Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

find a closed form for the sum of the zeta function $\zeta(k)$ for $k$ runs from $1$ to $n$. I need this to find the sum of an infinite series involving the zeta function at the natural numbers. Any help is nice.

share|cite|improve this question
up vote 3 down vote accepted

Since you aren't very specific with what you're asking, I only have a vague idea of what you might be looking for so I will give you a couple things. I think what you're asking for is the following:

Faulhaber's Formula: $$ \sum_{k=1}^n k^x=\frac{1}{x+1}\sum_{k=0}^x \binom{x+1}{k}B_kn^{x+1-k} $$ where $B_k$ are the Bernoulli numbers. Of course, these provide good approximations when the powers are in the integers. Generally, you'd need Euler-Maclaurin. These sums also have expressions using generalized harmonic numbers.

Depending on the context in which you are working, you may find one or more of the following useful:

$$ \sum_{n=2}^\infty \zeta(n)-1=1 $$ $$ \sum_{n=1}^\infty \zeta(2n+1)-1=\frac{1}{4} $$ $$ \sum_{n=1}^\infty \zeta(2n)-1=\frac{3}{4} $$

And of course, I'm assuming you know the Hurwitz Zeta function: $$ \zeta(s,q)=\sum_{n=0}^\infty \frac{1}{(q+n)^s} $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.