Relation of the Cyclotomic Equation with the Harmonic and Derived Series

We associate some (old) convergent series related to definite integrals with the cyclotomic equation x m − 1 = 0, for several natural numbers m; for example, for m = 3, x 3 − 1 = (x − 1)(1 + x + x 2) leads to ∫01dx(1/(1+x+x2))=π/(33)=(1-1/2)+(1/4-1/5)+(1/7-1/8)+⋯. In some cases, we express the results in terms of the Dirichlet characters. Generalizations for arbitrary m are well defined but do imply integrals and/or series summations rather involved.

Later, in 1668 Mercator (= Kremer) proved [1, page 185] the convergence of the alternative series (of even/odd numbers) and summed it (i.e., taking the → ∞ limit): Apparently, some work from India preceded the (even later) so-called Gregory-Leibniz formula ( [1] again, page 184) for another alternative series (of inverse of odd numbers): In this paper we interpret (4) and (5) as arising from the cyclotomic equation of roots of the unity (Gauss; see, e.g., [5]):  In our cases, = 2,4, respectively, for (4) and (5), and then perhaps we can write anew the results in terms of some natural arithmetic functions. In (6), the roots ̸ = ±1 are complex conjugate pairs, as (6) describes a real equation, so the roots have modulus 1, and all of them lie in the complex unit circle ≈ 1 .
The main purpose of this paper is to generalize these results for other (generic) natural numbers ∈ N. We will try to relate the series to some definite integrals.
We anticipate already here part of the workings for = 2 and 4. For = 2, 2 − 1 = ( − 1)( + 1); we always leave out the = 1 root. The inverse of ( + 1) enters into (4) and we proceed to the four operations: Then we obtain the above result for = 2: in terms of the arithmetic modulated "sign" function (= even/odd) ( ) := (−1) +1 / . Note that after the integration, the limit = 1 can be taken. Equation (7) involves the series expressed as a genuine definite integral. Similarly, for = 4, it is 4 − 1 = ( 2 − 1)( 2 + 1), separating the roots ±1 and ± ; now (1 + 2 ) −1 enters into (5), which becomes, again after inversion, expansion, integration, and taking limits, which can be written as the (alternative) difference between the two series and of course also as one integral And it can be given now in terms of the so-called Dirichlet characters (see below).
In this paper, as said, we will generalize the constructions above (for = 2, 4) for any integer ∈ N (in principle) and include (when possible) the appropriate Dirichlet character.
The convergence or divergence of the above series is usually self-explained: convergence occurs always for the decreasing alternative series (Leibniz), but as the convergence is not absolute, but conditional, the ordering in the series should be maintained. The higher divergence behaves, if at The Scientific World Journal 3 all, with a factor ∝ log( ): divergences are no higher than logarithmic. We will try to be careful in subtracting two divergent series. We use a philosophy close to physics (in particular, to quantum electrodynamics or q. e. d.): we first truncate the series, taking a fixed upper bound N ≫ 1 (called the cut-off ; in physics this process is called regularization). Then we subtract other series, also divergent but with a similar type of divergence (this is called renormalization): the result should be convergent (radiative corrections); see, for example, the book by Schwinger [6].
For a modern treatment of the Dirichlet series consult [7].

The Case for =2: Four Methods
Here we repeat the log(2) result for the case = 2 (7) by four different methods, because eventually the four might be useful.
Of the four methods, the most common and "easy" is the second (integration); it can, in principle (i.e., if the integrand is known and the integration feasible), always be used. The Mathematica Computer Programs give many integrals and double sums also directly.
As recapitulation, the series are obtained from the polynomial of roots ̸ =1: after INVersion, EXPansion, INTegration, and TAKing = 1.
In principle, the result of the series summation can be also obtained from the Hansen formula (22). Finally for this = 3 case, we quote the four diverging series for completeness (the signal → meaning just the limit for ≫ 1): as the complete solution for the = 3 case. We quote the following four divergent series for later use, still in this = 3 case (the limit → meaning just ≫ 1): Now, for = 4, we have first the natural cyclotomic expression (we write (I) because this is not the only possible factorization to be used). Repeating the steps as before for = 3, our first final result is here: (1, 2, 3, 4, 5, 6, 7, 8) = (1, 0, −1, 0, 1, 0, −1, 0) being periodic mod 4 (and restricted multiplicative). Note also why do we get = 1 and 3 in ∑ 4 (not 2!) mod 4: the expansion is for 1/(1 + 2 ) ≈ 1 − 2 + 4 − 6 , and so forth, so it is with even powers only, so with only odd powers after integration! We are done with this, as ∑ 4 , ∑ 0 4 , and ∑ 2 4 are automatically obtainable.
The second factorization of 4 −1 is obtained by separating only the = 1 root: As := (1 + + 2 + 3 ) contains the = −1 root, one writes = ( + 1)(1 + 2 ), where the integral can be computed at once (indeed, it is indicated above). The final result for factorization (II) is The Scientific World Journal 5 which is a redundant result, as defines ∑ 1 4 − ∑ 2 4 computable from the above calculation (I). In fact, we end up this = 4 case by writing the analogy to (35): The nonprime structure of implies only a difference calculation, as it was also the case for = 3, prime.
So the full solution for the = 4 case has one redundancy (two factorizations), and it is done (solved) once a single calculation is made; for example, ∫ As (1 + 3 ) = (1 + )(1 − + 2 ), the integral is easy, with this factoring: The operations INV, EXP, INT, and TAK = 1 applied to the polynomial 3 ( ) = 1+ 3 yield the infinite but convergent sum Notice again the jump, now by three: it is due to the cubic 3 terms in 1/(1 + 3 ).
For the next prime, namely, = 7, we have three couples of complex roots, plus the = 1 value: the sextic integral has not been attempted, but the summation can be again done; we refrain from elaborating. This is the general trend for prime numbers ; there are ( − 1)/2 pairs of complex conjugate roots; "a priori, " the only integral versus series is the simplest case, generalizing the above result: The number ( − 1)/2 coincides also with the Euler number.
As general conclusion, we have shown a remarkable relation between the cyclotomic equation − 1 = 0 and some series and definite integrals; they go from the simplest integrals (and series) in the literature (like ∫ (1/(1 + )) = log(2)) to very complicated cases, still feasible: the integrals have denominators factoring in quadratic ones, and the series are of the type ∑(1/( 2 + + )), computable, in principle, by means of the Hansen's formula.
There are, however, some questions left in our work: for example, the series for 1/(1 + + 2 + 3 +⋅ ⋅ ⋅+ ) we identify it with the series ∑ 1 +1 ; this is correct, but we have checked it "case by case, " offering no general proof, and so forth. Also we feel that some new series might perhaps appear, whenever the quadratic components offer an integer expansion: those are two questions for the future.