In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887.
The Lerch zeta function is given by
A related function, the Lerch transcendent, is given by
The transcendent only converges for any real number , where:
or
and .
The two are related, as
The Lerch transcendent has an integral representation:
The proof is based on using the integral definition of the Gamma function to write
and then interchanging the sum and integral. The resulting integral representation converges for Re(s) > 0, and Re(a) > 0. This analytically continues to z outside the unit disk. The integral formula also holds if z = 1, Re(s) > 1, and Re(a) > 0; see Hurwitz zeta function.
A contour integral representation is given by
where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.
A Hermite-like integral representation is given by
for
and
for
Similar representations include
and
holding for positive z (and more generally wherever the integrals converge). Furthermore,
The last formula is also known as Lipschitz formula.
The Lerch zeta function and Lerch transcendent generalize various special functions.
The Hurwitz zeta function is the special case
The polylogarithm is another special case:
The Riemann zeta function is a special case of both of the above:
Other special cases include:
The Dirichlet eta function:
The Dirichlet beta function:
The Legendre chi function:
The polygamma function:
For λ rational, the summand is a root of unity, and thus may be expressed as a finite sum over the Hurwitz zeta function. Suppose with and . Then and .
Various identities include:
and
and
A series representation for the Lerch transcendent is given by
(Note that is a binomial coefficient.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, ... by This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ(s,1). The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882. The Hurwitz zeta function has an integral representation for and (This integral can be viewed as a Mellin transform.
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Li_s(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral.
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. This function is strictly increasing and strictly concave on , and it asymptotically behaves as for large arguments () in the sector with some infinitesimally small positive constant . The digamma function is often denoted as or Ϝ (the uppercase form of the archaic Greek consonant digamma meaning double-gamma).
P-adic numbers are a number theoretic analogue of the real numbers, which interpolate between arithmetics, analysis and geometry. In this course we study their basic properties and give various applic
Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calcul ...
Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, an ...
We study the elliptic curves given by y(2) = x(3) + bx + t(3n+1) over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve u(3) ...