Adumi Kazuki or Azumi Kazuki ?

It should technically be 'Adumi' in romaji. The character 'づ' is 'du', and 'ず' is 'zu', so 'あづみ かずき' is 'Adumi Kazuki'.

But in Japanese alphabet, 'づ' and 'ず' has the same phoneme, they are all pronounced as 'zu'. The only difference here is meaning.

There is no "technically correct" one, there are multiple versions of romanization of Japanese. There is indeed an official one which the Japanese government uses, but almost no one uses that (especially about how to represent the long vowels).

fireattack said:
There is no "technically correct" one, there are multiple versions of romanization of Japanese. There is indeed an official one which the Japanese government uses, but almost no one uses that (especially about how to represent the long vowels).

Quite agree. One of my friend who's learning Japanese told me it's ok here to write 'zu' as 'づ'.

fireattack said:
There is no "technically correct" one, there are multiple versions of romanization of Japanese. There is indeed an official one which the Japanese government uses, but almost no one uses that (especially about how to represent the long vowels).

乐观的食用盐 said:
Quite agree. One of my friend who's learning Japanese told me it's ok here to write 'zu' as 'づ'.

https://ja.wikipedia.org/wiki/%E3%81%A5

上代の「づ」の発音は、舌先の破裂音を子音とする /du/ であったと考えられるが、室町時代末には摩擦音を伴う /dzu/ に転じていた。さらに江戸時代に入ると「づ」（/dzu/）と「ず」（/zu/）の間で発音上の区別が失われ、仮名遣いの混乱を生じるようになった[1]。なお現在でもごく一部の方言に /du/ の発音が残る。

Looks like its "standard" pronunciation changes overtime: /du/ --> /dzu/ --> /zu/

But some of the dialects still pronounce it as the ancient /du/.