We prove that the Ramanujan integral I_R(x)=∫ _0^∞e^-xt dt/t(π ^2+log ^2t), x>0, is a Stieltjes function by finding an explicit, strictly positive inverse Laplace transform of its kernel. Its normalized primitive is a complete Bernstein function, answering two questions of Mishra and Swaminathan [5]. We also establish determinate moment representations and strict Hankel determinant inequalities for the signed derivatives. For the associated Turán expression, we distinguish the optimal non-negativity threshold from the strictly smaller complete-monotonicity threshold, give an exact convolution characterization of the latter, and improve its explicit lower bound. The normalized thresholds have the same limit, c_∞ =0.6718937116849245… , which we evaluate by radicals over ℚ(π ) and prove to be transcendental. Uniform weighted-gamma concentration controls the global infima and gives the asymptotic location of every positivity minimizer.
We determine Fourier coefficients of the Siegel–Eisenstein series E_m+1^(2m)(Z) in the boundary weight m+1 . The constant term is calculated for all odd m≥ 3 , whereas the non-zero index terms are treated for m≡ 1 4 and m≥ 5 . In the latter case the series is not holomorphic in general, and its Fourier expansion contains coefficient functions depending non-trivially on the imaginary part of Z . Starting from Mizumoto’s Fourier expansion formula, we isolate all terms that can contribute to the value at s=0 , calculate the constant term, and describe the exceptional non-zero Fourier coefficients. The constant term is expressed in terms of the Koecher–Maass zeta function of rank one and the determinant of the imaginary part. For non-zero indices, the possible exceptional contributions are shown to arise only from a finite list of ranks, together with the possible pole of the non-degenerate Siegel series when the discriminant is a square. The middle-rank terms that occur in degree 6 are treated separately and shown to cancel. These formulas give higher-degree analogues of the known degree-2 computations for the weight-two Siegel–Eisenstein series.
Let K=ℚ(√(D)) be a real quadratic field, where D>1 is squarefree, and let p_K denote the partition function on totally positive integers of K. We give an elementary explicit construction which realizes any prescribed value of the partition function on infinite families of real quadratic fields. More precisely, for every integer R≥ 1 , if the fractional part of the standard parameter ξ _D=-ω _D' satisfies {ξ _D} <1/R , then α _R=(⌈ Rξ _D⌉ +1)+Rω _D =(⌊ Rξ _D⌋ +2)+Rω _D has exactly ⌊ R/2⌋ +2 partitions. Consequently, for every integer m≥ 2 , there are infinitely many real quadratic fields K for which one can explicitly write down an element α∈𝒪_K^+ with p_K(α )=m . The construction gives a simple coefficient-counting method which produces concrete elements and complete lists of partitions.
For a numerical semigroup S, we denote by F (S) the Frobenius number of S. For numerical semigroups generated by modified arithmetic sequences A, we characterize (i) a ∈ A for which F(⟨ A ∖{a}⟩ )=F(⟨ A ⟩ ) , and (ii) characterize 3-subsets A^' of A for which F(⟨ A^'⟩ )=F(⟨ A ⟩ ) , and explore some 4-subsets and some subsets of cardinality roughly √(k) for some choices of k for which the Frobenius number is the same as that of A.
Cho, Huh, Nam, and Sohn gave explicit formulas for cc(t, m) and scc(t, m), the numbers of t-core partitions and of self-conjugate t-core partitions with m corners, respectively. In this paper, we establish congruences for these two quantities. For an odd prime p, we completely determine cc(p, m) modulo p and scc(2p,m)=scc(2p+1,m) modulo 2p, and we evaluate scc(t, 2p) and scc(t,2p+1) modulo p for every t.
