linear combinations of Dirichlet series whose coefficients are terms of the Thue-Morse sequence give rise to identities involving the Riemann Zeta function (Tóth, 2022 ). For instance: where is the term of the Thue-Morse sequence. In fact, for all with real part greater than , we have Numerical algorithms A classical algorithm, in use prior to about 1930, proceeds by applying the Euler-Maclaurin formula to obtain, for n and m positive integers, where, letting denote the indicated Bernoulli number, and the error satisfies with σ = Re(s). A modern numerical algorithm is the Odlyzko–Schönhage algorithm. Applications The zeta function occurs in applied statistics (see Zipf's law and Zipf–Mandelbrot law). Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. In one notable example, the Riemann zeta function shows up explicitly in one method of calculating the Casimir effect. The zeta function is also useful for the analysis of dynamical systems. Musical Tuning In the theory of musical tunings, the zeta function can be used to find equal divisions of the octave (EDOs) that closely approximate the intervals of the harmonic series. For increasing values of , the value of peaks near integers that correspond to such EDOs. Examples include popular choices such as 12, 19, and 53. Infinite series The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants. In fact the even and odd terms give the two sums and Parametrized versions of the above sums are given by and with and where and are the polygamma function and Euler's constant, respectively, as well as all of which are continuous at . Other sums include where denotes the imaginary part of a complex number. There are yet more formulas in the article Harmonic number. Generalizations There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function. These include the Hurwitz zeta function (the convergent series representation was given by Helmut Hasse in 1930, cf. Hurwitz zeta function), which coincides with the Riemann zeta function when (the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet -functions and the Dedekind zeta function. For other related functions see the articles zeta function and -function. The polylogarithm is given by which coincides with the Riemann zeta function when