I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to read Davenport's Multiplicative Number Theory, and the treatment of L-functions in there requires to understand convergence/absolute convergence of infinite products, which I know little about. Most importantly I'd like to know why

$$ \prod (1+|a_n|) \to a < \infty \quad \Longrightarrow \quad \prod (1+ a_n) \to b \neq 0. $$

I believe I'll need more properties of products later on, so just a proof of this would be appreciated but I'd also need the reference.

Thanks in advance,

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    $\begingroup$ By the way, working over $\mathbb C$. $\endgroup$ – Patrick Da Silva Jun 14 '12 at 3:45
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    $\begingroup$ Is your question why does $\prod (1+a_n)$ converge? or why is the limit not $0$? or both? $\endgroup$ – user17762 Jun 14 '12 at 3:48
  • $\begingroup$ @Marvis : Both! This property is used when treating convergence of the product expression for $L(s,\chi)$. $\endgroup$ – Patrick Da Silva Jun 14 '12 at 3:52
  • $\begingroup$ There must also be an additional assumption that none of the terms $a_n$ are $-1$ for the product to be non-zero i.e. something like $\prod (1+a_n) = 0$ iff at-least one $a_n = -1$. $\endgroup$ – user17762 Jun 14 '12 at 3:55
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    $\begingroup$ I don't have any books with me, but the basics of infinite products are often covered in introductory complex analysis textbooks (because they are useful in any number of places in complex analysis, and arguably, rigorous development of the formal use of infinite products was one of the main historical motivations for complex analysis / analysis in general). Ahlfors's Complex analysis and Conway's Functions of one complex variable I both come to mind as possible sources for this. $\endgroup$ – leslie townes Jun 14 '12 at 4:02

I will answer your question

"Most importantly I'd like to know why $$ \prod (1+|a_n|) \to a < \infty \quad \Longrightarrow \quad \prod (1+ a_n) \to b \neq 0. "$$

We will first prove that if $\sum \lvert a_n \rvert < \infty$, then the product $\prod_{n=1}^{\infty} (1+a_n)$ converges. Note that the condition you have $\prod (1+|a_n|) \to a < \infty$ is equivalent to the condition that $\sum \lvert a_n \rvert < \infty$, which can be seen from the inequality below. $$\sum \lvert a_n \rvert \leq \prod (1+|a_n|) \leq \exp \left(\sum \lvert a_n \rvert \right)$$

Further, we will also show that the product converges to $0$ if and only if one of its factors is $0$.

If $\sum \lvert a_n \rvert$ converges, then there exists some $M \in \mathbb{N}$ such that for all $n > M$, we have that $\lvert a_n \rvert < \frac12$. Hence, we can write $$\prod (1+a_n) = \prod_{n \leq M} (1+a_n) \prod_{n > M} (1+a_n)$$ Throwing away the finitely many terms till $M$, we are interested in the infinite product $\prod_{n > M} (1+a_n)$. We can define $b_n = a_{n+M}$ and hence we are interested in the infinite product $\prod_{n=1}^{\infty} (1+b_n)$, where $\lvert b_n \rvert < \dfrac12$. The complex logarithm satisfies $1+z = \exp(\log(1+z))$ whenever $\lvert z \rvert < 1$ and hence $$ \prod_{n=1}^{N} (1+b_n) = \prod_{n=1}^{N} e^{\log(1+b_n)} = \exp \left(\sum_{n=1}^N \log(1+b_n)\right)$$ Let $f(N) = \displaystyle \sum_{n=1}^N \log(1+b_n)$. By the Taylor series expansion, we can see that $$\lvert \log(1+z) \rvert \leq 2 \lvert z \rvert$$ whenever $\lvert z \rvert < \frac12$. Hence, $\lvert \log(1+b_n) \rvert \leq 2 \lvert b_n \rvert$. Now since $\sum \lvert a_n \rvert$ converges, so does $\sum \lvert b_n \rvert$ and hence so does $\sum \lvert \log(1+b_n) \rvert$. Hence, $\lim_{N \rightarrow \infty} f(N)$ exists. Call it $F$. Now since the exponential function is continuous, we have that $$\lim_{N \to \infty} \exp(f(N)) = \exp(F)$$ This also shows that why the limit of the infinite product $\prod_{n=1}^{\infty}(1+a_n)$ cannot be $0$, unless one of its factors is $0$. From the above, we see that $\prod_{n=1}^{\infty}(1+b_n)$ cannot be $0$, since $\lvert F \rvert < \infty$. Hence, if the infinite product $\prod_{n=1}^{\infty}(1+a_n)$ is zero, then we have that $\prod_{n=1}^{M}(1+a_n) = 0$. But this is a finite product and it can be $0$ if and only if one of the factors is zero.

Most often this is all that is needed when you are interested in the convergence of the product expressions for the $L$ functions.

  • $\begingroup$ Hm. So absolutely convergent products defined in this sense (the $\prod (1 + |a_n|)$ way) makes more sense to me now. Thanks! This is precisely what I needed to get over the point I was stuck at. +1 & check! $\endgroup$ – Patrick Da Silva Jun 14 '12 at 4:46
  • $\begingroup$ won't believe what it is valuable to me, Marvis. Thanks @Martin. Thanks Marvis. I want to read every words in it instantly. many +1 :) $\endgroup$ – mrs Nov 24 '12 at 8:11
  • $\begingroup$ I learned something! $\endgroup$ – Arturo don Juan Feb 7 '15 at 1:19
  • $\begingroup$ Could you please tell me where the following inequality came from? $$\sum \lvert a_n \rvert \leq \prod (1+|a_n|) \leq \exp \left(\sum \lvert a_n \rvert \right)$$. $\endgroup$ – Arturo don Juan Feb 7 '15 at 1:22
  • $\begingroup$ The first inequality follows from $\prod(1 + |a_n|) = \sum |a_n| + \sum |a_n||a_m| + \sum |a_n||a_m||a_l| \ldots$. The second is trivial if you take logarithms $$ \log\left(\prod (1+|a_n)\right) = \sum \log(1+|a_n) \leq \sum |a_n| $$ $\endgroup$ – gerd Jun 5 '15 at 12:26

I am making this CW, so that other people can add further references.


  • Konrad Knopp: Infinite Sequences and Series; see p.92. I believe Knopp's books can be considered a classical references for this.

  • Konrad Knopp: Theorie und Anwendung der unendlichen Reihen; or the English translation Theory and application of infinite series. There is a whole chapter devoted to infinite products, see p.218. The book is freely available here. This is given as reference in Wikipedia article. (You can read this on Talk page of that article: This article is probably very non-ideal, but when I needed this material a while ago it was hard to find, and so I figured wikipedia would be a good place to hold it. So it seems you are not the there are other people who had problems with finding references about infinite products.)

  • Earl David Rainville: Infinite Series. This book was mentioned in connection with infinite products in this answer.

  • Reinhold Remmert: Classical Topics in Complex Function Theory, Graduate Texts in Mathematics, Volume 172, translated from German.

    Part A of this outstanding book is dedicated to infinite products. It has six chapters on 140+ pages and covers a lot of classical material, from very basic convergence theory to quite advanced material on functions in one . Highlights include the sine product, partition products, a detailed treatment of the $\Gamma$-function and the $\mathrm{B}$-function, the Weierstraß product theorem, Iss'sa's theorem, Mittag Leffler's theorem and much more. In addition the book has a lot of historical references and remarks and recommendations for further reading.

  • J. N. Sharma: Infinite Series and Products, see p.129. I did not know about this book, I found it using Google Books - see below.



The reason I've included this part is that I only knew about Knopp's book(s) offhand, but it seemed very probable that there are plenty of notes available online. This is how I found Payne's and Chen's notes; you can check the search results for yourself to see, whether you find some other interesting things.

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    $\begingroup$ Now I realized that I am not sure which of the above references deal with infinite product of real numbers only (obviously, you want to know about complex numbers). But I believe that some of the proofs (Cauchy criterion, absolute convergence) are similar for the real and complex case. $\endgroup$ – Martin Sleziak Jun 14 '12 at 6:19

Complex Analysis, Princeton Lectures in Analysis by Stein and Shakarchi. p 140-141.


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