The starting definition that I am using is: ζ ( s ) = 1 s − 1 + ∑ n = 0 ∞ γ n ⋅ ( − 1 ) n n ! ( s − 1 )

Question

The starting definition that I am using is:  ζ  (  s  )  =      1          s      −      1        +      ∑          n      =      0        ∞        γ    n    ⋅            (      −      1              )        n                    n      !        (  s  −  1  )

ryancameron52 · Accepted Answer

If as usual   ψ  (  x  )  =  x  −  [  x  ]  −  1      /    2 a standard formula (see below after the answer) gives:  ζ  (  s  )  =      1          s      −      1        −      ∫                  1        −                    ∞            u          −      s        d  ψ  (  u  )  ,  ℜ  s  &amp;gt;  0so with your notation      γ    k    =  (  −  1      )    k              d      k              d              s        k              (  ζ  (  s  )  −      1          s      −      1        )            |              s      =      1        =  −      ∫                  1        −                    ∞            u          −      1        (  log  ⁡  u      )    k    d  ψ  (  u  )But      ∫                  1        −                    ∞            u          −      1        (  log  ⁡  u      )    k    d  ψ  (  u  )  =      lim          N      →      ∞            ∫                  1        −                    N            u          −      1        (  log  ⁡  u      )    m    (  d  u  −  d  [  u  ]  )  =      lim          N      →      ∞        (            (      log      ⁡      u              )                  k          +          1                            k      +      1                  |        1    N    −      ∑          m      =      1        N              (      log      ⁡      m              )        k              k    )so indeed we get the required formula:      γ    k    =      lim          N      →      ∞        (  −            (      log      ⁡      N              )                  k          +          1                            k      +      1        +      ∑          m      =      1        N              (      log      ⁡      m              )        k              k    )

The starting definition that I am using is: ζ ( s ) = 1

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