A Separation in Modulus Property of the Zeros of a Partial Theta Function
We consider the partial theta function θ ( q , z ) : = ∑ j = 0 ∞ q j ( j + 1 ) / 2 z j , where z ∈ ℂ is a variable and q ∈ ℂ, 0 < | q | < 1, is a parameter. Set α 0 : = 3 / 2 π = 0.2756644477... We show that, for n ≥ 5, for | q | ≤ 1 − 1/(α 0 n ) and for k ≥ n there exists a unique zero ξ k of...
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| Published in | Analysis mathematica (Budapest) Vol. 44; no. 4; pp. 501 - 519 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.12.2018
Springer Nature B.V Springer Verlag |
| Subjects | |
| Online Access | Get full text |
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| Summary: | We consider the partial theta function
θ
(
q
,
z
)
:
=
∑
j
=
0
∞
q
j
(
j
+
1
)
/
2
z
j
, where
z
∈ ℂ is a variable and
q
∈ ℂ, 0 < |
q
| < 1, is a parameter. Set
α
0
:
=
3
/
2
π
=
0.2756644477...
We show that, for
n
≥ 5, for |
q
| ≤ 1 − 1/(α
0
n
) and for
k
≥
n
there exists a unique zero ξ
k
of
θ
(
q
,.) satisfying the inequalities |
q
|
−
k
+1/2
< |ξ
k
| < |
q
|
−k−1/2
; all these zeros are simple ones. The moduli of the remaining
n
−1 zeros are ≤ |
q
|
−n+1/2
. A
spectral value
of
q
is a value for which
θ
(
q
,.) has a multiple zero. We prove the existence of the spectral values 0.4353184958... ±
i
0.1230440086... for which
θ
has double zeros −5.963... ±
i
6.104... |
|---|---|
| ISSN: | 0133-3852 1588-273X |
| DOI: | 10.1007/s10476-018-0308-8 |