Exploration of integrals of powers of functions that have a simple pole

Authors

1. Rudolph Dwars
2. Kalpesh Muchhal

Links to papers

Currently, we have explored integrals of powers of two functions :

1. reciprocal of the log function (direct link to paper)
2. zeta function (direct link to paper)

Summary of key results

1) Discovery of new constants that can be interpreted as generalizations of the Ramanujan-Soldner constant (RS=1.451369..)

 | exponent q | root of L(q,x)=0 | root of Zi(q,x)=0 | 
 |------------|------------------|-------------------|
 | 0.51       | 1.000669 | 1.000694 |
 | 0.55       | 1.012343 | 1.012832 |
 | 0.6        | 1.039038 | 1.040602 |
 | 0.8        | 1.211222 | 1.217270 |
 | 1.0        | 1.451369 | 1.447265 |
 | 1.5        | 2.349101 | 2.052350 |
 | 2.0        | 3.846468 | 2.476428 |
 | 2.5        | 6.319568 | 2.746947 |
 | 4.5        | 46.451592| 3.579452 |
 |------------|----------|----------|
 In the L case, we also observe root_(q+1)/root_q -> exp(1) = 2.71828.. as q gets larger
 In the Zi case, we observe root_(q+1)/root_q -> 1 as q gets larger 

2) Analytical continuation and roots

We analytically continue L(q,z) and Zi(q,z) to the complex plzne z and again observe interesting properties. For eg. real(L(q,z)) shows q negative roots for q integer, apart from the one root on the positive x axis that was shown earlier, hence a total of q+1 roots.

Similarly we explore the roots of symmetric sums like M(n,z) = L(n,z) + L(n,1-z) or M(n,z) = Zi(n,z) + Zi(n,1-z), and numerically it seems all the roots of such M(n,z) are on the critical line.