Distribution of values of Fourier coefficients for modular forms of weight 1

On cusp forms associated with binary theta series

“…R e m a r k s. 1) Theorem 1 extends the result T 1 (f, x) x(log x) γ (1)−1 of Fomenko [4]. 2) It is easy to see that X f = ∅ if and only if either C ∼ = (Z/2Z) k for some k ∈ N 0 or C ∼ = (Z/2Z) k × (Z/4Z) and C f is not ambiguous (i.e.…”

Principal forms X 2+nY 2 representing many integers

“…(3) We let E be the set of integers that are represented by G, but not by f , and E(x) the associated counting function (for example, in the above example 2 and 7 are represented by G 1 , but not by [1,0,14]). Bernays showed that E(x) = o(x log −1/2 x) (this result was later sharpened by Fomenko [7] to E(x) ≪ x log −2/3 x). We finally conclude that…”

“…If D = −4n, then [1, 0, n] is the principal form. Finally, we say that an integer m is represented by the genus G if it is represented by at least one class in G. For example, if n = 14 (and thus D = −56 and h(D) = 4), then we have two genera G 1 and G 2 where, say, (the class of) [1,0,14] and [2,0,7] belong to G 1 while [3, −2, 5] and [3,2,5] belong to G 2 .…”

“…We prove (7) by constructing a set Σ = Σ(x) of primes q ≤ x, q ≡ c (mod d), with S = |Σ| and showing…”

“…We prove (7) by constructing a set Σ = Σ(x) of primes q ≤ x, q ≡ c (mod d), with S = |Σ| and showing x ≤ q ≤ x, q ≡ l (mod k)}. Then every q ∈ Σ satisfies q ≡ c (mod d) and χ(p i ) = 1 for i = r since…”