How do you solve an equation consisting of a logarithmically chopped subsets of the truncated geometric series, equal to zero?

Question

The Fourier transform of the Riemann zeta function looks like this:

The polynomial equation I am interested in solving has the following exponents and coefficients:

that is, it is a subset of the geometric series with gaps in it.

Superimposing the polynomial coefficients on the plot of t Fourier transform of Riemann zeta, one gets a match:

The starting point is that Mathematica can transform this equation:

 Reduce[1/(E^log1)^x + 1/(E^log2)^x + 1/(E^log3)^x == 0, x]

into logarithms of polynomial roots when the variables log1, log2, log3 are rational numbers. One finds this same starting point some number theory books. Reduce is another version of Solve.

With the help of Jeffrey Stopple's answer here I found the polynomial above here.

The polynomial whose coefficients fit into the plot of Fourier transform above has the following definition:

$$P(x)=\sum _{q=1}^{\frac{m}{2}} \left(\sum _{k=\frac{\text{Round}[h \log (m-(2 q-1))]}{\gcd h}}^{\frac{\text{Round}[h \log (m-(2 q-2))]}{\gcd h}-1} x^k\right)$$

where $m$ is some even number greater than $4$.

An abbreviated form of this particular polynomial is:

$$P(x)=1+x+\text{...}+x^{277}+x^{440}+\text{...}+x^{553}+x^{644}+\text{...}+x^{715}+\text{...}+x^{831}+x^{878}+\text{...}+x^{921}=0$$ where dot dot dot means that all the exponents inbetween the numbers are included.

So it is geometric series with holes in it.

Question:

Is there a way of finding solution to equations that have the truncated geometric series with gaps in it?

In case I got it all wrong, here is the associated Mathematica program that is meant to describe what I am looking for. gcd is the greatest common divisor of the rational approximations of the logarithms.

(*start*)(*Associated Mathematica program*)
Clear[h, x, k, s, s1, nn, m];
$MaxRootDegree = 1000;
m = 10;(*m must be an even integer and greater than 4*)h = 400;
gcd = GCD @@ Table[Round[Log[n]*h]/h, {n, 1, m}]
Table[Round[Log[n]*h]/h, {n, 1, m}]/gcd
r = 100;
integer = 0;
s = 1/gcd*(2*Pi*I*integer - 
    Log[Root[
      Sum[(-1)^(n + 1) #1^(Round[Log[n]*h]/h/gcd), {n, 1, m}] &, r]])
N[s, 80]
N[Sum[(-1)^(n + 1)/(E^(Round[Log[n]*h]/h))^s, {n, 1, m}]]
s1 = 1/gcd*(2*Pi*I*integer - 
    Log[Root[
      polynomial = 
       Sum[Sum[#1^k, {k, Round[Log[m - (2*q - 1)]*h]/h/gcd, 
           Round[Log[m - (2*q - 2)]*h]/h/gcd - 1}], {q, 1, m/2}] &, 
      r - 1]])
N[s1, 80]
N[Sum[(-1)^(n + 1)/(E^(Round[Log[n]*h]/h))^s1, {n, 1, m}]]
(*end*)

(*start*)
m = 20;
h = 200;
sort = Sort[
  Flatten[Table[
    Table[k, {k, Round[Log[m - (2*q - 1)]*h]/h/gcd, 
      Round[Log[m - (2*q - 2)]*h]/h/gcd - 1}], {q, 1, m/2}]]]
"Plot of exponents of polynomial"
ListLinePlot[sort]
"Plot of coefficients of polynomial"
ListPlot[Sum[
  Table[If[sort[[n]] == k, 1, 0], {k, 1, Max[sort]}], {n, 1, 
   Length[sort]}], Filling -> 0]
(*end*)