You can do it cheaper through a publication company, because they've already negotiated rates with the publishers. Though $1000 seems excessive. In Westchester County, I paid $300. There is a large spread in fees that will be charged.
I wrote to my state assembly person; she wrote back saying A8125 was introduced to repeal the requirement, but no action taken. I would encourage you to write to your assembly person and senator, because this is just a drag on NY businesses.
”unless I pay them $1000 to publish the name of my LLC in two public newspapers for six weeks, one daily newspaper and one weekly newspaper”
Reading that as “one daily newspaper and one weekly newspaper”, that would be 6 (weeks) times at least (6 + 1) = 42 adverts in a newspaper (I can also read it as “2 public, one daily and one weekly”. That would increase the number of adverts). Price per advert would be about $25.
I guess most LLCs get incorporated in an at least somewhat large city, so the circulation of the paper will be somewhat high. $25 per advert then doesn’t sound that bad to me.
I know zilch of advertising prices, though, so let’s google it. Reading https://fitsmallbusiness.com/newspaper-advertising-costs/ (a link that google gave me, but that I don’t know the quality of), I see the price per “column inch” go down from about €18 to about €13 when you advertise more often.
⇒ it could well be that $1,000 is a fair price, giving the higher price you would have to pay and the amount of work you would have to do if you did it alone.
(Still seems a weird law, though. I can imagine that one should give people to object if your company will be noisy, stinky, or the like, but then, the city, IMO, should require a permit)
I think the law made sense in the past so that the formation of a company would be well-known and not a secret. Now that everything is available online, less so. That the publication servicing companies know you exist, advertise directly to you because of that, and need no other information than your payment to do the publication is proof that the law is no longer useful.
I think part of it is the assumption that affected citizens (who might object to the new shop/cafe/…) are more likely to read such papers than government publications.
That likely still is true, but less so (fewer people subscribe to papers), and the assumption that people who read the paper would spot your advert probably only held back in the time when papers had reasonably small coverage (so that your advert wouldn’t appear in a large section of the paper, but stand on its own), and were about the only source of news (which made them much better read)