Hi everyone.
Could you plz tell me where the zeros of $f(s)$ in the strip $\{0 < \Re s < 1 \}$ are ?
Do they all have $\Re s= 1/2$ ?
$$f(s) = 1 - 2^{-s} - 3^{-s} + 4^{-s} - 5^{-s} + ...$$
$$= \sum a(n)/n^s$$
with $a(n) = -1\\ $ if $n = -1 \mod 3\\ $ or $n = -1 \mod 4,\\ $ $a(n) = 1\\ $ otherwise.
$$f(s) = \zeta(s) - 2 \big[ 3^{-s} \zeta(s,-1/3) + 4^{-s} \zeta(s,-1/4) - 12^{-s} \zeta(s,-1/12) \big]$$
Could you plz give me some zeros in the strip $\{0 < \Re z < 1 \}$?
Thanks.
Eta.
The question for the case of a linear combination of Dirichlet L-series is actually easier than the case of a single L-function (Since RH is not known). In fact in each strip $1/2 \leq \sigma_1 <\Re(s)<\sigma_2 \leq 1$ there exists $\gg T$ zeroes for $-T < \Im(s) < T$. This follows by e.g. the Joint Voronin universality theorem for Dirichlet L-functions of Bagchi (A good reference for these results is Jörn Steuding's SLN 1877 "Value Distribution of L-functions").
Update Nov 14. I found the recent paper of Saias and Weingartner "Zeros of Dirichlet series with periodic coefficients", Acta Arithmetica 2009 where they get the same results that I indicated above, but also that there exists zeros to the right of the critical strip. Namely there exists some $\eta>0$ such that there are $\gg T$ zeros in any strip $1 \leq \sigma_0 <\Re(s) < \sigma_1 \leq 1+\eta$. This is actually simpler to prove since the Dirichlet series is absolutely convergent and the joint universality result is not needed, and more classical results of Bohr can be used instead.
Regarding zeros on the left of the critical line. The same result should hold in that in any vertical strip there exists $\gg T$ zeros. While this is not done in Saias-Weingartner as far as I can see it follows from the functional equation and using joint universality for the L-series in $1/2<\Re(s)<1$. Now we have two different functional equations depending on whether the L-series is odd or even it differs slightly in the Gamma-factors (this is the reason why the argument in my first answer is not applicable. See below). However Stirlings formula should imply that they do not differ sufficiently for this argument not to hold.
Further results we can get unconditionally is that there are about $T \log T$ with imaginary part less than $T$. It is not too difficult to prove that if we have a closed vertical strip that does not include the critical line, that the right order of magnitude actually is $T$, from which it would follow that for any open vertical strip including the critical line would have $T\log T$, i.e. the majority of the zeros, so the zeros should cluster around the critical line. Explicit results in this direction are included in the paper of Jörn Steuding "On Dirichlet series with periodic coefficients", Ramanujan Journal 2002 where he proves these results, i.e. clustering around $\Re(s)=1/2$, as well as other estimates (Another related paper is Garunkstis-Steuding "On the zero distribution of the Lerch zeta-function" where they prove corresponding results for the Lerch zeta function).
However to prove that they lie exactly on the critical line I believe that they must satisfy the same functional equation (see below) so an analogue of the Hardy function can be found and worked with. Therefore it is not clear (I am not sure about this though) that there should be any zeros exactly on the critical line (at least in order for there to be zeros on any particular line there should be a reason for it, since the zeros are countable, but the reals in an intervals are uncountable). Numerical experiments are welcome (I am not doing them though.).
Edit after comment of John below: I had originally thought that Bombieri and Hejhal's, and Hejhal's and Selberg's later results on linear combinations of L-functions would have applied on this problem, but as John pointed out below, this should not be the case, since the L-functions have to have the same Functional equation. Selberg's latest (unpublished) result would have yielded a positive proportion (order of $T \log T$ of zeros on the critical line), and Bombieri-Hejhal's (conditional on RH and weak Montgomery pair correlation conjecture) would have yielded the true asymptotics, if this would have been the case.
I checked one of Hejhal's papers on this subject, and John is right in his comment below. The condition to apply this method is that the Gamma-factors in the functional equation and the modulus are the same. When we consider a linear combination of Dirichlet L-functions we have to have a combination of only odd or even Dirichlet characters and the same modulus. For the Hurwitz zeta-function of rational parameters all Dirichlet characters will appear and thus this example is not of this type.
Thus I do not have an answer regarding zeros on the critical line. However the argument that shows that we have at least the order of $T$ zeros in any vertical strip $1/2 \leq \sigma_1<\Re(s)<\sigma_2 \leq 1+\eta$ with imaginary part less than $T$ still holds. Thus at least we know that Riemann hypothesis is not true for this function.
Johan