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ZEROS OF THE ESTERMANN ZETA FUNCTION
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 94 / Issue 1 / February 2013
- Published online by Cambridge University Press:
- 01 March 2013, pp. 38-49
- Print publication:
- February 2013
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- Article
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In this paper we investigate the zeros of the Estermann zeta function
$E(s; k/ \ell , \alpha )= { \mathop{\sum }\nolimits}_{n= 1}^{\infty } {\sigma }_{\alpha } (n) \exp (2\pi ink/ \ell ){n}^{- s} $ as a function of a complex variable 
$s$, where 
$k$ and 
$\ell $ are coprime integers and 
${\sigma }_{\alpha } (n)= {\mathop{\sum }\nolimits}_{d\vert n} {d}^{\alpha } $ is the generalized divisor function with a fixed complex number 
$\alpha $. In particular, we study the question on how the zeros of 
$E(s; k/ \ell , \alpha )$ depend on the parameters 
$k/ \ell $ and 
$\alpha $. It turns out that for some zeros there is a continuous dependency whereas for other zeros there is not.
