Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form \(F(t)=\sum _{n=1}^\infty f(n)e^{2\pi i nt}/n\) , for general coefficient functions f. Our main result states that if, for some constants C and \(\alpha\) with \(0<\alpha <1\) , we have \(|\sum _{1\le n\le x}f(n)e^{2\pi i nt}|\le C x^{\alpha }\) uniformly in \(x\ge 1\) and \(t\in \mathbb R\) , then the series F(t) is Hölder continuous with exponent \(1-\alpha\) , and the graph of |F(t)| on the interval [0, 1] has box-counting dimension \(\le 1+\alpha\) . As applications we recover the best-possible uniform Hölder exponents for the Weierstrass functions \(\sum _{k=1}^\infty a^k\cos (2\pi b^k t)\) and the Riemann function \(\sum _{n=1}^\infty \sin (\pi n^2 t)/n^2\) . Moreoever, under the assumption of the Generalized Riemann Hypothesis, we obtain nontrivial bounds for Hölder exponents and dimensions associated with series of the form \(\sum _{n=1}^\infty \mu (n)e^{2\pi i n^kt}/n^k\) , where \(\mu\) is the Möbius function.
