# Volume 30 - 2007

### 1. Carmichael number with three prime factors.

Let $C_3(x)$ be the number of Carmichael numbers $n\le x$ having exactly 3 prime factors. It has been conjectured that $C_3(x)$ is of order $x^{1/3}(\log x)^{-1/3}$ exactly. We prove an upper bound of order $x^{7/20+\varepsilon}$, improving the previous best result due to Balasubramanian and Nagaraj, in which the exponent $7/20$ was replaced by $5/14$. The proof combines various elementary estimates with an argument using Kloosterman fractions, which ultimately relies on a bound for the Ramanujan sum.

### 2. The number of imaginary quadratic fields with a given class number

This paper formulates some conjectures for the number of imaginary quadratic fields of a given class number. It establishes an asymptotic formula for the number of such fields with class number below $H$, and also shows that many fields have class numbers lying outside a very thin set.

### 3. Rational triangles with the same perimeter and the same area.

This paper is concerned with the problem of finding two or more rational triangles with the same perimeter and the same area. As the problem of finding two isosceles rational triangles with the same perimeter and same area has already been solved, in this paper we obtain several parametric solutions to the problem of finding a pair of rational triangles with the same area and perimeter when at least one of the triangles is scalene. We also show that, with certain exceptions, given an arbitrary scalene rational triangle, a second scalene rational triangle with the same perimeter and same area may be constructed, and by repeated application of this process, we may obtain an arbitrarily large number of scalene rational triangles with the same perimeter and same area.

### 4. Contributions to the theory of the Hurwitz zeta-function

We give various contributions to the theory of Hurwitz zeta-function. An elementary part is the argument relating to the partial sum of the defining Dirichlet series for it; how much can we retrieve the whole from the part. We also give the sixth proof of the far-reaching Ramanujan -- Yoshimoto formula, which is a closed form for the important sum $\sum^\infty_{m=2} \frac{\zeta(m,\alpha)}{m+\lambda} z^{m+\lambda}$. This proof, incorporating the structure of the Hurwitz zeta-function as the principal solution of the difference equation, seems one of the most natural ones. The formula may be applied to deduce almost all formulas in H.~M.~Srivastava and J.~Choi. The same is applied to obtain closed form for the integral of the Euler psi function and give Espisona-Moll results. %In this paper we shall give various contributions to the theory of the Hurwitz zeta-function. In \S1 we shall continue our previous study and give integral representations (for the derivatives as well) which give another basis of the theory of gamma and related functions. In \S2 we shall give the sixth proof of the Ramanujan formula with two examples which supersede those results presented in the book of Srivastava and Choi. In \S3 we shall give two more proofs of the closed formula for the integral of the psi-function, thus recovering the recent result of Episona and Moll. Finally, in \S4 we shall give another proof of the functional equation. Hereby we put all existing literature in the hierarchical and […]

### 5. On the Barban-Davenport Halberstam theorem: XIX

The present paper deals with the dispersion $G(x,Q)=\sum_{k\leq Q}\sum_{0<a\leq k}\{S(x;a,k)-f(a,k)x\}^2$ for large $Q$, and improves the lower bound by proving that $G(x,Q)>\frac{1}{12}\{\Gamma(C)+o(1)\}Q^2+O(x \log_{-A}x)$ when $x/Q\rightarrow\infty$ where $\Gamma(C)$ is an explicitly defined function of $C$.