I guess this is a known result but I could not find it on the Internet. Consider these equations formed from the reciprocals of the divisors of $n$ raised to a complex number $s=a+ib$ :
$\displaystyle 1$
$\displaystyle 1 + \frac{1}{2^s}=0$
$\displaystyle 1 + \frac{1}{3^s}=0$
$\displaystyle 1 + \frac{1}{2^s} + \frac{1}{4^s}=0$
$\displaystyle 1 + \frac{1}{5^s}=0$
$\displaystyle 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{6^s}=0$
$\displaystyle 1 + \frac{1}{7^s}=0$
$\displaystyle 1 + \frac{1}{2^s} + \frac{1}{4^s} + \frac{1}{8^s}=0$
$\displaystyle 1 + \frac{1}{3^s} + \frac{1}{9^s}=0$
$\displaystyle 1 + \frac{1}{2^s} + \frac{1}{5^s} + \frac{1}{10^s}=0$
$\displaystyle 1 + \frac{1}{11^s}=0$
$\displaystyle 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \frac{1}{6^s} + \frac{1}{12^s}=0$
$...$
Is it true that the zeros have real part equal to $0$ ?
Wolfram Alpha example, sums of reciprocals of divisors of 30
This Mathematica program finds some of the zeros up to the number $42$.
Clear[nn, s]; Do[ nn = i; Total[1/Divisors[nn]^(s)]; Print[s /.Table[FindRoot[Total[1/Divisors[nn]^(s)] == 0, {s, 5*n*I}], {n, 1, 30}]], {i, 2, 42}]
There are cases when the FindRoot command in Mathematica gives solutions that have real part not equal to zero, but these appear to be accompanied with warning messages and I have checked numerically that those solutions are not zeros.