Theorem connecting partial sums and the values of the Riemann zeta function

In the course of some studies relating to our first paper on analytic number theoretic sums and quantum field theory, I came across a simple yet seemingly not too well known theorem (as far as I have seen in the literature; however I do note the likeness to, and manner from which the result can be directly extrapolated from, Hans Rademacher’s 1973 book Topics in Analytic Number Theory). The theorem can be found in a short paper by Jan Minac from 1994. The main result gives a slightly different perspective on the relation between the values of the Riemann zeta function and the partial sums of the sums of integer powers.

The origin of the Riemann zeta function ${\zeta(s)}$ goes back to the study of many-particle systems and how they may be encoded in a single partition function. In 1737, Euler proved the identity

$\displaystyle \prod \limits_p (1 - \frac{1}{p^s})^{-1} = \sum \limits_{n=1}^{\infty} = \zeta(s), \ \ \ \ \ (1)$

which holds for all real numbers ${s > 1}$ and, amazingly, refers to all prime numbers ${p}$ . After Euler’s introduction of ${\zeta (s)}$ over the real numbers, it was Bernhard Riemann in his seminal 1859 paper who took this formula as his starting point and extended it to a complex variable. In the process, he proved among other things the relation between its zeros and the distribution of prime numbers. From this comes the Riemann zeta function, which can be written as the sum of the reciprocals of the positive integers

$\displaystyle \zeta(s)=\sum \limits_{n=1}^{\infty}\frac{1}{n^{s}}=1+\frac{1}{2^s}+\frac{1}{3^s}+ \dots, \ \ \ \ \ (2)$

where ${s}$ is a complex number with ${Re(s) > 1}$ . For negative integers (and also zero), the Riemann zeta function can be related to the Bernoulli numbers ${\zeta(-s) = (-1)^s B_{s+1} / (s+1)}$ . Another remarkable fact about this, and about the Bernoulli numbers in general, is that all the values ${\zeta(-s)}$ are rational numbers: ${\zeta(-1) = - 1/12}$ , ${\zeta(-2) = 0}$ , ${\zeta(-3) = 1/120}$ , and so on.

From the definition of Faulhaber’s formula, heuristically there is a relation ${\sum \limits_{n=1}^{\infty} n^s = \frac{(-1)^s B_{s+1}}{s+1}}$ that can be deduced, which agrees with the value of the Riemann zeta function for negative integers ${s = -p >0}$ . However, in general, when one expands the partial sums for the respective sums of integer powers, there is no obvious relationship between the partial sum form and the values obtained by the zeta function as indicated in the above equality.

Interestingly, from Minac’s paper, we see that we can define the partial sums ${S_a (N) := \sum \limits_{n=1}^{N-1} n^a}$ such that they are expressible as polynomials in ${N}$ of degree ${a+1}$ with rational coefficients (again, see Rademacher’s 1973 book). One can then ask whether any connection between ${\zeta(-a)}$ and ${S_a(N)}$ can be made. This leads us to the main theorem of Minac’s paper, for which I also repeat the proof.

Theorem 1 : Consider the power sum ${S_a (N) = \sum \limits_{n=1}^{N-1} n^a}$ . For positive integers ${a}$ there exists the relation ${\zeta(-a) = \int_0^1 dx \ S_a (x)}$ .

Proof: Let ${B_k}$ be the kth Bernoulli number for ${k = 0,1,2,\dots}$ , and ${B_0(x) = 1}$ . For each ${m \in \mathbb{N}}$ we define the mth Bernoulli polynomial ${B_m (x) = \sum \limits_{k=0}^{m} \begin{pmatrix} m \\ k \\ \end{pmatrix} B_k x^{m-k}}$ . Importantly, we note ${B_m(1) = (-1)^m B_m = (-1)^m B_m(0)}$ , and the derivative of the Bernoulli polynomial takes the form ${d/dx \ B_m (x) = mB_{m-1}(x)}$ . Finally, we shall use the fact ${S_a(N) = \frac{B_{a+1}(N) - B_{a+1}(1)}{a+1}}$ , which is based on one of the most important properties of Bernoulli polynomials $B_m (x + 1) - B_m (x) = mx^{m-1}$ , as well as ${\zeta(-a) = (-1)^a \frac{B_{a+1}}{a+1} = - \frac{B_{a+1}}{a+1}}$ for ${a \in \mathbb{N}}$ .

Since ${S_a(N)}$ is polynomial in ${N}$ of degree ${a + 1}$ , the important observation is that we can exploit these basic properties of Bernoulli polynomials to establish the integration of the polynomial representation ${S_a(N)}$ of the partial sums from ${0}$ to ${1}$ . Let ${a \in \mathbb{N}}$ , then from the above it follows

$\displaystyle \int_0^1 dx \ S_a(x) = \int_0^1 \frac{B_{a+1}(x) - B_{a+1}(1)}{a+1}$

$\displaystyle = \frac{B_{a+2}(1) - B_{a+2}(0)}{(a+1)(a+2)} + (-1)^a \frac{B_{a+1}}{a+1} = 0 + (-1)^a \frac{B_{a+1}}{a+1} = \zeta(-a). \ \ \ \ \ (3)$

$\Box$

As an example, let us consider one of my favourite cases: the sum of natural numbers corresponding formally to ${zeta(-1)}$ . Theorem 1 gives

$\displaystyle 1 + 2 + 3 + 4 + \dots = \zeta(-1) = \int_0^1 dx \ \frac{x(x - 1)}{2}= \int_0^1 dx \ \left(\frac{x^2}{2} - \frac{x}{2}\right) = \frac{1}{6} - \frac{1}{4} = -\frac{1}{12}. \ \ \ \ \ (4)$

Very cool!

[1] Ján Mináč, “A remark on the values of the Riemann zeta function,” Expositiones Mathematicae, 12: 459-462, 1994.

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