We break the convexity bound in the t-aspect for L-functions attached to cusp forms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s,f⊗χ) by Grossencharacters χ, from our previous paper on integral moments.

In this paper, we consider certain classes of Eisenstein-type series involving hyperbolic functions, and prove some formulas for them which can be regarded as relevant analogues of our previous results. We can also regard these formulas as certain generalizations of the famous formulas for the ordinary Eisenstein series given by Hurwitz.

Let K be a real quadratic number field and let p be a prime number which is inert in K. We denote the completion of K at the place p by Kp. We propose a p-adic construction of special elements in Kp× and formulate the conjecture that they should be p-units lying in narrow ray class fields of K. The truth of this conjecture would entail an explicit class field theory for real quadratic number fields. This construction can be viewed as a natural generalization of a construction obtained by Darmon and Dasgupta who proposed a p-adic construction of p-units lying in narrow ring class fields of K.

Let q≥2 and N≥1 be integers. W. Zhang recently proved that for any fixed ε>0 and qε≤N≤q1/2−ε, where the sum is taken over all nonprincipal characters χ modulo q, L(1,χ) denotes the L-functions corresponding to χ, and αq=qo(1) is some explicit function of q. Here we improve this result and show that the same asymptotic formula holds in the essentially full range qε≤N≤q1−ε.

We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.

We prove a joint universality theorem in the Voronin sense for the periodic Hurwitz zeta-functions.

In this article, a discrete mean value of the derivative of the Riemann zeta function is computed. This mean value will be important for several applications concerning the size of ζ′(ρ), where ζ(s) is the Riemann zeta function and ρ is a non-trivial zero of ζ(s).

Suppose that {tn} is the sequence of positive roots of ζ (½ + it) counted according to multiplicity and arranged in non-decreasing order; in my paper [6] I proved that

and my main objective here is to improve this bound.

In this paper, we give certain analytic functional relations for the double harmonic series related to the double Euler numbers. These can be regarded as continuous generalizations of the known discrete relations obtained by the author recently.

This article studies the non-homogeneous quadratic Bessel zeta function ζRB(s, v, a), defined as the sum of the squares of the positive zeros of the Bessel function Jv(z) plus a positive constant. In particular, explicit formulas for the main associated zeta invariants, namely, poles and residua ζRB(0, v, a) and ζRB(0, v, a), are given.

It is shown that a twist of a Dirichlet series by a Dirichlet character may lead to very different analytic properties.

In this paper we derive a relation between character sums and partial sums of Dirichlet series.

We are interested in the distribution of those zeros of the Riemann zeta-function which lie on the critical line ℜs = ½, and the maxima of the function between successive zeros. Our results are to be independent of any unproved hypothesis. Put

This is an expanded version of two lectures given at the conference held at Sydney University in December 1997 on the 50th anniversary of the death of G. H. Hardy.

The Redheffer matrix An = (aij)n×n defined by aij = 1 when i|j or j = 1 and aij = 0 otherwise has many interesting number theoretic properties. In this paper we give fairly precise estimates for its eigenvalues in punctured discs of small radius centred at 1.

Let

Asymptotic formulae for Ik(T) have been established for the cases k=1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of Ik(T) remains unknown for any other value of k (except the trivial k = 0, of course). Heath-Brown, [6], and Ramachandra, [10], [11], independently established that, assuming the Riemann Hypothesis, when 0≤K≤2, Ik(T) is of the order T(log T)k2 One believes that this is the right order of magnitude for Ik(T) even when k = 2 and indeed expects an asymptotic formula of the form

where Ck is a suitable positive constant. It is not clear what the value of Ck should be.

Let |θ| < π/2 and . By refining Selberg's method, we study the large values of as t → ∞ For σ close to ½ we obtain Ω+ estimates that are as good as those obtained previously on the Riemann Hypothesis. In particular, we show that

and

Our results supplement those of Montgomery which are good when σ > ½ is fixed.

The best current bounds for the proportion of zeros of ζ(s) on the critical line are due to Conrey [C], using Levinson's method [Lev]. This method can also be used to detect simple zeros on the critical line. To apply Levinson's method one first needs an asymptotic formula for the meansquare from 0 to T of ζ(s)M(s) near the -line, where

where μ(n) is the Möbius function, h(x) is a real polynomial with h(0) = 0, and y=Tθ for some θ > 0. It turns out that the parameter θ is critical to the method: having an asymptotic formula valid for large values of θ is necessary in order to obtain good results. For example, if we let κ denote the proportion of nontrivial zeros of ζ(s) which are simple and on the critical line, then having the formula valid for 0 < θ < yields κ > 0·3562, having 0 < θ < gives κ > 0·40219, and it is necessary to have θ > 0·165 in order to obtain a positive lower bound for κ. At present, it is known that the asymptotic formula remains valid for 0 < θ < , this is due to Conrey. Without assuming the Riemann Hypothesis, Levinson's method provides the only known way of obtaining a positive lower bound for κ.

A well-known theorem of Hardy and Littlewood gives a three-term asymptotic formula, counting the lattice points inside an expanding, right triangle. In this paper a generalisation of their theorem is presented. Also an analytic method is developed which enables one to interpret the coefficients in the formula. These methods are combined to give a generalisation of a “heightcounting” formula of Györy and Pethö which itself was a generalisation of a theorem of Lang.