For a vector $\mathbf a=(a_1,\ldots,a_r)$ of positive integers we prove formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ and its polynomial part.Comment: 21 pages, to appear in The Ramanujan Journa

Let {K/\mathbb{Q}} be a finite Galois extension. Let {\chi_{1},\ldots,\chi_{r}} be {r\geq 1} distinct characters of the Galois group with the associated Artin L-functions {L(s,\chi_{1}),\ldots,L(s,\chi_{r})}. Let {m\geq 0}. We prove that the derivatives {L^{(k)}(s,\chi_{j})}, {1\leq j\leq r}, {0\leq k\leq m}, are linearly independent over the field of meromorphic functions of order {<1}. From this it follows that the L-functions corresponding to the irreducible characters are algebraically independent over the field of meromorphic functions of order {<1}.

Let KaQ be a ®nite Galois extension. For the character w of a representation of the Galois group G X GalKaQ on a ®nite dimensional complex vector space, let LsY wY KaQ be the corresponding Artin L-function ([2], p. 296). Looking for multiplicative relations between the Dedekind zeta functions of the sub®elds of K, Artin discovered ([1], Satz 5, p. 106) that between the L-functions to the irreducible characters of G does not exist any multiplicative relation. For abelian extensions KaQ the Artin L-functions to the irreducible characters, i.e. Dirichlet L-functions, were proved by Voronin to be functionally independent over the ®eld C of complex numbers ([4]).

We show that 1, 2 and 3 are the only Fibonacci numbers whose Euler functions are also Fibonacci numbers.

Let K/Q be a finite Galois extension, s 0 ∈ C \ {1}, Hol(s 0 ) the semigroup of Artin L-functions holomorphic at s 0 . If the Galois group is almost monomial then Artin's L-functions are holomorphic at s 0 if and only if Hol (s 0 ) is factorial. This holds also if s 0 is a zero of an irreducible L-function of dimension ≤ 2, without any condition on the Galois group.

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AbstractIf {K/\mathbb{Q}} is a finite Galois extension with an almost monomial Galois group and if {s_{0}\in\mathbb{C}\setminus\{1\}} is not a common zero for any two Artin L-functions associated to distinct complex irreducible characters of the Galois group, then all Artin L-functions of {K/\mathbb{Q}} are holomorphic at {s_{0}}. We present examples and basic properties of almost monomial groups.

Let f (z) = ∞ n=1 a(n)e 2πinz be a normalized Hecke eigenform in S new 2k (Γ 0 (N )) with integer Fourier coefficients. We prove that there exists a constant C(f ) > 0 such that any integer is a sum of at most C(f ) coefficients a(n). It holds C(f ) ≪ ε,k N 6k−3 16 +ε .