We study the moments of a general number field, completely solving the moment problem upto the leading term in the asymptotic.

We study the shifted convolution sums associated to completely multiplicative functions taking values in {pm 1} and give combinatorical proofs of two recent results in the direction of Chowla's conjecture. We also determine the corresponding "spectrum"

Motivated by weighted partitions of n that vanish if and only if n is a prime, Craig, van Ittersum, and Ono conjectured a classification of quasimodular forms which detect primes in the sense that the n-th Fourier coefficient vanishes if and only if n is a prime. In this paper, we prove this conjecture by showing that Fourier coefficients of quasimodular cusp forms exhibit infinitely many sign changes.

Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ0(N) with normalized eigenvalues λf (n) and let X > 1 be a real number. We consider the sum and show that for every k ⩾ 1 and ε > 0. The same problem was considered for the case N = 1, that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k ⩾ 5. Since the result is valid for arbitrary level, we obtain, as a corollary, estimates on sums of the form , where the sum involves restricted coefficients of some suitable half integral weight modular forms.

Let K be a number field over and let denote the number of integral ideals of K of norm equal to . In this paper we obtain asymptotic formulae for sums of the form thereby generalizing the previous works on the problem. Previously such asymptotics were known only in the case when K is Galois or when K was a non normal cubic extension and . The present work subsumes both these cases.

We study the class numbers of imaginary quadratic fields and connect them to certain digits in the base n expansions of certain rational numbers.

We study the sign changes in restricted coefficients in Hilbert modular forms for an arbitrary number field. We consider sign changes in arithmetic progressions and square-free integral ideals.