Geodesic
The critical values of generalizations of the Hurwitz zeta function
Documenta mathematica, Tome 15 (2010), pp. 489-506
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We investigate a few types of generalizations of the Hurwitz zeta function, written Z(s,a) in this abstract, where s is a complex variable and a is a parameter in the domain that depends on the type. In the easiest case we take a∈R, and one of our main results is that Z(−m,a) is a constant times Em​(a) for 0≤m∈Z, where Em​ is the generalized Euler polynomial of degree n. In another case, a is a positive definite real symmetric matrix of size n, and Z(−m,a) for 0≤m∈Z is a polynomial function of the entries of a of degree ≤mn. We will also define Z with a totally real number field as the base field, and will show that Z(−m,a)∈Q in a typical case.
DOI : 10.4171/dm/303
Classification : 11B68, 11M06, 30B50, 33E05 
@article{10_4171_dm_303,
     author = {Goro Shimura},
     title = {The critical values of generalizations of the {Hurwitz} zeta function},
     journal = {Documenta mathematica},
     pages = {489--506},
     year = {2010},
     volume = {15},
     doi = {10.4171/dm/303},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/303/}
}
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Goro Shimura. The critical values of generalizations of the Hurwitz zeta function. Documenta mathematica, Tome 15 (2010), pp. 489-506. doi: 10.4171/dm/303

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