http://www.civfanatics.com/civ4/strategy/tech_research.php (link to the article which has been copied below for the teams convenience, I make no claim to it at all)
One important note.
FLOOR means rounding DOWN to the nearest integer AND it has higher precedence than all other operators EXCEPT FOR parentheses. You will see it used a lot in the article.
The basics (of generating beakers)
1) Each city generates a raw amount of commerce via the squares that are being worked and the trade routes in the cities.
2) This commerce is in turn translated into beakers (research), coins (gold), or notes (culture) via the technology and culture sliders. The actual amount of beakers and notes are gained by using the specified slider rate in the following equation:
FLOOR (total commerce * the specified slider rate).
The remainder is then given in gold.
So, yes, you WILL get AT LEAST one gold from each city that has a commerce that is not a multiple of 10, if your science or culture is not 100%.
3) Each city then takes each amount and adds in any additional sources of research, gold, or culture (via shrines, production processes, and specialists, mainly).
4) Each city then multiplies it by whatever modifiers it has depending on what buildings it has built (e.g., beakers are multiplied by 1.25 if it has a library, gold is multiplied by 1.5 if it has a bank, etc.). NOTE: multipliers are ADDITIVE (i.e., if you have a library and university, your beakers will be multiplied by 1.5 (1 + 0.25 + 0.25) NOT 1.5625 (1 * 1.25 * 1.25)).
5) The beaker totals are then added up across the cities and the sum is your base beaker total.
Technology Costs
To calculate the technology cost of a technology:
1) Take the base cost of the technology. (This can be found for each technology in Assets\XML\Technologies\CIV4TechInfo.xml).
2) Take the Difficulty Modifier and add 0.5 * the Number of teammates you have to it. (The difficulty modifier can be found in Assets\XML\GameInfo\CIV4HandicapInfo.xml.)
3) Multiply 1) by 2) and take the FLOOR of the product.
4) Multiply 3) by the Map Modifier and take the FLOOR of the product. (The map modifier can be found in Assets\XML\GameInfo\CIV4WorldInfo.xml.)
5) Multiply 4) by the Speed Modifier and take the FLOOR of the product. This generates the actual cost of the technology. (The speed modifier can be found in Assets\XML\GameInfo\CIV4GameSpeedInfo.xml.)
Thus the formula to calculate a Technology's cost is:
Technology Cost = FLOOR (FLOOR (FLOOR (Base Cost * (Difficulty Modifier + 0.5 * # of teammates)) * Map Modifier) * Speed Modifier)
Applying the Research to the Technology
The amount of beakers applied to the technology cost is dependant upon the number of civilizations which know the tech and the number of requirements you have fulfilled for the technology.
Calculating the Known Civilizations w/ Tech modifier
1) Multiply 0.30 by the number of KNOWN, LIVING civilizations, who have the tech.
2) Divide by the number of civilizations which STARTED on the map and ROUND DOWN to the hundredth place (0.01) of the quotient.
3) Add 2) to 1
Tech Known by Civilizations modifier = 1 + RDDW (0.30 * # known Civs who have the tech / # of Civs who started the game
Calculating the Prerequisites modifier
1) Start with 1. If the technology does NOT have a minimum requirement (i.e., the starting techs = Fishing, The Wheel, Agriculture, Hunting, Mysticism, Mining), then use 1 AS the modifier.
2) Add 0.2 to 1) if a Technology has a MINIMUM Requirement that the player has met. Note: Even if a tech has MANY MANDATORY PREREQUISITES, it will still only give you a boost of 1.2 because you MUST have all those prerequisites to research the tech. (IOW, the minimum requirement IS all of those prerequisites.)
3) Add 0.2 to 2) for EACH ADDITIONAL OPTIONAL Prerequisite that the player has met.
Requirements modifier = 1 + (0.2 * MINIMUM Req. met) + (0.2 * # of Optional Prereq. met)
Calculating the Actual Amount of Beakers Generated Toward a Tech
To calculate the amount of beakers which you generate toward the technology:
1) Take the total sum of beakers generated by all your cities (the result from the basics part).
2) Add 1 to it if you have a city.
3) Multiply 2) by the Known Civilizations w/ Tech modifier and take the FLOOR of the product.
4) Multiply 3) by Prerequisites modifier and take the FLOOR of the product.
The formula to calculate the amount of beakers applied to your technology is:
Beakers applied to Technology = FLOOR (FLOOR ((Total Base Beakers + 1) * KCwT modifier) * Prerequisites modifier)
An Example:
I always seem to understand better with an example, so here's a basic one.
Say you want to research Code of Laws which has a base cost of 350 beakers, the mandatory requirement of Writing and the optional (meaning you must research one of them) requirements of (Priesthood or Currency).
You know Writing and Priesthood
You produce 37 net beakers per turn and you have met 3 other civilizations who know it out of 7 who started the map.
You started on a standard map (1.4 modifier) at the monarch difficulty level (1.15 modifier) at epic game speed (1.5 modifier) and have 1 teammate.
Your cost would be:
FLOOR (FLOOR (FLOOR (350 * (1.15 + 0.5 * 1)) * 1.4) * 1.5)
FLOOR (350 * 1.65) = 577
FLOOR (577 * 1.4) = 807
FLOOR (807 * 1.5) = 1210
Total cost = 1210
The amount of beakers applied to code of laws would be:
FLOOR (FLOOR ((37 + 1) * (1 + RDDW (0.30 * 3 / 7))) * (1 + 0.2 + (0.2 * 0))
1 + RDDW (0.30 * 3 / 7) = 1.12
FLOOR (38 * 1.12) = 42
FLOOR (42 * 1.2) = 50
Beakers applied to Code of Laws/turn = 50
Note that having Writing AND Priesthood only gives you a bonus modifier of 1.2 because you must have at LEAST Writing PLUS one of the optional prerequisites to be able to research Code of Laws. If you also had Currency, then the modifier would be 1.4.
The next bit in the spoiler tags is part of the article, but according to a reputable source has been fixed.
Repercussions
1) It can be VERY important to establish contact with all the other civilizations in order to speed your beakers applied count. You could potentially be making 28% (17 known civs w/ tech/ 18 civs total) more beakers than you normally could on a tech. This is especially true on harder difficulties. If you just meet half of the other civs, your research on techs known by the others would improve by 14%. This is probably the biggest repercussion of the calculation.
2) If you're going to research several technologies and one is an optional requirement of another, research the optional one first as that will boost your research toward the other one unless you have a specific reason for researching the other one.
3) If you're researching the starting techs (no prerequisites modifier), try to reduce the amount of overflow you have, as you're multiplying it by the lowest amount possible. It would be better spent earning some gold which may be necessary to run a deficit research job.
4) If you're researching a tech which is known by a lot of people and which has many optional prerequisites which you already have, try to time it so that you end a turn with 1 beaker left to research that tech. The next turn INCREASE the amount of commerce you generate in order to increase the research overflow on the next turn to the max amount possible. You could potentially generate 104.8% ((1 + 17 known civs w/ tech/18 civs) * 1.6 [tech w/ 3 optional prerequisites]) more beakers of overflow than you normally would for that tech (assuming it's from Writing to a Starting Tech which no one knows. I know, it's unlikely. ). However, for a realistic overflow difference getting almost 46% (1.25 * 1.4 / 1.2) more beakers in overflow is very feasible.
[First tech = 6 known civs w/ tech out of 7 civs and 1 additional optional prerequisite obtained]
[Second tech = no known civs with tech and no additional optional prerequisites obtained]
like Paper w/ Civil Service and Theology to Education.
5) Pre-Alphabet (and possibly in non-tech trading games), you will be able to determine the number of known players with a certain tech but not who has this specific tech unless all known or no known civs have it. (thx to DaviddesJ for pointing this out).
6) Similar to repercussion #5, if the AI is ahead of you in tech, you will be able to determine the number of known players with a technology which is BEYOND your research ability (i.e., those you do not have the prerequisite technologies for) but not who has this specific tech unless all known or no known civs have it. NOTE: there would also be times where you would NOT know whether another known civ has the tech as the civ modifier would be small enough that no difference could be detected. (thx to Roland Johanson for pointing this out.)
7) If you don't choose a tech as soon as you found your first city, you are probably losing a bonus (if you don't choose a starting tech). This might not be that significant, but when every beaker counts, losing two beakers in the beginning might mean the difference between getting that iron rush 1 turn faster.
8) If you have a neighbor who you're thinking of killing off who has techs which you have not researched yet, it might not be worth it to kill them off UNTIL you've researched the tech (1 neighbor who knows the tech in a 7 player game will provide a 1.04 boost).
Additional Modifier for knowing 1 more civilization who has the tech out of....
2 Starting Civilizations = 0.15
3 Starting Civilizations = 0.10
4 Starting Civilizations = 0.075
5 Starting Civilizations = 0.06
6 Starting Civilizations = 0.05
7 Starting Civilizations = 0.0428...
8 Starting Civilizations = 0.0375
9 Starting Civilizations = 0.0333...
10 Starting Civilizations = 0.03
11 Starting Civilizations = 0.02727...
12 Starting Civilizations = 0.025
13 Starting Civilizations = 0.0230769...
14 Starting Civilizations = 0.0214...
15 Starting Civilizations = 0.02
16 Starting Civilizations = 0.01875
17 Starting Civilizations = 0.017647...
18 Starting Civilizations = 0.01666....
Thanks to:
* LordTerror for his work dealing with the teammate modifier
* Zombie69, Roland Johanson, kryszcztov, jesusin, and LordTerror for suggesting ways to improve and/or clarify the article
* DaviddesJ for suggesting repercussion #5.
* Roland Johanson for suggesting repercussion #6.
* Arthog for his idea about dead civilizations possibly contributing to the modifier which led to repercussion #8 and for providing a save file to test the idea on.
baby! yeah!
on the last turn of 
reaserch it could be best to run at 70% to finish the tech instead of dropping to 0 
cap.
