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The left cosets ${\Gamma }_{\infty }\backslash \Gamma (1)$ correspond to primitive bottom rows $(c,d)\in {\mathbb{Z}}^2$, up to simultaneous sign. PutFor $\tau =x+iy$, the identitiesshow that the absolute value of a summand isThe hypothesis $\mathrm{Re}s>(2-k)/2$ says precisely that $r>2$.

If $\tau$ ranges over a compact subset $K\subset \mathfrak{h}$, the positive-definite quadratic form $|c\tau +d{|}^2$ has a uniform lower boundfor some $C_K>0$. The summands are therefore bounded uniformly on $K$ by a constant timesThe corresponding two-dimensional lattice sum converges for $r>2$. The Weierstrass M-test proves absolute and locally uniform convergence. This is the absolute convergence of a weight-k real-analytic Eisenstein series.

For $\sigma =\Re s>1$,On every compact subset of $\Re s>1$, the terms are bounded by a convergent series $\sum n^{-1-\delta }$. The Weierstrass M-test gives locally uniform convergence, and the theorem on locally uniform convergence of holomorphic functions shows that the limit is an analytic function. Hence $D_f$ is analytic throughout $\Re s>1$.

For positive summable coefficients, the Weierstrass M-test gives a continuous uniform sum. The partial sum over $3^k\le n$ is the unique best uniform approximation of degree at most $n$. If $3^K$ is the first omitted frequency, every tail term equals $(-1{)}^j$ at $x_j=\cos (j\pi /3^K)$ because its frequency ratio is odd. The entire tail therefore attains alternating extrema equal in magnitude to its coefficient sum at more than $n+1$ points. The Chebyshev alternation theorem proves the assertion, including the empty partial sum at $n=0$.