Integrals, series,
and the ideas between.

Generating functions for squared harmonic numbers provide evaluations of Euler sums and logarithmic integrals.

A contour-integral approach to an infinite trigonometric series, with an application of equidistribution.

An alternating Euler sum leads to harmonic-number generating functions, logarithmic integrals, and polylogarithms.

Use Ramanujan's master theorem to connect power-series coefficients with Mellin transforms of Bessel functions.

Derive Kummer's Fourier series for the logarithm of the gamma function, then use it to evaluate Vardi's integral.

From the defining recursion for Bernoulli polynomials to their Fourier expansions and an associated integral.

Weyl's criterion connects uniform distribution modulo one with exponential sums and averages of Riemann-integrable functions.

Eisenstein's trigonometric proof of quadratic reciprocity, starting with Euler's criterion and Gauss's lemma.

Two incomplete beta integrals, evaluated through substitutions and the addition formulas for the Weierstrass elliptic function.