We consider the space X of all analytic functionsof two complex variables s1 and s2, equipping it with the natural locally convex topology and using the growth parameter, the order of f as defined recently by the auth...We consider the space X of all analytic functionsof two complex variables s1 and s2, equipping it with the natural locally convex topology and using the growth parameter, the order of f as defined recently by the authors. Under this topology X becomes a Frechet space Apart from finding the characterization of continuous linear functionals, linear transformation on X, we have obtained the necessary and sufficient conditions for a double sequence in X to be a proper bases.展开更多
Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example, may be defined so that the iterated limits and exist and are equal for all x, and ye...Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example, may be defined so that the iterated limits and exist and are equal for all x, and yet the Pringsheim limit does not exist. The sequence is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an everywhere discontinuous limit. We explore whether the limit of this sequence in the Pringsheim sense equals the iterated result and derive an interesting property of cosines as a byproduct.展开更多
The Cauchy integral formula expresses the value of a function f(z), which is analytic in a simply connected domain D, at any point z0 interior to a simple closed contour C situated in D in terms of the values of on C....The Cauchy integral formula expresses the value of a function f(z), which is analytic in a simply connected domain D, at any point z0 interior to a simple closed contour C situated in D in terms of the values of on C. We deal in this paper with the question whether C can be the boundary ∂Ω of a fundamental domain Ω of f(z). At the first look the answer appears to be negative since ∂Ω contains singular points of the function and it can be unbounded. However, the extension of Cauchy integral formula to some of these unbounded curves, respectively arcs ending in singular points of f(z) is possible due to the fact that they can be obtained at the limit as r → ∞ of some bounded curves contained in the pre-image of the circle |z| = r and of some circles |z-a| = 1/r for which the formula is valid.展开更多
Let p and q be two distinct primes, epq（n） denotes the largest exponent of power pq which divides n. In this paper, we study the mean value properties of function epq（n）, and give some hybrid mean value formulas f...Let p and q be two distinct primes, epq（n） denotes the largest exponent of power pq which divides n. In this paper, we study the mean value properties of function epq（n）, and give some hybrid mean value formulas for epq（n） and Dirichlet divisor function d（n）. Key words： largest exponent; asymptotic formula; hybrid mean value; Dirichlet divisor function d（n）展开更多
In this paper, some conclusions related to the prime number theorem, such as the Mertens formula are improved by the improved Abelian summation formula, and some problems such as “Dirichlet” function and “W(n)” fu...In this paper, some conclusions related to the prime number theorem, such as the Mertens formula are improved by the improved Abelian summation formula, and some problems such as “Dirichlet” function and “W(n)” function are studied.展开更多
The eta function is examined over the critical strip 0σ1and there is an investigation of the statement that all zeros of the zeta function must lie on the critical line Re(s)=1/2. A further investigation is made into...The eta function is examined over the critical strip 0σ1and there is an investigation of the statement that all zeros of the zeta function must lie on the critical line Re(s)=1/2. A further investigation is made into the claim that there are no other zeros of the zeta or eta functions within the critical strip.展开更多
文摘We consider the space X of all analytic functionsof two complex variables s1 and s2, equipping it with the natural locally convex topology and using the growth parameter, the order of f as defined recently by the authors. Under this topology X becomes a Frechet space Apart from finding the characterization of continuous linear functionals, linear transformation on X, we have obtained the necessary and sufficient conditions for a double sequence in X to be a proper bases.
文摘Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example, may be defined so that the iterated limits and exist and are equal for all x, and yet the Pringsheim limit does not exist. The sequence is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an everywhere discontinuous limit. We explore whether the limit of this sequence in the Pringsheim sense equals the iterated result and derive an interesting property of cosines as a byproduct.
文摘The Cauchy integral formula expresses the value of a function f(z), which is analytic in a simply connected domain D, at any point z0 interior to a simple closed contour C situated in D in terms of the values of on C. We deal in this paper with the question whether C can be the boundary ∂Ω of a fundamental domain Ω of f(z). At the first look the answer appears to be negative since ∂Ω contains singular points of the function and it can be unbounded. However, the extension of Cauchy integral formula to some of these unbounded curves, respectively arcs ending in singular points of f(z) is possible due to the fact that they can be obtained at the limit as r → ∞ of some bounded curves contained in the pre-image of the circle |z| = r and of some circles |z-a| = 1/r for which the formula is valid.
文摘Let p and q be two distinct primes, epq（n） denotes the largest exponent of power pq which divides n. In this paper, we study the mean value properties of function epq（n）, and give some hybrid mean value formulas for epq（n） and Dirichlet divisor function d（n）. Key words： largest exponent; asymptotic formula; hybrid mean value; Dirichlet divisor function d（n）
文摘In this paper, some conclusions related to the prime number theorem, such as the Mertens formula are improved by the improved Abelian summation formula, and some problems such as “Dirichlet” function and “W(n)” function are studied.
文摘The eta function is examined over the critical strip 0σ1and there is an investigation of the statement that all zeros of the zeta function must lie on the critical line Re(s)=1/2. A further investigation is made into the claim that there are no other zeros of the zeta or eta functions within the critical strip.
