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.

Processing

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 Riemman integral and exchange integral with the summation.

2nd Objective:

Exchange integral with the summation:

Adding functions omit the first integral:

On arrive at the final mathematical formula:

in the second column Faulhaber’s

Processing functions:

formula in the third column, the last formula highlighted:

Text reference

[1] Rademacher H. (Springer 1973). “Topics in Analytic Number Theory