Let Zn(s;a1,,an) be the Epstein zeta function defined as the meromorphic continuation of the function kZn\{0}(i=1n[aiki]2)s,Res>n/2 to the complex plane. We show that for fixed sn/2, the function Zn(s;a1,,an) as a function of (a1,,an)(R+)n with fixed i=1nai has a unique minimum at the point a1==an. When i=1nci is fixed, the function (c1,,cn)Zn(s;ec1,,ecn) can be shown to be a convex function of any (n1) of the variables {c1,,cn}. These results are then applied to the study of the sign of Zn(s;a1,,an) when s is in the critical range (0,n/2). It is shown that when 1n9, Zn(s;a1,,an) as a function of (a1,,an)(R+)n can be both positive and negative for every s(0,n/2). When n10, there are some open subsets In,+ of s(0,n/2), where Zn(s;a1,,an) is positive for all (a1,,an)(R+)n. By regarding Zn(s;a1,,an) as a function of s, we find that when n10, the generalized Riemann hypothesis is false for all (a1,,an).

1.
P.
Epstein
,
Math. Ann.
56
,
615
(
1903
).
2.
P.
Epstein
,
Math. Ann.
65
,
205
(
1907
).
3.
C. L.
Siegel
,
Ann. Math.
44
,
143
(
1943
).
4.
M.
Koecher
,
J. Reine Angew. Math.
192
,
1
(
1953
).
5.
C. L.
Siegel
,
J. Indian Math. Soc., New Ser.
20
,
1
(
1956
).
6.
A.
Selberg
,
Report of the Institute in Theory of Numbers
(
University of Colorado
,
Boulder, CO
,
1959
), pp.
207
210
.
7.
A.
Terras
,
Nagoya Math. J.
42
,
173
(
1971
).
8.
S.
Chowla
,
Proc. Nat. Inst. Sci. India
13, No. 4
1
(
1947
).
9.
E.
Elizalde
,
S. D.
Odintsov
,
A.
Romeo
,
A. A.
Bytsenko
, and
S.
Zerbini
,
Zeta Regularization Techniques with Applications
(
World Scientific
,
River Edge, NJ
,
1994
).
10.
E.
Elizalde
,
Ten Physical Applications of Spectral Zeta Functions
(
Springer-Verlag
,
Berlin
,
1995
).
11.
K.
Kirsten
,
Spectral Functions in Mathematics and Physics
(
Chapman & Hall
,
London
/
CRC
,
Boca Raton, FL
,
2002
).
12.
J.
Ambjørn
and
S.
Wolfram
,
Ann. Phys.
147
,
1
(
1983
).
13.
14.
G.
Denardo
and
E.
Spallucci
,
Nucl. Phys. B
169
,
514
(
1980
).
15.
A.
Actor
,
Class. Quantum Grav.
7
,
663
(
1990
).
17.
E.
Elizalde
and
K.
Kirsten
,
J. Math. Phys.
35
,
1260
(
1994
).
18.
S. C.
Lim
and
L. P.
Teo
,
J. Phys. A: Math. Theor.
41
,
145403
(
2008
).
19.
R. A.
Rankin
,
Proc. Glasgow Math. Assoc.
1,
149
(
1953
).
20.
J. W. S.
Cassels
,
Proc. Glasg. Math. Assoc.
4,
73
(
1959
);
J. W. S.
Cassels
,
Proc. Glasg. Math. Assoc.
6,
116
(
1963
).
21.
V.
Ennola
,
Proc. Cambridge Philos. Soc.
60
,
855
(
1964
).
22.
P. H.
Diananda
,
Proc. Glasg. Math. Assoc.
6,
202
(
1964
).
23.
S. S.
Ryškov
,
Sibirsk. Mat. Z.
14,
1065
(
1973
).
24.
S. Š.
Šušbaev
,
Dokl. Akad. Nauk UzSSR
9
,
15
(
1976
).
25.
S. Š.
Šušbaev
,
Voprosy Vyčisl. i Prikl. Mat. (Tashkent)
No. 46,
3
(
1977
).
26.
S. Š.
Šušbaev
,
Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk
No. 1,
33
(
1978
).
27.
K. M.
Èndibaev
,
Voprosy Vychisl. i Prikl. Mat. (Tashkent)
No. 51,
178
(
1978
).
28.
A.
Terras
,
J. Number Theory
12
,
258
(
1980
).
29.
K. M.
Èndibaev
,
Dokl. Akad. Nauk UzSSR
4
,
18
(
1978
).
30.
K. L.
Fields
,
Mathematika
27
,
17
(
1980
).
31.
Z. D.
Lomakina
and
S. S.
Ryshkov
,
Zap. Nauchn. Semin. LOMI
151
,
95
(
1986
).
32.
S. Sh.
Shushbaev
,
Voprosy Vychisl. i Prikl. Mat. (Tashkent)
No. 82,
108
(
1987
).
33.
S. Sh.
Shushbaev
,
Mat. Zametki
45, No. 1,
123
(
1989
).
34.
S. Sh.
Shushbaev
,
Sibirsk. Mat. Zh.
31, No. 5,
157
(
1990
).
35.
S. Sh.
Shushbaev
,
Izv. Vyssh. Uchebn. Zaved., Mat.
5
,
78
(
1990
).
36.
S. Sh.
Shushbaev
,
Sibirsk. Mat. Zh.
35, No. 6,
1397
(
1994
).
37.
E. V.
Orlovskaya
,
Vestnik S.-Peterburg. Univ. Mat. Mekh. Astronom.
149, No. 3,
23
(
1994
).
38.
E. V.
Orlovskaya
,
Zap. Nauchn. Semin. POMI
211
,
150
(
1994
).
39.
W.
Jenkner
,
J. Thero. Nombres Bordeaux
7, No. 1,
1
(
1995
).
40.
S. Sh.
Shushbaev
and
G. A.
Kalybaeva
,
Uzbek. Mat. Zh.
No. 2,
80
(
2002
).
41.
42.
P.
Sarnak
and
A.
Strömbergsson
,
Invent. Math.
165
,
115
(
2006
).
43.
S.
Chowla
and
A.
Selberg
,
Proc. Natl. Acad. Sci. U.S.A.
35
,
371
(
1949
).
44.
A.
Selberg
and
S.
Chowla
,
J. Reine Angew. Math.
227,
86
(
1967
).
45.
I. S.
Gradshteyn
and
I. M.
Ryzhik
,
Table of Integrals, Series, and Products
(
Academic
,
San Diego, CA
,
2000
).
46.
A.
Fujii
,
Math. Univ. St. Pauli
49, No. 2,
195
(
2000
).
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