But first, let's discuss pausal forms. Those who know from pausal forms are invited to skip down to the paragraph that begins "Gashem is the pausal form of geshem."
Using the siddur, I noticed some strange stuff some years ago. For example, in the Retzeh paragraph that's added to Birkat Hamazon on Shabbat, we find "kemitzvat retzonékha [new clause] uvirtzonekhá haniah ["hanah" in Birnbaum] lanu..." What's with the two different endings for the same word? And similarly in the weekday Amidah, we find "vekarno tarum bishu'atékha [new clause] ki lishu'atekhá kivinu..."
And in the post-drinking blessing for wine, what's with all the gefen and gafen? "Gafen" must be the basic way of saying it, right? I mean, everyone's heard "borei peri hagafen." So you look up vine in an English-Hebrew dictionary--it's gefen.
And at the end of Birkat Hamazon, we have "vezar'o mevakesh lahem" instead of "lehem," and in Ashrei "ugdol-hased" instead of "hesed."
What's going on? What's going on is pausal forms. In biblical Hebrew, some words undergo a change if they're immediately followed by a major pause. One pausal form is the "-ekhá" (meaning "your") that becomes "-ékha" in pausation. Thus lishu'atekhá and bishu'atékha. Another occurs in nouns with three consonants and two segols (a segol is the three-dot vowel that sounds like a short "e"), such as lehem, melekh, and hesed; the first segol becomes a qamatz (the vowel that looks like a squushed-down capital T). Thus "vezar'o mevakesh lahem" and "ugdol-hased."
"But wait!," you may be saying. "You said the pausal forms are in biblical Hebrew. While some of the things you've quoted are from the Bible--viz., Ashrei and the verse near the end of Birkat Hamazon--others are not. So, not to put to fine a point on it, one needs to ask 'Wha?'!"
If you're saying this, you're right. Much of Ashkenazi liturgy was recast into biblical style in recent centuries, and we sometimes go hog wild, as it were, with the pausalities. Consider the Berakhah Aharonah: "al Yisra'el amékha, ve'al Yerushalayim irékha, ve'al Tsion mishkan kevodékha, ve'al mizbehékha, ve'al heykhalékha." On this and on this and on this and on this and on this. This is a lot of pausativity. (Although I like this one; it makes it more lively than "on this and on this and on this and on this and on this.") But yes, it’s overdone.
Gashem is the pausal form of geshem. So which one to use? I used to be a gashemite, to use the word of Dr. Shnayer Leiman. Every siddur I've seen has a full stop after "hag*shem" (either a Western-style period or a Hebrew sof pasuk, which looks like a Western colon and is more or less equivalent to a period). This is true even of those siddurim that have "hageshem." And in general, even those shelihei tzibbur who say "hageshem" also pronounce a full stop. So "hagashem" made sense to me. According to Dr. Leiman, this is in fact the rationale of the gashemites.
There are some who are very intolerant of what they consider the wrong pronunciation. Even if I had no other reason to like the virtual person known as Mississippi Fred MacDowell, I would still be eternally grateful to him for pointing me toward Dr. Leiman. Specifically, to this shiur by Dr. Leiman on the history and halakhah of the geshem-vs.-gashem controversy. I recommend the shiur highly, even if you're not a vocalization geeq. It's as much history as it is halakhah, and Dr. Leiman's presentation is fascinating, educational, and entertaining. You don't need a yeshiva background to understand it--I, for example, am an am ha'aretz with no training in rabbinics.
This comment from Siddur Tzelota d'Avraham summarizes both the geshemite position and, so that it can be refuted, the gashemite position (you can enlarge the image by clicking on it):
Morid hageshem: The gimmel takes a segol [i.e., it’s geshem]. This is how it’s printed in all the old Ashkenazic siddurim and mahzorim, and also in the Sephardic siddur that is available to me, as well as in the siddur of the Holy Luminescent Rabbi Who Wrote the Tanya (may the memory of the righteous one be a blessing). I’ve heard that a grammarian in Berlin [Isaac Satanow] published a siddur, Vaye'ater Yitzhak, in which he prints it as "hagashem," in accordance with the rules concerning pauses. And in fact Zechariah 14:15 [sic; it seems to me to be verse 17] ends with "yihyeh hagashem." After Satanow, it was published as hagashem in several siddurim. But this is just a complete scrambling of the old books; there is no pause here at all. In siddurim, it's printed on a line by itself in order to teach that it's said not all year 'round, but only in the winter. But it is connected to the language that follows it; by making the rain fall (morid hageshem), God kindly sustains the living (mekhalkel hayyim behesed). As the Tur (Orah Hayyim 114) notes, "morid hageshem" supports "mekhalkel hayyim" because the rains provide livelihood and sustenance (kalkalah, which has the same root as mekhalkel). It has nothing at all to do with "mehayyeh hametim" (giver of life to the dead), which precedes it.Note that in the same siddur that makes this argument that there is no pause at all after hageshem, hageshem is followed by a period. As I was saying. This is why I was a gashemite.
I’m not at all competent to evaluate the halakhic arguments for either position. I have become a geshemite for nonhalakhic reasons. When I started thinking about it, I realized that the phrase is parallel to those that follow--they all begin with participles (or what are in Modern Hebrew called present-tense verbs):
mashiv haruah umorid hag*shem
mekhalkel hayyim behesed
mehayyeh metim berahamim rabim
umkayyem emunato lisheney afar
OK. So it’s associated with the following phrases. This doesn’t necessarily mean it should be geshem. After all, Ashkenazic liturgy is heavy on the pauses--recall the Berakhah Aharonah. Maybe it should be pausal gashem. But note the next phrase; it ends with hesed, not hased. Because I’m claiming that the two lines have parallel construction, I choose to say geshem (which is not the same as claiming that gashem is incorrect).
Judging by Dr. Leiman’s lecture, each pronunciation is advocated by bunches of rabbis, and these guys are always right, so you’re probably OK either way. On the other hand, since some of these rabbis say the other pronunciation is downright wrong, it’s possible that you’re non-OK either way.