ARITHMETICAL FOURIER SERIES AND THE MODULAR RELATION

We consider the zeta functions satisfying the functional equation with multiple gamma factors and prove a far-reaching theorem,an intermediate modular relation, which gives rise to many (including many of the hitherto found) arithmetical Fourier series as a consequence of the functional equation.Typical examples are the Diophantine Fourier series considered by Hardy and Littlewood and one considered by Hartman and Wintner, which are reciprocals of each other, in addition to our previous work. These have been thoroughly studied by Li, Ma and Zhang. Our main contribution is to the effect that the modular relation gives rise to the Fourier series for the periodic Bernoulli polynomials and Kummer’s Fourier series for the log sin function, thus giving a foundation for a possible theory of arithmetical Fourier series based on the functional equation.