Analytic number theory, sum of powers of integers

ANALYTIC NUMBER THEORY

Sum of powers of integers

Palmioli Luca

The purpose of this study was to rewrite the formulas for

the sum of powers of integers in a subsequent general

mathematical formula independent of Bernoulli

polynomials and numbers, starting from the formula of

Faulhaber.

Abstract:

The domain of functions throughout the work are the

exponents “r” natural numbers N+.

The history:

The sum of powers of integers is defined:

In 1631 Johann Faulhaber published in the journal "Algebra

Academiae" a general formula which was later proved by

Carl Jacobi in 1834 where we used the Bernoulli

polynomials and numbers.

P

ROCESSING

To replace the binomial formula and the

I ° Objective:

Bernoulli numbers, respectively, with mathematical

formulas containing the Gamma and the Zeta function.

The Bernoulli numbers can be written as a function of

ζ(k) and extrapolate from Euler's formula to find the

integer values of (2k)

ζ

(2k) =

ζ

In this formula, Euler considered the absolute value of the

Bernoulli numbers as in Faulhaber’s formula are used for

each:

For each k > = 2 the formula is the same:

The addition operator or have the

function take the value of the Bernoulli numbers used in

Faulhaber’s formula.

Processing the formula:

Binomial formula can be rewritten in terms of Gamma

function:

Substituting these two functions in Faulhaber’s formula

The mathematical formula so constructed will not work

because the function ζ (k) does not converge for k = 1

(Series Harmonica -> ∞) then we can decompose the sum

in:

The Bernoulli numbers assume the value 1 when k = 0 and

(- 1/ 2) when k = 1 in the second column Faulhaber's

Processing functions:

formula in the third column, the last formula:

....................... Replacing the function ζ (k) with the

2nd Objective:

Riemman integral and exchange integral with the

summation.