Pattern in Dirichlet regularizations of divergent series.

Question

So, I was trying to find a general pattern/formula for the Dirichlet Regularization of a Divergent series. I hadn't faced any problems until I reached:

$$\sum_{n=0}^\infty n^x$$

So, let's make a function $\operatorname{Di}$, which basically gives the Dirichlet regularization for any summation put in (if valid). The value $x$ here is nothing but a constant, and plays a similar role as $a$ in $ax^2 + bx + c = 0$.

Now, if I try to do this calculation for:

$$\operatorname{Di}\left(\sum_{n=0}^\infty n^x\right)$$

Here's where I arrive at the problem. For even values of $x$, the pattern seems to be simple, it is just $\frac{-1}{n+1}$, the problem arrives for odd values of $x$.

For even values, the final regularized result seems to be:

$$\frac{-1}{3}, \frac{-1}{5}, \frac{-1}{7}, \ldots, \frac{-1}{x+1}$$

However, there does not seem to be some set pattern for odd values of $x$.

The values seem to be chaotic, and below is a list of results for all odd values $1$ to $20$:

x  |  Regularized Sum
---|----------------------
1  |  5/12
3  |  31/120
5  |  41/252
7  |  31/240
9  |  61/660
11 |  3421/32760
13 |  -1/84
15 |  4127/8160
17 |  43069/14364
19 |  174941/6600

I would like to know the pattern, a general formula based on the value of $x$ for this, and the reason why it isn't negative unlike the values for when $x$ is even, or when $x$ is $13$.

Answer 1

Take a look at this post of the famous mathematician Terry Tao : https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

You will find that the denominators of the fractions you find agree with the formula

$$\frac{B_{s+1}}{s+1}$$

of this document (where the $B_n$ are the Bernoulli numbers) but there is no such agreement with the numerators.

Edit:

In fact, with your OEIS finding, the right formula :

$$\frac{1-B_{n+1}}{n+1}$$

is the good one ; we can check it on the first results (in red the successive Bernoulli numbers).

$$\begin{array}{|r|r|r|} \hline 1&\tfrac{5}{12}&\left(1-\left(\color{red}{+\tfrac{1}{6}}\right)\right)\tfrac{1}{2} \\ 3&\tfrac{31}{120}&\left(1-\left(\color{red}{-\tfrac{1}{30}}\right)\right)\tfrac{1}{4}\\ 5&\tfrac{41}{252}&\left(1-\left(\color{red}{+\tfrac{1}{42}}\right)\right)\tfrac{1}{6}\\ 7&\frac{31}{240}&\left(1-\left(\color{red}{-\tfrac{1}{30}}\right)\right)\tfrac{1}{8}\\ 9&\tfrac{61}{660}&\left(1-\left(+\tfrac{5}{66}\right)\right)\tfrac{1}{10}\\ 11&\tfrac{3421}{32760}&\left(1-\left(-\tfrac{691}{2730}\right)\right)\tfrac{1}{12}\\ 13&-\tfrac{1}{84}&\left(1-\left(+\tfrac{7}{6}\right)\right)\tfrac{1}{14}\\ 15&\tfrac{4127}{8160}&\left(1-\left(-\tfrac{3617}{510}\right)\right)\tfrac{1}{16}\\ \hline \end{array}$$