What is known about the journal Notes on Number Theory and Discrete Mathematics?
I have never heard of this journal, but from the website it seems legitimate based on its publication history (extending back to 1995) and its institution of origin.
What your nephew should do now is just settle in and wait, and don't bother emailing the editors. As a commenter said, one month is not nearly enough time for a journal to process and review an article. It's usually more like 3-4 months and could possibly go longer than that. And the editors of these journals probably get so many emails from authors requesting status updates that usually those emails are simply ignored -- or if you have a nice editor there'll be an automated reply that says, in so many words, "Please stop emailing me."
The frequent emailing could even backfire. Recently I submitted an article to a journal, with a student co-author, and was told they'd get back with me in 9-12 weeks. Six months passed and I had heard nothing. I emailed the editor and asked to make sure she had everything she needed (= polite way of bugging her for an update). The editor said she would check with the reviewer. One day later I received the review -- a three-line rejection letter that indicated clearly that the reviewer had not even read the article. To me, there is a strong likelihood that the article was rejected directly because the reviewer was annoyed at being bugged. This is clear malpractice, but what are you going to do about it? Welcome to our wonderful academic publishing culture.
So, tell your nephew to move on to his next project and let this simmer on the back burner until the end of the summer.
The fact that your nephew hasn't heard anything yet is a good thing. There are really only two reasons a mathematics journal would get back to you within a month:
- the paper clearly isn't good enough for the journal (typically a "desk reject" where the editor makes that decision without a full review); or
- it is a "predatory journal" which doesn't actually do proper peer review (and likely will charge you to publish your article).
So probably this journal is legitimate, and the editor thinks your nephew's paper might be worth publishing, and has sent it out to reviewers. This stage can easily take six months to a year (maybe towards the lower end of that for a journal with "notes" in the name).
I couldn't find a ranking for this journal, and MathSciNet no longer indexes it. Both of these are bad signs in terms of quality, but it looks like a real journal.