R)\sum _{n\geq 1}f(n)\). The theory comes from the works of Srinivasa Ramanujan. The summation technique is based on the Euler-Maclaurin summation formula and uses the partial sums of a given series. In some cases, one has the formula \((R)\sum _{n\geq 1}f(n) =\sum _{n=1}^{\infty }\left(f(n)-\int _{n}^{n+1}f(x)\,dx \right)\). It is also shown that, for functions of moderate growth, \((R)\sum _{n\geq 1}f(n) =\mathop{\lim }\limits_{z\to 0+} \left(\sum _{n=1}^{\infty }f(n)e^{-nz} -\int _{1}^{\infty }f(x)e^{-zx}\, dx \right)\). Here are several examples: \((R)\sum _{n\geq 1}\frac{1}{n} =\gamma \), \((R)\sum _{n\geq 1}\frac{1}{n^{s} } =\zeta (s)-\frac{1}{s-1} \quad (\operatorname{Re}s>1)\), \((R)\sum _{n\geq 1}(n-1)^{k} =\zeta (-k)\). (\(\zeta (s)\) is Riemann's zeta function and \(\gamma \) is Euler's constant.) When \(k=1\), the last example reduces to the popular paradoxical evaluation \((R)(1+2+3+\dots)=-\frac{1}{12} \) . The book contains a number of interesting formulas and makes several important connections to classical analysis.
scientific article; zbMATH DE number 6755646