Given a sentence in Tagalog language, translate the sentence to English language keeping the meaning of the original sentence intact.
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Question: Tagalog sentence: Sa kanyang sanaysay Heilbronn din inilapat ang kanyang mga resulta sa primes sa isang arithmetic paglala at sa estima ng sum ng Möbius function.

Answer: In his thesis Heilbronn also applied his result to primes in an arithmetic progression and to estimates of the sum of the Möbius function.


Question: Tagalog sentence: Magsisimula kami sa identity Samakatuwid, mayroon kaming

Answer: We begin with the identity Thus, we have


Question: Tagalog sentence: Isa kawili-wili epekto ay maaaring mas Observed Mas maaga, ang ilang mga araw bago panuluyan (damaging, pagpapahina), sa ngayon siya Observed pangunahing barley, iba pang mga cereal na may marahil ay hindi Magpakita Halos bilang magkano (barley).

Answer:
One very interesting effect may be observed much earlier, several days before lodging (damaging, weakening), so far observed mainly on barley, with other cereals probably does not demonstrate nearly as much.