PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 140, Number 12, December 2012, Pages 4097–4103
S 0002-9939(2012)11275-4
Article electronically published on April 10, 2012

A NOTE ON SIMPLE a-POINTS OF L-FUNCTIONS
S. M. GONEK, S. J. LESTER, AND M. B. MILINOVICH(Communicated by Matthew A. Papanikolas)
Abstract. We prove, subject to certain hypotheses, that a positive proportion of the a-points of the Riemann zeta-function and Dirichlet L-functions
with primitive characters are simple and discuss corresponding results for otherfunctions in the Selberg class. We also prove an unconditional result of thistype for the a-points in ﬁxed strips to the right of the line s = 1/2.

1. Introduction
Let Fa (s) = ζ(s)−a, where ζ(s) is the Riemann zeta-function, s = σ+it, and a is
a nonzero complex number. The zeros of Fa (s), which we denote by ρa = βa + iγa ,are called the a-points of ζ(s), and their distribution has long been an object ofstudy (cf. [6, 9, 11] for the principal results and further references). The resulting
value distribution theory is quite beautiful and not as widely known as it should
be. Some of the most basic facts are these. First, there exists a number n0 (a)such that Fa (s) has a zero quite close to s = −2n for all integers n ≥ n0 (a), andthere are at most ﬁnitely many other zeros in s ≤ 0. (Levinson [6] states this fors ≤ −2, but it is not diﬃcult to see that it holds for s ≤ 0.) We call these zerosof Fa (s) trivial a-points. The remaining zeros all lie in a strip 0 < s < A, where
A depends on a, and we call these nontrivial a-points. For these we have

T
T
T
log
−
+ Oa (log T )1=(1.1)
Na (T ) =2π2π2π0<γa <Tβa >0if a = 1; if a = 1, there is an additional term − log 2(T /2π) on the right-handside of the above equation (cf. [6] or [9]). The zeros of Fa (s) cluster near the lines = 1/2 (Levinson [6]), as do the zeros of ζ(s) (Titchmarsh [11]), of which there
are approximately the same number in the strip 0 < s < T (see (2.1) below).
However, Fa (s) has  T zeros with βa > σ and 0 < γa < T for each ﬁxed σ with
1/2 < σ ≤ 1, whereas ζ(s) has at most O(T θ(σ) ) zeros in such a region, whereθ(σ) < 1 (Titchmarsh [11]).
By analogy with the case a = 0, it is natural to ask whether all or almost all of the
a-points ρa are simple. By simple we mean that Fa (ρa ) = 0, but Fa (ρa ) = ζ  (ρa ) =
0. The trivial zeros near the numbers −2n with n ≥ n0 (a) are known to be simple.
Also, for almost every a, each a-point is simple, for there are an uncountable number
Received by the editors March 18, 2011 and, in revised form, May 18, 2011 and May 24, 2011.
2010 Mathematics Subject Classiﬁcation. Primary 11M06, 11M26.
Key words and phrases. a-points, simple zeros, Riemann zeta-function, L-functions, Selberg
class.
Research of the ﬁrst author was partially supported by NSF grant DMS-0653809.
c2012
American Mathematical Society
Reverts to public domain 28 years from publication

4097

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4098

S. M. GONEK, S. J. LESTER, AND M. B. MILINOVICH

of a’s but only a countable number of zeros of ζ  (s). Garunkštis and Steuding [5]have recently shown that for every a an inﬁnite number of a-points are simple. On
the other hand, for some a there certainly are multiple zeros of Fa (s): if ζ  (ρ ) = 0
and ζ(ρ ) = a, then ρ is an a-point of order at least two.
The purpose of this paper is to investigate the density of simple a-points bothto the left and right of the critical line s = 1/2. Our methods also apply to the
a-points of a number of other functions in the Selberg class [9], such as Dirichlet
L-functions with primitive characters, but we primarily focus on the Riemann zetafunction and later indicate the modiﬁcations necessary to handle other functions.
2. a-points to the left of the line s = 1/2
Analogously to Na (T ) in (1.1), we deﬁne N (T ) and N  (T ) to be the number ofcomplex zeros in the half-strip σ > 0, 0 < t < T of ζ(s) and ζ  (s), respectively.
Then (see [2, 7])(2.1)

N (T ) =

T
T
T
log
−
+ O(log T )
2π
2π 2π

and
T
T
T
log
−
+ O(log T ).
2π
4π
2π
We also let Na (σ1 , σ2 , T ), N (σ1 , σ2 , T ), and N  (σ1 , σ2 , T ) denote the number of
a-points of ζ(s), zeros of ζ(s), and zeros of ζ  (s), respectively, with real part in(σ1 , σ2 ) and imaginary part in (0, T ).
We ﬁrst consider the question of simple a-points under the assumption that the
Riemann Hypothesis is true. By a theorem of Speiser [10], the Riemann Hypothesis
implies that ζ  (s) has no zeros in the strip 0 < σ < 1/2. Furthermore, Selberg [9](see also Tsang [12]) has shown that on the Riemann Hypothesis at least 1/2 ofthe nontrivial a-points lie to the left of s = 1/2. It follows that these must all be
simple. Thus, we easily have

(2.2)

N  (T ) =Theorem 1. Assume the Riemann Hypothesis. Then at least


T
1
T
+ o(1)
log
22π
2π
of the nontrivial a-points of ζ(s) with ordinates in (0, T ) are simple and lie to theleft of the line s = 1/2.
It was pointed out by Selberg that independently of the Riemann Hypothesis,
one can show that if a = |a|eiθ = 0 for almost all θ ∈ [0, 2π), then at least half the
a-points lie to the left of s = 1/2.
Next we estimate the number of simple a-points under another hypothesis. In [9]Selberg says it is likely that for a given a = 0, three quarters of the a-points are to
the left of the line s = 1/2 (see Tsang [12] and Montgomery’s review1 of [6]). We
state this as
Conjecture 1 (Selberg’s a-Point Conjecture). For all a = 0 we have
3
TT
Na (0, 1/2, T ) =+ o(1)
log
.
4
2π
2π
1 MathSciNet: MR0406952 (53#10737)

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A NOTE ON SIMPLE a-POINTS OF L-FUNCTIONS

4099

To support the conjecture, Selberg notes that if the ratio of two complex numbersa and a is negative, one can show that 3/4 of the zeros of (ζ(s) − a)(ζ(s) − a ) lie
to the left of s = 1/2 for almost all values of the argument of a.
Now let N (0) (T ) denote the number of zeros of ζ(s) with real part equal to 1/2
and imaginary part in (0, T ), and set
N (0) (T ).
T →∞N (T )(2.3)

θ = lim inf

Our next theorem asserts that if Selberg’s a-Point Conjecture is true and enoughzeros of ζ(s) are on the line s = 1/2, then a positive proportion of the a-points
are still simple.
Theorem 2. Assume Selberg’s a-Point Conjecture and let θ be as in (2.3). Then
at least


T
T
1
log
θ − + o(1)42π2πof the nontrivial a-points of ζ(s) with ordinates in (0, T ) are simple and lie to theleft of the line s = 1/2.
Proof. Levinson and Montgomery [7] have shown unconditionally that
(2.4)

N  (0, 1/2, T ) = N (0, 1/2, T ) + O(log T ).

By the functional equation for ζ(s), the nontrivial zeros of ζ(s) are symmetric about
the line s = 1/2. Thus, at most (1 − θ)/2 of these zeros lie to the left of s = 1/2,
and at most (1 − θ)/2 lie to the right of it. By (2.1), (2.2), and (2.4), it therefore
follows that
T
 1−θ
T
+ o(1)log.
(2.5)
N  (0, 1/2, T ) ≤22π
2π
For z ∈ C, let ma (z) = 0 if z is not an a-point of ζ(s); if z is an a-point, letma (z) denote its multiplicity as an a-point. Similarly, let m (z) = 0 if z is not azero of ζ  (s), and, if it is, let m (z) denote its multiplicity. Then when z = ρa isan a-point, we clearly have m (ρa ) = ma (ρa ) − 1. Let R(T ) denote the rectangularregion (0, 1/2) × (0, T ) ⊂ C. Thenma (ρa ) =(m (ρa ) + 1)
ρa ∈R(T )ma (ρa )≥2(2.6)

ρa ∈R(T )m (ρa )≥1≤2m (ρa )ρa ∈R(T )m (ρa )≥1≤2m (ρ ) = 2N  (0, 1/2, T ).



ρ ∈R(T )m (ρ )≥1

Here and below our sums are over distinct a-points ρa and zeros ρ if ζ  (s). We also
have


1 +
ma (ρa ).
Na (0, 1/2, T ) =ρa ∈R(T )ma (ρa )=1

ρa ∈R(T )ma (ρa )≥2

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4100

S. M. GONEK, S. J. LESTER, AND M. B. MILINOVICH

Combining this with (2.6), we ﬁnd that the number of simple a-points in R(T )satisﬁes(2.7)
1 ≥ Na (0, 1/2, T ) − 2N  (0, 1/2, T ).
ρa ∈R(T )ma (ρa )=1

Thus, by (2.5) and Conjecture 1 we haveρa ∈R(T )ma (ρa )=1

1≥

3
4

− (1 − θ) + o(1)

T
2π

log


T
T
1
T
= θ − + o(1)
log
.
2π
4
2π
2π

This completes the proof of the theorem.

Bui et al. [4] have recently shown that θ ≥ .4105. Using this in Theorem 2, we
obtainCorollary 1. Assume Selberg’s a-Point Conjecture. Then at least


4
T
T
+ o(1)
log25
2π
2π
of the nontrivial a-points of ζ(s) with ordinates in (0, T ) are simple and lie to theleft of the line s = 1/2.
We state the case θ = 1 of Theorem 2 separately as
Corollary 2. Assume Selberg’s a-Point Conjecture and that 100 percent of thezeros of ζ(s) lie on the line s = 1/2. Then at least


3
T
T
+ o(1)
log
42π
2π
of the nontrivial a-points of ζ(s) with ordinates in (0, T ) are simple and lie to theleft of the line s = 1/2.
In particular, the conclusion of Corollary 2 holds under the assumption of Selberg’s a-Point Conjecture and the Riemann Hypothesis.
Our arguments extend to Dirichlet L-functions with a ﬁxed primitive characterχ as well. To see this we note the following:(i) L(s, χ) and Fa (s, χ) = L(s, χ) − a both have ∼ (T /2π) log T /2π nontrivial
zeros with real part greater than 0 and imaginary part in (0, T ).
(ii) Levinson and Montgomery’s result that ζ(s) and ζ  (s) have the same number of zeros to within O(log T ) in the rectangular region R(T ) holds forL(s, χ) and L (s, χ) (see Powell [8]).
(iii) Selberg’s result that at least 1/2 of the a-points of the zeta-function lie
to the left of the line s = 1/2 is true for L(s, χ) as well, provided the
Riemann Hypothesis is true for L(s, χ) (see Selberg [9]).
(iv) Selberg actually asserts the a-Point Conjecture for every primitive functionin the Selberg class, so in particular for L(s, χ).

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A NOTE ON SIMPLE a-POINTS OF L-FUNCTIONS

4101

It follows from (i)–(iv) that the analogues of Theorems 1 and 2 are true for any
Dirichlet L-function L(s, χ) with a primitive character χ. We state this as
Theorem 3. Let L(s, χ) denote a Dirichlet L-function with a primitive character
χ. If the Riemann Hypothesis is true for L(s, χ), then at least


1
T
T
+ o(1)
log
22π
2π
of the nontrivial a-points of L(s, χ) with ordinates in (0, T ) are simple and lie tothe left of the line s = 1/2. If instead of the Riemann Hypothesis we assume thatSelberg’s a-Point Conjecture holds for L(s, χ), and if θχ is the analogue for L(s, χ)of θ (cf. (2.3)), then at least


T
T
1
log
θχ − + o(1)42π2πof the nontrivial a-points of L(s, χ) with ordinates in (0, T ) are simple and lie tothe left of the line s = 1/2.
Bauer [1] has shown that θχ ≥ .365815 for primitive Dirichlet L-functions. Using
this in the second part of Theorem 3, we obtain
Corollary 3. Assume Selberg’s a-Point Conjecture holds for a Dirichlet L-function
L(s, χ) with χ a primitive character. Then at leastTT.1158 + o(1)
log
2π
2π
of the nontrivial a-points of L(s, χ) with ordinates in (0, T ) are simple and lie tothe left of the line s = 1/2.
Of course, the result corresponding to Corollary 2 holds for L(s, χ) as well.
Analogues of (i) and (iii) above are known for all functions in the Selberg class.
We have not found the analogue of (ii) for the Selberg class in the literature, but it
is likely that it can be proved by making minor modiﬁcations to the argument of
Levinson and Montgomery in [7]. If that is the case and the Riemann Hypothesis
is true for the functions of the Selberg class, then at least 1/2 of their a-points are
simple. Furthermore, if the analogue of Selberg’s a-Point Conjecture also holds,
then 3/4 of the a-points are simple.
3. a-points to the right of the line s = 1/2
Let σ1 and σ2 be ﬁxed with 1/2 < σ1 < σ2 < 1, and recall that Na (σ1 , σ2 , T )is the number of a-points of ζ(s) in the rectangular region (σ1 , σ2 ) × (0, T ). Then
there is a positive constant Ca (σ1 , σ2 ) such that
Na (σ1 , σ2 , T ) ∼ Ca (σ1 , σ2 )T(see Borchsenius and Jessen [3], Theorem 14). Thus a natural question is whether
a positive proportion of these a-points are simple. In this section we show unconditionally that this is indeed the case. The proof uses Voronin’s universality theorem
for the Riemann zeta-function [13].

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4102

S. M. GONEK, S. J. LESTER, AND M. B. MILINOVICH

Theorem 4. Let 1/2 < σ1 < σ2 < 1 with σ1 , σ2 ﬁxed, and let a = 0. Then there isa positive constant ca (σ1 , σ2 ) such that at least ca (σ1 , σ2 ) T of the a-points of ζ(s)in the region (σ1 , σ2 ) × (0, T ) are simple.
Proof. Set s0 = (σ1 + σ2 )/2, let 0 < R < (σ2 − σ1 )/2 < 1/4, and let DR be theclosed disc of radius R centered at s0 . Consider the function|a|(s − s0 )2
on DR . By Voronin’s universality theorem (see [13]), for any > 0 there is asequence {τn }∞n=1 such that the discs DR + iτn are nonoverlapping,f (s) = a +|{τn : 0 < τn ≤ T }|  T,
andmax |ζ(s + iτn ) − f (s)| <s∈∂DRfor all n. By our choice of f (s) we also havemin |f (s) − a| =

s∈∂DR

|a|R
.
2

Taking = |a|R2 /100, we see that < |a|R/2, since R < 1/4. Thus, by Rouché’s
Theorem, ζ(s + iτn ) − a and f (s) − a have the same number of zeros inside DR foreach n, namely one counting multiplicities. It follows that the zero of ζ(s + iτn ) − a
is a simple a-point of ζ(s). Therefore, there are  T simple a-points of ζ(s) in(σ1 , σ2 ) × (0, T ). This completes the proof.
Analogues of Voronin’s Universality Theorem are known to hold for many functions in the Selberg class and, in particular, for Dirichlet L-functions. The above
proof therefore extends to them.
Our theorems leave open the question of the simplicity of a-points on or slightly
to the right of the line s = 1/2. It is not diﬃcult to show that if ζ  ( 21 + it0 ) = 0,then ζ( 21 + it0 ) = 0. Hence, every a-point of ζ(s) on the line s = 1/2 is simple.
However, as Selberg suggests (see [9], p. 58, footnote 10), there are probably
only ﬁnitely many such a-points. This together with Selberg’s a-Point Conjecture
implies that (1/4 + o(1))(T /(2π)) log T /(2π) of the a-points of ζ(s) lie to the rightof s = 1/2. We conjecture that almost all of these are simple too. It would beinteresting if one could show that a positive proportion of them are.
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A NOTE ON SIMPLE a-POINTS OF L-FUNCTIONS

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Department of Mathematics, University of Rochester, Rochester, New York 14627
E-mail address: gonek@math.rochester.edu
Department of Mathematics, University of Rochester, Rochester, New York 14627
E-mail address: lester@math.rochester.edu
Department of Mathematics, University of Mississippi, University, Mississippi 38677
E-mail address: mbmilino@olemiss.edu

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