Assuming a conjecture on distinct zeros of Dirichlet $L$-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of $L$-functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

Two results related to the mixed joint universality for a polynomial Euler product $\unicode[STIX]{x1D711}(s)$ and a periodic Hurwitz zeta function $\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$, when $\unicode[STIX]{x1D6FC}$ is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.

We study the values of the zeta-function of the root system of type G2 at positive integer points. In our previous work we considered the case when all integers are even, but in the present paper we prove several theorems which include the situation when some of the integers are odd. The underlying reason why we may treat such cases, including odd integers, is also discussed.

The Cosmic Infrared Background ExpeRiment (CIBER) is a rocket-borne absolute photometry imaging and spectroscopy experiment optimized to detect signatures of first-light galaxies present during reionization in the unresolved IR background. CIBER-I consists of a wide-field two-color camera for fluctuation measurements, a low-resolution absolute spectrometer for absolute EBL measurements, and a narrow-band imaging spectrometer to measure and correct scattered emission from the foreground zodiacal cloud. CIBER-I was successfully flown in February 2009 and July 2010 and four more flights are planned by 2014, including an upgrade (CIBER-II). We propose, after several additional flights of CIBER-I, an improved CIBER-II camera consisting of a wide-field 30 cm imager operating in 4 bands between 0.5 and 2.1 microns. It is designed for a high significance detection of unresolved IR background fluctuations at the minimum level necessary for reionization. With a FOV 50 to 2000 times larger than existing IR instruments on satellites, CIBER-II will carry out the definitive study to establish the surface density of sources responsible for reionization.

We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions from which some generalizations of the classical sum formula can be deduced.

In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types A2, A3, B2, B3 and C3. In this paper, we consider the case of G2-type. We define certain analogues of Bernoulli polynomials of G2-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of G2-type. Next, we consider the meromorphic continuation of the zeta-function of G2-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.

A general mean value theorem for Dirichlet series, with a sharp error estimate near the boundary of the critical strip, is proved. Applications of this theorem to various automorphic $L$-functions are discussed. Moreover, sharp upper bounds of mean square values of $L$-functions are obtained when they are attached to lifted forms, such as Doi–Naganuma and Ikeda lifts in the theory of Siegel modular forms.

As the first step of research on functional equations for multiple zeta-functions, we present a candidate of the functional equation for a class of two variable double zeta-functions of the Hurwitz–Lerch type, which includes the classical Euler sum as a special case.

The present paper contains three main results. The first is asymptotic expansions of Barnes double zeta-functions, and as a corollary, asymptotic expansions of holomorphic Eisenstein series follow. The second is asymptotic expansions of Shintani double zeta-functions, and the third is the analytic continuation of n-variable multiple zeta-functions (or generalized Euler-Zagier sums). The basic technique of proving those results is the method of using the Mellin-Barnes type of integrals.

Let $\zeta(s, \alpha)$ be the Hurwitz zeta function with parameter $\alpha$ . Power mean values of the form $\sum^q_{a=1}\zeta(s,a/q)^h$ or $\sum^q_{a=1}|\zeta(s,a/q)|^{2h}$ are studied, where $q$ and $h$ are positive integers. These mean values can be written as linear combinations of $\sum^q_{a=1}\zeta_r(s_1,\ldots,s_r;a/q)$ , where $\zeta_r(s_1,\ldots,s_r;\alpha)$ is a generalization of Euler–Zagier multiple zeta sums. The Mellin–Barnes integral formula is used to prove an asymptotic expansion of $\sum^q_{a=1}\zeta_r(s_1,\ldots,s_r;a/q)$ , with respect to $q$ . Hence a general way of deducing asymptotic expansion formulas for $\sum^q_{a=1}\zeta(s,a/q)^h$ and $\sum^q_{a=1}|\zeta(s,a/q)|^{2h}$ is obtained. In particular, the asymptotic expansion of $\sum^q_{a=1}\zeta(1/2,a/q)^3$ with respect to $q$ is written down.

The main object of this paper is the mean square Ih(s) of higher derivatives of Hurwitz zeta functions ζ(s, α). We shall prove asymptotic formulas for Ih(1/2 + it) as t → +∞ with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for Ih(1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for Ih(1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product ζ(u, α)ζ(v, α) with the independent complex variables u and v.

Refined expressions are given for the error terms in the asymptotic expansion formulas for double zeta and double gamma functions, proved in the author's former paper [2]. Some inaccurate claims in [2] are corrected.

The joint universality theorem for Lerch zeta-functions L(λl, αl, s) (1 ≤ l ≤ n) is proved, in the case when λls are rational numbers and αls are transcendental numbers. The case n = 1 was known before ([12]); the rationality of λls is used to establish the theorem for the “joint” case n ≥ 2. As a corollary, the joint functional independence for those functions is shown.

We study Δ(x; ϕ), the error term in the asymptotic formula for [sum ]n[les ]xcn, where the cns are generated by the Rankin–Selberg series. Our main tools are Voronoï-type formulae. First we reduce the evaluation of Δ(x; ϕ) to that of Δ1(x; ϕ), the error term of the weighted sum [sum ]n[les ]x(x−n)cn. Then we prove an upper bound and a sharp mean square formula for Δ1(x; ϕ), by applying the Voronoï formula of Meurman's type. We also prove that an improvement of the error term in the mean square formula would imply an improvement of the upper bound of Δ(x; ϕ). Some other related topics are also discussed.

Asymptotic expansions of the Barnes double zeta-function

formula here

and the double gamma-function Γ2(α, (1, w)), with respect to the parameter w, are proved. An application to Hecke L-functions of real quadratic fields is also discussed.

Let q be a positive integer, and L(s, χ) the Dirichlet L-function corresponding to a Dirichlet character χ mod q. We put

where χ runs over all Dirichlet characters mod q except for the principal character χ0.

The existence of the asymptotic probability measure of the Riemann zeta-function was proved in Bohr-Jessen’s classical paper [3] [4].

Let s = σ + it be a complex variable, ζ(s) the Riemann zeta-function, and R an arbitrary rectangle with the edges parallel to the axes. Then, for any σ0 > ½ and T > 0, the set

is Jordan measurable, and we denote the Jordan measure of this set by V(T, R; ξ). Then, Bohr-Jessen’s main result asserts the existence of the limit

Let dk(n) be the number of the factorizations of n into k positive numbers. It is known that the following asymptotic formula holds:

where r and q are co-prime integers with 0 < r < q, Pk is a polynomial of degree k − 1, φ(q) is the Euler function, and Δk(q; r) is the error term. (See Lavrik [3]).