# Need help in understanding meaning of a notation and theorem used in research paper due to a reference being in German Language

I thought of utilizing this lockdown period to study research papers in number theory by myself.

I began reading the research paper By T Estermann ->" On Goldbach Problem : Proof that Almost all Even Positive Integers are sum of two primes". I have read and understood all the proof except one equation and that too due to reason that it uses a equation as a reference from German Book " E Landau, Vorlesungen über Zahlentheorie I ( Liepzig, 1927) . I don't understand German.

I thought of posting this question here in hope that someone must have studied this research paper and also in case a researcher understanding German language wishes to help.

Image of the equation I don't understand ( equation 49 on Page 7 ) ! Equation 49]1

The problem-> 1.I am unable to understand what ( 2|m) means in (49) ?(Notation (2|m) is not used in research paper before) 2.what is the statement of theorem is used to derive (49)? . . Everything except this theorem is Understood in paper.

Images from Original German book->  What i did to resolve the issue -> I tried to find English Edition but couldn't. 2.Then I tried to translate whole pdf book which didn't materalized due to book being a bit long. 3.When I split the pdf I wanted to translate theb it Shows Error 403. 4. I changed the screenshot images( from archive.org) to pdf but again error in translation.

It is my humble request if anybody can tell me meaning of notation (2|m) ?

And it will be very much more helpful if someone tell what is the statement of theorem / result used to derive equation 49.

I shall be really thankful for any help offered.

• $m \mid n$ means that $m$ divides $n$. In particular, $2 \mid m$ means that $m$ is even. "m gerade" in the original German means "m is even". May 3, 2020 at 16:21
• Type it in google translator... the words only, not the math. I have done this with some success.
– efs
May 3, 2020 at 16:22
• You should better use "deepl.com", which in the opinion of everybody I know of is lightyears ahead of google translate for translations between English, German, and French. May 3, 2020 at 16:48
• Estermann refers to formula (239) which is just a formula, no words (except the assumption that $m$ is even). Of course you also need to understand how he proves it. May 3, 2020 at 17:38
• @KlausNiederkrüger thank you very much for telling about deepl. I am using it now. It is good May 4, 2020 at 5:05

• @user155294 currently I am reading the German edition of this book by translating line by line using deepl. com. I have a question does on page 226 ( please see image 1) D (m, n) > $c_{4, 3}$ means D( m, n) is greater than some constant? May 4, 2020 at 7:45
• @Tim Green Yes. Moreover $S(m)\approx D(m,n)$ for $n>c_{4,2}$ and $m$ is even, with some big value for $m$ and $c_{4,2}$ and $0<m\leq n$. May 4, 2020 at 9:26
First, the notation $$2\mid m$$ should be very common, and it is definitely very common in German books. As an example, I own a copy of Algebra from Siegfried Bosch, first edition from 1992, and it uses this notation as well.