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A complete solution of Mumford's second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found. An explicit formula for computing such an expression for theta derivative with an arbitrary rational characteristic is derived, and illustrated with examples. This approach fails only on four characteristics: 1 / 2 1 / 6 , 1 / 6 1 / 2 , 1 / 6 1 / 6 and 1 / 6 5 / 6 ; however expressions for theta derivatives with these four characteristics are found by other authors. Expressions for theta derivatives appear to be homogeneous of degree 3 with respect to theta constants. [ABSTRACT FROM AUTHOR]

In this paper, we deduce the asymptotic formulas of parametric digamma function Ψ (− s ; a) at the integers and poles. Then using these identities and residue theorem, we establish a large number of formulas of double series involving parametric harmonic numbers. Some illustrative special cases as well as immediate consequences of the main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, using an elementary algebraic-combinatorial approach, we derive expressions that allow us for a given Markoff-Hurwitz equation to find the number of its solutions over ℤ/pkℤ if the number of solutions over ℤ/pℤ is known. We also calculate the corresponding Poincaré series. [ABSTRACT FROM AUTHOR]

We study the number of divisors in residue classes modulo m and prove, for example, that the exact equidistribution holds for almost all natural numbers coprime to m in the sense of natural density if and only if m = 2kp1p2...ps, where k and s are non-negative integers and pj are distinct Fermat primes. We also provide a general and exact lower bound for the proportion of divisors in the residue class 1 mod m. The same combinatorial technique using Davenport's constant leads to exact lower bounds for the number of representations of a natural number by a given binary quadratic form with a negative discriminant. [ABSTRACT FROM AUTHOR]