I found the solutions of the equations D_1(z)=0, D_3(z)=0, D_5(z)=0, D_7(z)=0 and D_9(z)=0 which are x=2logφ=0.9624, x=2.03185, x=2.89218, x=3.68896, x=4.46019 respectively. D_odd(z) functions are the well known Bloch Wigner Ramakrishnan functions introduced by Zagier and z=exp(-x). Do you know if these solutions are related to each other?
I have found that with accuracy over 99%:
1)x=0.9624=log(φ^2) for the solution of D_1(exp(-x))=0
2)x=2.03185=log(3*φ^2) for the solution of D_3(exp(-x))=0
3)x=2.89218=log(7*φ^2) for the solution of D_5(exp(-x))=0
4)x=3.68896=log(15*φ^2) for the solution of D_7(exp(-x))=0
5)x=4.46019=log(33*φ^2) for the solution of D_9(exp(-x))=0
......
n)x=log(a_n*φ^2) for the solution of D_d(z)=0, where D_d(z) is the odd indexed Bloch Wigner Ramakrishnan polylogarithm function and a_n=1,3,7,15,33,71,151,... an odd number which obeys a sequence I have already submitted to OEIS and has the growth rate of Fibonacci numbers.
See also my sequence A363503 - OEIS and [2306.14652] Strongly coupled fermions in odd dimensions and the running cut-off $Λ_d$ (arxiv.org)
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