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Let me try to summarize this long discussion in the comments. There are many free resources.

1. arXiv. It is true that not all mathematicians post their papers on the arXiv, for various reasons. But some of those who don't, post them on their personal sites. There are also other depositories, for example in Europe (see 5 below).

2. NSF depository. NSF's stated policy is that all results of NSF-sponsored research "must be available to the public at most in 1 year since their publication". I checked: the site is somewhat confusing but it works.

3. Many journals are freely available, and many more make their papers available after some time (usually 4-5 years). When you choose a journal to publish your paper, take this into account! Here is a convenient catalog of online journals. I am sure many other libraries have similar catalogs; I find this one convenient. The journals with free access are marked green, with partial access yellow and red. Access may depend on your location or on the date of publication.

4. Finally there are "pirate" sites. Some of them may have huge collections, larger than many university libraries. They frequently change their names and location. Some keywords may be "bookfi" or "genesis" for books, and "sci-hub" for journal articles. (Some of them may be illegal in some countries).

5. A simple search on Google and especially on Google Scholar sometimes finds what you need. It could be a place you do not expect. Some saved/cached copy. Some preprint depository that you do not know. Some personal web site, etc.

6. The very important resource is MathScinet, which unfortunately has no free version. But its German competitor Zentralblatt Math is partially free. Whatever you search there, it gives you only 3 items for free. But by clever choice of search criteria you can obtain amazing results. For really old items, there is also Jahrbuch which is free (it is a subset of Zentralblatt). For new papers, Google Scholar is excellent, especially if you know the author's name and title of the paper. It also sometimes finds you a free copy when available.

EDIT. I asked NSF, and they explained that all NSF-supported papers older than 1 year are really available, though the site is somewhat confusing. One has to click on the title of the paper, and then on a little square which says "pdf".

EDIT 2: I collected some links to free resources on my web page.

Let me try to summarize this long discussion in the comments. There are many free resources.

1. arXiv. It is true that not all mathematicians post their papers on the arXiv, for various reasons. But some of those who don't, post them on their personal sites. There are also other depositories, for example in Europe (see 5 below).

2. NSF depository. NSF's stated policy is that all results of NSF-sponsored research "must be available to the public at most in 1 year since their publication". I checked: the site is somewhat confusing but it works.

3. Many journals are freely available, and many more make their papers available after some time (usually 4-5 years). When you choose a journal to publish your paper, take this into account! Here is a convenient catalog of online journals. I am sure many other libraries have similar catalogs; I find this one convenient. The journals with free access are marked green, with partial access yellow and red. Access may depend on your location or on the date of publication.

4. Finally there are "pirate" sites. Some of them may have huge collections, larger than many university libraries. They frequently change their names and location. Some keywords may be "bookfi" or "genesis" for books, and "sci-hub" for journal articles. (Some of them may be illegal in some countries).

5. A simple search on Google and especially on Google Scholar sometimes finds what you need. It could be a place you do not expect. Some saved/cached copy. Some preprint depository that you do not know. Some personal web site, etc.

6. The very important resource is MathScinet, which unfortunately has no free version. But its German competitor Zentralblatt Math is partially free. Whatever you search there, it gives you only 3 items for free. But by clever choice of search criteria you can obtain amazing results. For really old items, there is also Jahrbuch which is free (it is a subset of Zentralblatt). For new papers, Google Scholar is excellent, especially if you know the author's name and title of the paper. It also sometimes finds you a free copy when available.

EDIT. I asked NSF, and they explained that all NSF-supported papers older than 1 year are really available, though the site is somewhat confusing. One has to click on the title of the paper, and then on a little square which says "pdf".

EDIT 2: I collected some links to free resources on my web page.

EDIT 3: Zentralblatt Math is now free for all, full version. Thanks to the German government.

Let me try to summarize this long discussion in the comments. There are many free resources.

1. arXiv. It is true that not all mathematicians post their papers on the arXiv, for various reasons. But some of those who don't, post them on their personal sites. There are also other depositories, for example in Europe (see 5 below).

2. NSF depository. NSF's stated policy is that all results of NSF-sponsored research "must be available to the public at most in 1 year since their publication". I checked: the site is somewhat confusing but it works.

3. Many journals are freely available, and many more make their papers available after some time (usually 4-5 years). When you choose a journal to publish your paper, take this into account! Here is a convenient catalog of online journals. I am sure many other libraries have similar catalogs; I find this one convenient. The journals with free access are marked green, with partial access yellow and red. Access may depend on your location or on the date of publication.

4. Finally there are "pirate" sites. Some of them may have huge collections, larger than many university libraries. They frequently change their names and location. Some keywords may be "bookfi" or "genesis". (Some of them may be illegal in some countries).

5. A simple search on Google and especially on Google Scholar sometimes finds what you need. It could be a place you do not expect. Some saved/cached copy. Some preprint depository that you do not know. Some personal web site, etc.

6. The very important resource is MathScinet, which unfortunately has no free version. But its German competitor Zentralblatt Math is partially free. Whatever you search there, it gives you only 3 items for free. But by clever choice of search criteria you can obtain amazing results. For really old items, there is also Jahrbuch which is free (it is a subset of Zentralblatt). For new papers, Google Scholar is excellent, especially if you know the author's name and title of the paper. It also sometimes finds you a free copy when available.

EDIT. I asked NSF, and they explained that all NSF-supported papers older than 1 year are really available, though the site is somewhat confusing. One has to click on the title of the paper, and then on a little square which says "pdf".

EDIT 2: I collected some links to free resources on my web page.

Let me try to summarize this long discussion in the comments. There are many free resources.

1. arXiv. It is true that not all mathematicians post their papers on the arXiv, for various reasons. But some of those who don't, post them on their personal sites. There are also other depositories, for example in Europe (see 5 below).

2. NSF depository. NSF's stated policy is that all results of NSF-sponsored research "must be available to the public at most in 1 year since their publication". I checked: the site is somewhat confusing but it works.

3. Many journals are freely available, and many more make their papers available after some time (usually 4-5 years). When you choose a journal to publish your paper, take this into account! Here is a convenient catalog of online journals. I am sure many other libraries have similar catalogs; I find this one convenient. The journals with free access are marked green, with partial access yellow and red. Access may depend on your location or on the date of publication.

4. Finally there are "pirate" sites. Some of them may have huge collections, larger than many university libraries. They frequently change their names and location. Some keywords may be "bookfi" or "genesis" for books, and "sci-hub" for journal articles. (Some of them may be illegal in some countries).

5. A simple search on Google and especially on Google Scholar sometimes finds what you need. It could be a place you do not expect. Some saved/cached copy. Some preprint depository that you do not know. Some personal web site, etc.

6. The very important resource is MathScinet, which unfortunately has no free version. But its German competitor Zentralblatt Math is partially free. Whatever you search there, it gives you only 3 items for free. But by clever choice of search criteria you can obtain amazing results. For really old items, there is also Jahrbuch which is free (it is a subset of Zentralblatt). For new papers, Google Scholar is excellent, especially if you know the author's name and title of the paper. It also sometimes finds you a free copy when available.

EDIT. I asked NSF, and they explained that all NSF-supported papers older than 1 year are really available, though the site is somewhat confusing. One has to click on the title of the paper, and then on a little square which says "pdf".

EDIT 2: I collected some links to free resources on my web page.

Let me try to summarize this long discussion in the comments. There are many free resources.

1. arXiv. It is true that not all mathematicians post their papers on the arXiv, for various reasons. But some of those who don't, post them on their personal sites. There are also other depositories, for example in Europe (see 5 below).

2. NSF depository. NSF's stated policy is that all results of NSF-sponsored research "must be available to the public at most in 1 year since their publication". I checked: the site is somewhat confusing but it works.

3. Many journals are freely available, and many more make their papers available after some time (usually 4-5 years). When you choose a journal to publish your paper, take this into account! Here is a convenient catalog of online journals. I am sure many other libraries have similar catalogs; I find this one convenient. The journals with free access are marked green, with partial access yellow and red. Access may depend on your location or on the date of publication.

4. Finally there are "pirate" sites. Some of them may have huge collections, larger than many university libraries. They frequently change their names and location. Some keywords may be "bookfi" or "genesis". (Some of them may be illegal in some countries).

5. Simple search on Google and especially on Google Scholar sometimes finds what you need. It could be a place you do not expect. Some saved/cached copy. Some preprint depository that you do not know. Some personal web site, etc.

6. The very important resource is MathScinet, which unfortunately has no free version. But its German competitor Zentralblatt Math is partially free. Whatever you search there, it gives you only 3 items for free. But by clever choice of search criteria you can obtain amazing results. For really old items, there is also Jahrbuch which is free (it is a subset of Zentralblatt). For new papers, Google Scholar is excellent, especially if you know the author's name and title of the paper. It also sometimes finds you a free copy when available.

EDIT. I asked NSF, and they explained that all NSF-supported papers older than 1 year are really available, though the site is somewhat confusing. One has to click on the title of the paper, and then on a little square which says "pdf".

EDIT 2: I collected some links to free resources on my web page.

Let me try to summarize this long discussion in the comments. There are many free resources.

1. arXiv. It is true that not all mathematicians post their papers on the arXiv, for various reasons. But some of those who don't, post them on their personal sites. There are also other depositories, for example in Europe (see 5 below).

2. NSF depository. NSF's stated policy is that all results of NSF-sponsored research "must be available to the public at most in 1 year since their publication". I checked: the site is somewhat confusing but it works.

3. Many journals are freely available, and many more make their papers available after some time (usually 4-5 years). When you choose a journal to publish your paper, take this into account! Here is a convenient catalog of online journals. I am sure many other libraries have similar catalogs; I find this one convenient. The journals with free access are marked green, with partial access yellow and red. Access may depend on your location or on the date of publication.

4. Finally there are "pirate" sites. Some of them may have huge collections, larger than many university libraries. They frequently change their names and location. Some keywords may be "bookfi" or "genesis". (Some of them may be illegal in some countries).

5. A simple search on Google and especially on Google Scholar sometimes finds what you need. It could be a place you do not expect. Some saved/cached copy. Some preprint depository that you do not know. Some personal web site, etc.

6. The very important resource is MathScinet, which unfortunately has no free version. But its German competitor Zentralblatt Math is partially free. Whatever you search there, it gives you only 3 items for free. But by clever choice of search criteria you can obtain amazing results. For really old items, there is also Jahrbuch which is free (it is a subset of Zentralblatt). For new papers, Google Scholar is excellent, especially if you know the author's name and title of the paper. It also sometimes finds you a free copy when available.

EDIT. I asked NSF, and they explained that all NSF-supported papers older than 1 year are really available, though the site is somewhat confusing. One has to click on the title of the paper, and then on a little square which says "pdf".

EDIT 2: I collected some links to free resources on my web page.