The Euler-Mascheroni constant, also known as Euler's constant or simply "gamma," is a constant that appears in many problems in analytic number theory and calculus. It is denoted by and the first few digits of this constant are as follows:
Gamma is strongly related to the natural logarithm function and the harmonic numbers, and is often defined in these terms. There is no closed form expression for the harmonic number, but gamma can be used to give an estimate of the harmonic number. Beyond this, the applications of gamma in mathematics and practical problems are wide and varied.
For as much as gamma has been studied and applied to problems in mathematics, not much is known about the properties of the number itself. It is not known if gamma is algebraic or transcendental. It is not even known if gamma is irrational like other important mathematical constants such as and
Harmonic numbers are a sequence of numbers formed by summing the reciprocals of positive integers.
Let be the harmonic number, then
can also be defined by the recurrence relation
Gamma is most commonly defined as the limit of the difference between the harmonic number and the natural logarithm of as approaches infinity.
Let be the Euler-Mascheroni constant, otherwise known as Euler's constant. It is defined as follows:
In integral notation,
Due to gamma's relation to harmonic numbers, it is used in many problems that require an estimation of a harmonic number. There is no closed form expression for the harmonic number, so it is necessary to have an accurate estimation that can be computed efficiently.
For relatively large
Note: Although converges to , the addition of gives a faster convergence, and thus a more accurate estimation.
Harmonic numbers are applicable in some famous mathematics problems:
Harmonic numbers are also applicable in some practical problems.
The amount of rain that falls in a certain town over the course of a year is recorded every year for 100 years. If the amount of rain each year is uniformly random, how many record rainfalls would you expect to see over that time period?
Assuming that the amount of rainfall every year is uniformly random,
- the year is guaranteed to set a record, because it is the only year recorded,
- the year has a chance to be higher than the 1st year,
- the year has a chance to be higher than the other 2 years,
- the year has a chance to be higher than the other 3 years,
- and so on.
Calculating the expected value of the number of record rainfalls involves summing these probabilities together, which gives the harmonic number. This can be estimated with the formula above:
Therefore, you would expect to see approximately record rainfalls over that time period.
Your company manufactures elevator cables. You are attempting to discover the maximum weight that one of your cables could safely hold. To do this, you test cables in the following way:
You gradually increase the weight on a cable until it breaks, then record the weight that broke the cable as .
You gradually increase the weight on the second cable to If the cable breaks before the weight reaches then you record this new maximum safe weight as Otherwise, if the second cable can hold , then you begin testing the next cable.
Each time a cable breaks under a lesser weight, you record this as the new maximum safe weight. You continue testing cables in this way until you have tested 500 cables.
If the weight that an individual cable can hold is uniformly random, how many cables would you expect to break during this testing process?
Assuming that the maximum weight a cable can hold is uniformly random,
- the cable will always be loaded to its maximum weight,
- the cable will have a chance to have a lesser maximum weight than the cable,
- the cable will have a chance to have a lesser maximum weight than the other 2 cables,
- the cable will have a chance to have a lesser maximum weight than the other 3 cables,
- and so on.
Calculating the expected value of the number of broken cables involves summing these probabilities together, which gives the harmonic number. This can be estimated with the formula above:
Therefore, you would expect to break approximately of the cables with this process.
Gamma is not just used in problems involving harmonic numbers. Like and , it finds a place in many areas of mathematics. The following are a few of the places in which gamma makes an appearance:
Gamma is used in the Laplace transform of the natural logarithm function.
Laplace transform of the natural logarithm function:
Digamma function's relation to harmonic numbers:
There are a number of other identities which contain the digamma function and gamma, and these can be seen on the linked page above.
Trigonometric integrals are functions based on integrals that involve trigonometric functions. The cosine integrals are related by gamma.
The cosine integral functions are defined as follows:
These functions are related by gamma:
Gamma is used to give bounds for the growth of the divisor function.
Growth rate of the divisor function:
where is the divisor function and is big O notation.
In other words, if one were to count the number of divisors for each positive integer up to and then sum those counts together, then the resulting sum, would satisfy the following inequality:
Show that the inequality above holds true for
A table with the values of the divisor function for all is given below:
The sum of all these values is
This is more than but it is less than
Gamma is also part of an inequality that gives an upper bound for the sum of divisors function.
Upper bound for sum of divisors function:
Assuming that the Riemann hypothesis is true, the sum of divisors function has an upper bound for sufficiently large :
Without the assumption that the Riemann hypothesis is true, the upper bound for the sum of divisors function is slightly more conservative (also for sufficiently large ):
where is the sum of divisors function.