Let ζ(s) be the Riemann zeta function, s=σ+it be a complex variable and a be a complex number. The zeros of ζ(s)-a, which are denoted by ρa=βa+iγa, are called a-points of ζ(s). First, we note that there is an a-point near any trivial zero s=-2n for sufficiently large n and apart from these a-points, there are only finitely many other a-points in the half-plane σ≤0 (see [4]). The a-points with βa≤0 are said to be trivial. All other a-points lie in a strip 0<σ<A, where A depends on a, and are called the nontrivial a-points. These points satisfy a Riemann-von Mangoldt type formula, namely

where

This is the well-known Riemann-von Mangoldt formula when a=0, which Bohr, Landau and Littlewood [1] generalized for all a∈C. We observe that these asymptotics are essentially independent of a, that is,

where N(T)=N0(T) denotes the number of nontrivial zeros ρ=β+iγ satisfying 0<γ<T. Levinson [8] showed that all but O(N(T)/loglogT) of the a-points with imaginary part in T<t<2T lie in Re(s)-12<(loglogT)2logT. So the a-points are clustered around the line Re(s)=12.

In [2], Conrey and Ghosh suggested the problem of estimating the average ∑0<γ0(k)≤Tζ(j)(ρ0(k)) for non-negative integers j and k, where ρ0(k)=β0(k)+iγ0(k) denote a zero of the k-th derivative ζ(k)(s). One of the first result on this topic was given by Fujii [3]. He gave an asymptotic formula of the sum ∑0<γ0≤Tζ'(ρ0)Xρ0 for a rational number X>0. The k=0 case was treated by Kaptan, Karabulut and Yildirim in [5]. Garunkštis and Steuding in [4] gave a generalization of Fujii's asymptotic formula with X=1 that if T⟶∞, we have

where Cn are the Stieltjes constants and

for any ε>0 and some constant C. Using formula (1.2), they concluded that the main term describes how the values ζ(1/2+it) approach the value a in the complex plane on average. In [4], Garunkštis and Steuding also proved that the set {(ζ(1/2+it),ζ'(1/2+it))|t∈R} is not dense in C2. This result tells us a value distribution on the critical line. Note that Voronin [14] proved that the set {(ζ(σ+it),ζ'(σ+it),…,ζ(n-1)(σ+it))|t∈R} is dense in Cn for all positive integers n and every fixed σ∈(1/2,1).

Recently, Karabulut and Yildirim in [7] studied Conrey and Ghosh's average and proved that for fixed j,k∈Z≥0 and large T, we have

where δj,0=1 if j=0 and 0 otherwise,

the sum over r being void in the case k=0 and zr(r=1,...,k) being the zeros of Pk(z)=∑j=0kzj/j!.

Let ρa(k)=βa(k)+iγa(k) denote an a-point of ζ(k)(s). Similar to the a-points of ζ(s), there is an a-point of ζ(k)(s) near any trivial zero s=-2n for sufficiently large n and apart from these a-points, there are only finitely many other a-points in the half-plane σ≤C for any C<0 (see Lemma 2.3)

In this paper, we give an asymptotic formula for the sum

The basic idea of the proof is to interpret the sum of ζ(j)(ρa(k)) as a sum of residues. By Cauchy's theorem, we have

where f(s) is ζ(j)(s) and R is the rectangle joining the points b+i,b+iT,-b'+iT and -b'+i with some constants b,b'>0.

Our main result is stated in the following.

Theorem 1.1. Let j and k be non-negative integers and a be a complex number. For sufficiently large T, we have

Here and in the sequel, the implicit constant in the error terms may depend on a.

Remark. By Theorem 1.1, we can deduce the average value of ζ(j)(ρa(k)), over the a-points ρa(k) of ζ(k)(s) with 1<Im(ρa(k))<T, i.e.,

where Nk(a,T) is the number of terms in the above sum. Because of the asymptotic formula Nk(a,T)∼(T/2π)logT (see [9]), the average is

So this tells us about the size of ζ(j)(s) at certain points (namely the a-points of ζ(k)(s)).

In this section, we prepare some lemmas and equations to prove Theorem 1.1.

Let k be a positive integer. We start with some results (see [9]) about the a-points of k-th derivative of the Riemann zeta function (see also [13]). For c>1, the following equation

holds in the region {s∈C σ>c,|t|≥1}. Equation (2.1) (see [9, Theorem 2.2]) yields an a-point free region for ζ(k)(s), that is, there exist real numbers E1k(a)≤0 and E2k(a)≥1 such that ζ(k)(s)-a≠0 for {s∈C σ≤E1k(a),|t|≥1} and {s∈C σ≥E2k(a)}. Moreover, in [9, Theorem 2.3] the second author proved that there exists N=Nk(a)∈N such that ζ(k)(s)=a has just one root in Cn:={s∈C -2n-1<σ<-2n+1,|t|<1} for each integer n≥N.

Let us recall that ζ(k)(s¯)=ζ(k)(s)¯, then if a∉R, there exist infinitely many a-points, or infinitely many a¯-points, of ζ(k)(s) in {s∈C 0<t<1}.

In the following lemma, we prove that equation (2.1) yields another a-point free region for ζ(k)(s).

Lemma 2.2. Let k be a non-negative integer. For any real number C<0, there exists a constant Tk,C>0 such that there are no a-points of ζ(k)(s) in {s∈C σ≤C,|t|≥Tk,C}.

Proof. By Stirling's formula, for |t|>1 and fixed σ≥1-C, we have

Moreover, one has

for fixed σ≥1-C. Using the last estimates and (2.1), we get

for |t|>1 and fixed σ≥1-C. Hence, there exists a constant Tk,C>0 such that |ζ(k)(s)|>|a| holds for all E1k(a)≤σ≤C and |t|≥Tk,C.

From Lemma 2.2, we deduce easily the following lemma.

Lemma 2.3. Let k be a non-negative integer. For any real number C<0, there are finitely many a-points of ζ(k)(s) in

For a positive integer k and a complex number a, we have (see [9, Theorem 1.1])

and for sufficiently large T, we also have

Now, using [9, Lemma 2.6], for any constants σ1,σ2 and s∈C with σ1<σ<σ2 and large t, we have

Lemma 4.1 in [9] states that, for a positive integer k and a sufficiently large σ≥E2k, we have (2.5) ζ(k+1)(s)ζ(k)(s)-a=∑l≥0 n0,...,nl≥2(-1)k(l+1)al+1(logn0)k+1(logn1...lognl)k1n0s...nls(a≠0),∑l≥0 n0≥2 n1,...,nl≥3-1(log2)kl+1(logn0)k+1(logn1...lognl)k2(l+1)sn0s...nls(a=0).

The right-hand side is complicated, so here we abbreviate it to ∑d≥1α(d)d-s. When k=0, ζ'(s)/(ζ(s)-a) also has a convergent Dirichlet series expansion (The case k=0 and a≠1 is mentioned in [11, (29)]). Note that α(1)=0 holds if k>0 and a≠0.

We finish this section by the following estimate

which holds as |t|→∞ for any small ε>0, where the function μ(σ) satisfies the inequalities

Let a be a complex number. We write s=σ+it,ρa(k)=βa(k)+iγa(k) with real numbers σ,t,βa(k) and γa(k). The case a=0 was already proved by Karabulut and Yildirim in [7], so here we assume a≠0. By the residue theorem, for a sufficiently large constant B and constant b∈(1,9/8), we have

where the integration is taken over a rectangular contour in counterclockwise direction denoted by R with vertices 1-b+i,B+i,B+iT,1-b+iT. Since there are finitely many a-points in {s∈C Re(s)≤1-b,Im(s)≥1}, we have

Hence, we have

The integral I1 is independent of T, so we have I1=O(1). Next, we consider I2. Using (2.5), we get

where we define (-logn)j=1 when n=1 and j=0. The integral factor can be calculated as

By these estimates, we obtain

From (2.4), we have

Now, we change the path of integration. If γa(k)<T, we change the path to the upper semicircle with center ρa(k) and radius 1. If γa(k)>T, we change the path to the lower semicircle with center ρa(k) and radius 1. Then, we have

on the new path. This estimate and the bound (2.6) yields

In view of the number of a-points (2.3), we obtain

This leads to I3≪T since 1<b<9/8.

Finally, we estimate I4. By (2.1) and Stirling's formula, for fixed 1<b<9/8 and large |t|>2, we have

Therefore, there exists a constant A such that

holds for any |t|≥A. We divide the path of the integral into two parts

Then, the second term is O(1) since it is independent of T. Using

we get

By (3.3), the integrand can be estimated as

Hence, each integral can be calculated as

for any small ε>0. It follows from the last estimate that the sum for n≥2 is bounded as

Similarly, one has

Therefore, we get

Karabulut and Yildirim [7] already studied K1, and they gave an estimate

It remains to evaluate K2. From (3.4), for k≥1, we have

so hereafter we consider the case k=0. In this case, we use Conrey and Ghosh's result (see [2, (16)])

for σ≤1/2 and |t|≫1, where χ(s):=2(2π)s-1Γ(1-s)sin(πs/2) and ℓ:=log(|t|/2π). Substituting this equation into the integrand of K2 with k=0, we have

Since the path of the integral satisfies Re(s)=1-b with 1<b<9/8, ζ(1-s)≍1 and ζ(j)(1-s)≪1 hold for any non-negative integer j. Therefore, we have

Combining K1 and K2, we obtain

From estimates of I1,I2,I3 and I4, we finally obtain Theorem 1.1.

In this section we present some problems that will be considered in a sequel to this note.

• Let L(s,χ) be the Dirichlet L-function associated with a primitive character χ mod q. We believe that we can extend Theorem 1.1 to higher derivatives of L(s,χ). To do so, we first extend Karabulut and Yildirim's result given by equation (1.5) (see [7]) using the same argument as in [6].

• The a-points of an L-function L(s) are the roots of the equation L(s)=a. We refer to Steuding book [12, chapter 7] and Selberg paper [10] for some results about a-points of L-functions from the Selberg class. Therefore, it is an interesting problem to extend Theorem 1.1 to other classes of L-functions (the Selberg class with some further conditions) and its higher derivatives.

The authors thank the anonymous referee for his careful reading of the manuscript and constructive suggestions which improved the paper.

In fact, in the proof of (1.2) Garunkštis and Steuding used the following formula established and corrected by Fujii in [3]

∑0<γ≤Tζ'(ρ)=T4πlog2T2π+C0-1T2πlogT2π + (1-C0-C02+3C1)T2π+OTe-ClogT,

where the summation is over all nontrivial zeros ρ=β+iγ of ζ(s).

The Stieltjes constants are given by the Laurent series expansion of ζ(s) at s=1,

ζ(s)=1s-1+∑n=0∞(-1)nCnn!(s-1)n.

For example, C0=limN→+∞∑n=1N1n-logN.