The function was introduced in Hurwitz (1882) and defined by the series expansion
25.11.1 | |||
, . | |||
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has a meromorphic continuation in the -plane, its only singularity in being a simple pole at with residue . As a function of , with () fixed, is analytic in the half-plane . The Riemann zeta function is a special case:
25.11.2 | |||
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For most purposes it suffices to restrict because of the following straightforward consequences of (25.11.1):
25.11.3 | |||
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25.11.4 | |||
. | |||
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Most references treat real with .
25.11.5 | |||
, , , . | |||
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25.11.6 | |||
, , . | |||
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25.11.7 | |||
, , , . | |||
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For see §24.2(iii).
25.11.8 | |||
, , . | |||
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25.11.9 | |||
if ; if . | |||
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25.11.10 | |||
, . | |||
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Throughout this subsection .
25.11.11 | |||
. | |||
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25.11.12 | |||
. | |||
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25.11.13 | |||
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25.11.14 | |||
. | |||
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25.11.15 | |||
, . | |||
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25.11.16 | |||
; integers, . | |||
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25.11.17 | |||
; . | |||
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In (25.11.18)–(25.11.24) primes on denote derivatives with respect to . Similarly in §§25.11(viii) and 25.11(xii).
25.11.18 | |||
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25.11.19 | |||
, , . | |||
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25.11.20 | |||
, , . | |||
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25.11.21 | |||
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where are integers with and .
25.11.22 | |||
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25.11.23 | |||
. | |||
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25.11.24 | |||
, . | |||
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25.11.25 | ||||
, . | ||||
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25.11.26 | ||||
, . | ||||
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25.11.27 | |||
, , . | |||
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25.11.28 | |||
, , . | |||
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25.11.29 | |||
, . | |||
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25.11.30 | |||
, , | |||
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where the integration contour (see Figure 5.9.1) is a loop around the negative real axis as described for (25.5.20).
Suggested 2021-08-23 by Gergő Nemes
25.11.31 | |||
, . | |||
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25.11.32 | |||
, , | |||
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where are the harmonic numbers:
25.11.33 | |||
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25.11.34 | |||
, . | |||
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25.11.35 | |||
, ; or , , . | |||
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When , (25.11.35) reduces to (25.2.3).
25.11.36 | Removed because it is just (25.15.1) combined with (25.15.3). | ||
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25.11.37 | |||
, . | |||
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25.11.38 | |||
, , . | |||
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25.11.39 | |||
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where is Catalan’s constant:
25.11.40 | |||
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As with fixed,
25.11.41 | |||
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As with fixed, ,
25.11.42 | |||
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uniformly with respect to bounded nonnegative values of .
As in the sector , with and fixed, we have the asymptotic expansion
25.11.43 | |||
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Similarly, as in the sector .
25.11.44 | |||
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and
25.11.45 | |||
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For the more general case , , see Elizalde (1986). For error bounds for (25.11.43), (25.11.44) and (25.11.45), see Nemes (2017a).