Pronunciation of 这 (ze4 or zen4?) in 天气这么好
'zhe4' is the most formal pronunciation. 'zhei4', 'zen4', and 'zhen4' are informal and dialectal.

I grew up in Beijing and local people use these variants frequently. (In contrast, 'zhe4' is actually not often heard in informal speech in the region.) In particular, 'zhei4' is common when people use 这 as a pronoun by itself (e.g., 这位 (this person)、这边儿 (this way)), and 'zen4' and 'zhen4' (less common than 'zen4') appear exclusively in 这么. Compare 怎么 'zen3me'.

The pronunciation of 'ze4' is not heard in the Beijing dialect, but could appear in other dialects of Mandarin.

Related note: When used alone to indicate location, 这儿, 'zher4' in pinyin, sounds more like 'zhar4' in the Beijing dialect - that is about somewhere between jerk sans k and jar in Midwestern American English.

Fun note: In very, very native Beijing dialect, 这儿 is sometimes pronounced like 'zhar4-har'. It might not be completely intelligible to people who grew up in regions in China farther away from Beijing.