The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

Euler is the only mathematician to have two numbers named after him: the immensely important Euler's Number in calculus, e, approximately equal to 2. 71828, and the Euler-Mascheroni Constant γ ( gamma ) sometimes referred to as just " Euler's constant ", approximately equal to 0. 57721.

It was not until the 18th century that Leonhard Euler proved that the formula 2 < sup > p − 1 </ sup >( 2 < sup > p </ sup >− 1 ) will yield all the even perfect numbers.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms.

The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity

where, by definition, the left hand side is ζ ( s ) and the infinite product on the right hand side extends over all prime numbers p ( such expressions are called Euler products ):

The name " transcendental " comes from Leibniz in his 1682 paper where he proved sin x is not an algebraic function of x. Euler was probably the first person to define transcendental numbers in the modern sense.

In number theory, the Euler numbers are a sequence E < sub > n </ sub > of integers defined by the following Taylor series expansion:

The Euler numbers appear as a special value of the Euler polynomials.

The odd-indexed Euler numbers are all zero.

Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and / or change all signs to positive.

An explicit formula for Euler numbers is given by:

The Euler numbers grow quite rapidly for large indices as

* Euler numbers calculator

These last four attempts assumed implicitly Girard's assertion ; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was a + bi for some real numbers a and b. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p ( z ).

* Euler numbers

There are no numbers which are Euler – Jacobi pseudoprimes to all bases as Carmichael numbers are.

He calculated the Euler – Mascheroni constant, perhaps somewhat eccentrically, to 236 decimal places and evaluated the Bernoulli numbers up to the 62nd.

It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base relatively prime to themselves.

This explains why, for example, the Euler angles appear to give a variable in the 3-torus, and the unit quaternions in a 3-sphere.

For comparison, in the equivalent Euler – Lagrange equations of motion of Lagrangian mechanics, the conjugate momenta also do not appear ; however, those equations are a system of N, generally second-order equations for the time evolution of the generalized coordinates.

One cannot use this equation to find as one does not know at The solution is then to use a Taylor series expansion exactly as if using the Euler method to solve for:

* Project Euler a series of challenging mathematical / computer programming problems

Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.

In the context of computing asymptotic expansions of sums and series, usually the most useful form of the Euler – Maclaurin formula is

Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

Notably, Euler directly proved the power series expansions for and the inverse tangent function.

Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of, and were used by mathematicians including Madhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.

The Riemann zeta function or Euler – Riemann zeta function, ζ ( s ), is a function of a complex variable s that analytically continues the sum of the infinite series which converges when the real part of s is greater than 1.

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to real s > 1.

The Meissel – Mertens constant is analogous to the Euler – Mascheroni constant, but the harmonic series sum in its definition is only over the primes rather than over all integers and the logarithm is taken twice, not just once.

The Fourier series is named in honour of Joseph Fourier ( 1768 – 1830 ), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d ' Alembert, and Daniel Bernoulli.

The Dirichlet series generating function is especially useful when a < sub > n </ sub > is a multiplicative function, when it has an Euler product expression in terms of the function's Bell series

Amongst the fruits of his industry may be mentioned a laborious investigation of the disturbances of Jupiter by Saturn, the results of which were employed and confirmed by Euler in his prize essay of 1748 ; a series of lunar observations extending over fifty years ; some interesting researches in terrestrial magnetism and atmospheric electricity, in the latter of which he detected a regular diurnal period ; and the determination of the places of a great number of stars, including at least twelve separate observations of Uranus, between 1750 and its discovery as a planet.

Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.

In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function.

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.

The Euler product attached to the Riemann zeta function, using also the sum of the geometric series, is

where ƒ is a smooth function, you could use the Euler – Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S ( x ), then for large values of a you could use " stationary phase " method to calculate the integral and give an approximate evaluation of the sum.

More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime p is the inverse of the Hecke polynomial, a quadratic polynomial in p < sup >− s </ sup >.

The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron.