analytic continuation of ζ(s) to a meromorphic function on the complex plane is achieved by the functional equation

SNEHA CHAUBEY, AMITA MALIK, NICOLAS ROBLES, AND ALEXANDRU ZAHARESCU

Abstract. We consider functions of the form F~c,a,T(s) =PM j=0

c_j(−1)^j

L^2j ξ^(a+2j)(s), with L = log_2π^T and cj real constants satisfying a certain constraint. We show that as T → ∞, the proportion of zeros of F~c,a,T(s) on the critical line Re(s) = 1/2 tends to 1.

1. Introduction Let ζ(s) =P∞

n=1n^−s for s = σ + it, σ > 1 and t ∈ R, denote the Riemann zeta-function. The analytic continuation of ζ(s) to a meromorphic function on the complex plane is achieved by the functional equation

ξ(s) = ξ(1 − s), where, the Riemann ξ-function is defined as

ξ(s) = 1

2s(s − 1)π^−s/2Γ s 2

 ζ(s).

For σ > 1, the Euler product is

ζ(s) =Y

p

(1 − p^−s)⁻¹,

where the product is taken over all the primes p. This links the Riemann zeta-function to multi- plicative number theory [17, §1 and §2]. It is well understood from the work of Riemann and von Mangoldt that the non-trivial zeros ρ = β + iγ of ζ(s) are located inside the critical strip 0 < β < 1, see [17, §3]. From the fact that Γ has no zeros, and has simple poles at the trivial zeros of ζ(s), it follows that the zeros of ξ are the same as the non-trivial zeros of ζ. The Riemann hypothesis states that β = ¹₂.

Now let N (T ) denote the number of zeros of ξ(s) in the rectangle 0 < σ < 1 and 0 < t ≤ T , each zero counted with multiplicity. It is well-known that

N (T ) = T 2π

 log T

2π− 1

 +7

8 + S(T ) + O 1 T

 , (1.1)

where

S(T ) = 1

π arg ζ 1 2+ iT



 log T,

as T → ∞, see [17, §9]. Let us now define N⁽⁰⁾(T ) to be the number of zeros of ζ(s) with β = ¹₂ on 0 < t ≤ T , where each zero is counted with multiplicity. We further set

κ = lim inf

T →∞

N⁽⁰⁾(T ) N (T ) .

In 1942, Selberg [14] showed that κ > 0, and later Levinson [10] showed that κ > 0.34. This was improved by Conrey [5] to κ > 0.4088. The history of these results and the current best bound can

2010 Mathematics Subject Classification. Primary: 11M26; Secondary: 11M06.

Keywords and phrases. Riemann ξ-function, normalized combinations, zeros, critical line, proportion, mollifier.

1

be found in [2, 7, 9, 13].

For a positive integer k, let ξ^(k)(s) denote the kth derivative of the Riemann ξ-function. The Riemann hypothesis implies that for any positive integer k, all the zeros of ξ^(k)(s) lie on the critical line. Suppose, in analogy to the above, that N_k(T ) denotes the number of zeros β + iγ of ξ^(k)(s) in the rectangle 0 < β < 1 and 0 < γ ≤ T and that N_k⁽⁰⁾(T ) denotes the number of zeros of ξ^(k)(s) with β = ¹₂ and 0 < γ ≤ T . A result of Conrey [3] states that if T is positive and sufficiently large, L = log_2π^T and U = T L⁻¹⁰, then

lim inf

T →∞ κ_k(T, U ) = 1 + O(k⁻²) (1.2)

as k → ∞ and where

κk(T, U ) = N_k⁽⁰⁾(T + U ) − N_k⁽⁰⁾(T ) N_k(T + U ) − N_k(T ) .

Moreover, in [4], following work from Anderson [1] and Heath-Brown [8], Conrey also established corresponding bounds for simple zeros. The coefficient of k⁻² was computed in [3] for zeros with multiplicity and in [4] for simple zeros. It was remarked that the proportion of simple zeros was always a bit smaller than that of zeros with potential multiplicity. Nonetheless, from (1.2) as the order of the derivative of ξ increases, the proportion of zeros on the critical line increases to one.

This strong result is due to Conrey [3].

Rezvyakova [11, 12] computed the coefficients of k⁻² in 2005 and her result holds uniformly for k in a certain range depending on T . In particular, Rezvyakova showed that the coefficient in front of k⁻² could be taken to be ³₅ for both simple as well as higher order zeros.

In the late 1990’s, Selberg considered combinations of Dirichlet L-functions on the critical line.

More specifically, let

L(s, χ) =

∞

X

n=1

χ(n)n^−s, σ > 1,

be a Dirichlet L-function of modulus q, where χ denotes a primitive character. The functional equation of L(s, χ) is given by

φ(s, χ) = επ^−s/2q^s/2Γ s + a 2



L(s, χ) = φ(1 − ¯s, χ), where

a= 1 − χ(−1)

2 and |ε| = 1,

see e.g. [6]. If we have n distinct even characters (a similar result holds for odd characters) and form the function

F (s) =

n

X

j=1

c_jε_jq^s/2_j L(s, χ_j) (1.3)

-for real c_j 6= 0, then

π^−s/2Γ s 2

 F (s)

is real for s = ¹₂ + it. In a series of unpublished notes [15], Selberg proved a beautiful result on the zeros of F (s). He derived a formula analogous to (1.1) for F (s), and also showed that N⁽⁰⁾(T, F ) > c(n)T log T for T > T₀(F ), where c(n) is a positive constant that depends only on n.

Moreover, in those lectures, there is mention of the conjecture that almost all the zeros have real

part equal to ¹₂.

To state our results, we need to introduce some further notations. For a fixed positive integer M , let us fix a vector ~c = (c0, c1, · · · , cM) such that cj ∈ R for all j and define

c^∗:=

M

X

j=0

(−1)^jcj

4^j . (1.4)

For all large numbers T , we set

L = log T

2π, and U = T L⁻¹⁰. Then for each positive integer a, we consider the function

F_~c,a,T(s) :=

M

X

j=0

cj(−1)^j

L^2j ξ^(a+2j)(s).

(1.5)

The presence of L^j has the effect of balancing the size of ξ^(j)(s) in F_~c,a,T(s), so that no one par- ticular term dominates the entire combination. The constants appearing in the error terms may depend on the vector ~c throughout the paper.

Inspired by the result of Selberg and the techniques of Levinson and Conrey, our object of study in this note is the number of zeros of F_~c,a,T(s) on the critical line σ = ¹₂ with imaginary part between T and T + U . With this in mind, we define the counting functions N_~c,a(T ) and N_~c,a⁽⁰⁾(T ) by

N_~c,a(T ) = X

F~c,a,T(ρ)=0 0<Im ρ≤T

1, and N_~c,a⁽⁰⁾(T ) = X

F~c,a,T(ρ)=0 Re ρ=1/2 0<Im ρ≤T

1.

Moreover, the proportion of zeros of F_~c,a,T(s) in the above rectangle on the critical line is given by the quotient

κ_~c,a,T := N_~c,a⁽⁰⁾(T + U ) − N_~c,a⁽⁰⁾(T ) N_~c,a(T + U ) − N_~c,a(T ). (1.6)

Now we are ready to state our main result.

Theorem 1.1. For any positive integer M , fix a vector ~c = (c₁, · · · , c_M) with real components such that c^∗ as defined in (1.4) is nonzero. Also, for F_~c,a,T(s) defined in (1.5), let κ_~c,a,T be as in (1.6).

Then

κ_~c,a,T ≥ 1 − e²+ 2

16a² + O_~c 1 a³

 , (1.7)

as a and T tend to infinity such that

a ≤ 1 2

log log T log log log T.

Some comments are in order. The above result maintains the quality of the bounds and the uniformity achieved in [11, 12]. The function F_~c,a,T(s) satisfies a functional equation given by

F_~c,a,T(s) = (−1)^aF_~c,a,T(1 − s).

(1.8)

Moreover if all the zeros of F_~c,a,T(s) satisfy σ₂ ≤ Re(s) ≤ σ₁ for some σ₁, σ₂ ∈ R, then so do the zeros of all its higher order derivatives.

In particular, when the linear combination in (1.5) consists of a single term, then under the Riemann hypothesis, all derivatives of ξ(s) will also have all their zeros on the critical line (see [11, Lemma 3]. The computations to follow show that the techniques from [3, 10, 11] can be applied in the same way to the function F_~c,a,T(s) defined in (1.5), and the proportion of zeros on the critical line tends to 1 in this case as well, even if, evidently, such functions F_~c,a,T(s) in general do not satisfy the Riemann hypothesis.

2. Zero free region, N~c,a(T ) and an inequality for κ~c,a,T

We first obtain a zero free region for our function F_~c,a,T(s) and then we find an asymptotic formula for the number of zeros N_~c,a(T ). Observe that the zeros of F_~c,a,T(s) are the same as the zeros of the function

F1(s) := i^a L^a

M

X

j=0

c_j(−1)^j

L^2j ξ^(a+2j)(s)

and so we use the above function F₁(s) to perform our computations. Let s = σ + it with σ > 1 and T ≤ t ≤ T + U . For some fixed β_~c large enough, we now prove that

F_~c,a,T(s) 6= 0 whenever σ_~c> β_~c or σ_~c< 1 − β_~c. To do this, we note that

F₁(s) =

M

X

j=0

cji^a+2j L^a+2j

a+2j

X

l=0

a + 2j l



ζ^(a+2j−l)(s)H^(l)(s),

where H(s) = ^s₂(s − 1)π^−s/2Γ ^s₂
. We multiply and divide the above equality by H(s) and use the following relation [11, Lemma 2]

H^(k)(s) = H(s)  1 2log s

2π + O 1

|t|

k

+ O

 2^k+2

    1 2log s

2π    

k−2

|t|⁻¹



. (2.1)

This gives

F₁(s) = H(s)

M

X

j=0

cji^a+2j L^a+2j

a+2j

X

l=0

a + 2j l



ζ^(a+2j−l)(s) (2.2)

× 1 2log s

2π+ O 1 t

l

+ O

    1 2log s

2π    

l−2

= H(s)

M

X

j=0

cji^a+2j L^a+2j

a+2j

X

l=0

a + 2j l



ζ^(a+2j−l)(s) 1 2log s

2π

l 1 + O

 1 log²t

 

= H(s)

M

X

j=0

cji^a+2j L^a+2j

 1 2log s

2π

a+2j a+2j

X

l=0

a + 2j l



ζ^(l)(s) 1 2log s

2π

−l

×

 1 + O

 1 log²t

 

= H(s)

M

X

j=0

cji^a+2j L^a+2j

 1 2log s

2π

a+2j

(1 + R_a+2j,~c(s))

= H(s)

M

X

j=0

cji^a+2j 2^a+2j



1 + O 1 T



(1 + R_a+2j,~c(s))

= H(s)i^a 2^a

M

X

j=0

cj(−1)^j 4^j



1 + R_a+2j,~c(s) + O 1 T



= H(s)i^a 2^ac^∗

 1 + 1

c^∗

M

X

j=0

cj(−1)^j

4^j R_a+2j,~c(s) + O 1 T

  , (2.3)

where the remainder term R_a+2j,~c is given by R_a+2j,~c(s) = ζ(s) − 1 +

a+2j

X

l=1

a + 2j l

 1 2log s

2π

−l

ζ^(l)(s)

 1 + O

 1 log²t



+ O

 1 log²t

 . Also note that

|R_a+2j,~c(s)| ≤

a+2j

X

l=0

a + 2j l

     1 2log s

2π    

−l(log 2)^l 2^β~c−2

 1 + O

 1 log²t



+ O

 1 log²t



≤ 1

2^β~c−2



1 + 2 log 2 log(t/2π)

a+2j 1 + O

 1 log²t



+ O

 1 log²t

 (2.4)

≤ 2^{a+2j−β~c+2}

 1 + O

 1 log²t



+ O

 1 log²t



< 1 2, (2.5)

β_~c> 1 large enough. Using this in (2.3), we conclude that F1(s) 6= 0 for Re(s) > β_~c,

which in turn implies from our earlier discussion about the zeros of F1(s) and F_~c,a,T being the same that

F_~c,a,T(s) 6= 0 for Re(s) > β_~c. (2.6)

The functional equation (1.8) yields,

F_~c,a,T(s) = (−1)^aF_~c,a,T(1 − s) 6= 0 for Re(s) < 1 − β_~c. (2.7)

This completes the argument for F_~c,a,T(s) 6= 0 for Re(s) > β_~cor when Re(s) < 1 − β_~c.

In a standard way and using the arguments above, one can compute N_~c,a(T + U ) − N_~c,a(T ), the number of zeros with imaginary part between T and T + U with U = T /L¹⁰, as claimed in the lemma below for which we omit the details. Note that it is enough to count the zeros in the rectangle with vertices as β_~c+ iT, 1 − β_~c+ iT, β_~c+ i(T + U ), and 1 − β_~c+ i(T + iU ). More precisely, one has the following.

Lemma 2.1. For T  0, and 0 < U ≤ T , we have N_~c,a(T + U ) − N_~c,a(T ) = T + U

π logT + U

2π −T + U π



− T π log T

2π −T π



+ O_~c(log T ) . This implies

N_~c,a(T + U ) − N_~c,a(T ) = U L

π + O (U ) .

We now show an inequality satisfied by κ_~c,a,T involving zeros of a certain arithmetic function V (s).

Lemma 2.2. With the notations as above, one has κ_~c,a,T ≥ 1 − 4π

U LN + O 1 U

 ,

where N is the number of zeros of V (s) inside the rectangular contour R = [1/2 + iT, B + iT, B + i(T + U ), 1/2 + i(T + U )] of the function V (s), for some fixed B such that B > β_c defined as

V (s) = 1 H(s)

M

X

j=0

cji^a+2j L^a+2j

^a+2j X

m=0

a + 2j m



H^(m)(s) Z

0.1

z^−se^πiz2

2i sin(πz)(− log z)^a+2j−m



1 −log z L

 dz

+

a+2j

X

m=0

(−1)^ma + 2j m



H^(m)(1 − s) Z

0&1

z^s−1e^−πiz2

2i sin(πz)(log z)^a+2j−mlog z L

 dz, where, as before H(s) is given by

H(s) = s(s − 1)

2 π^−s/2Γ s 2

 . The notation R

0&1 denotes an integral along a line directed from the upper right to lower left which is inclined at an angle of π/4 to the real axis and intersects it between 0 and 1, see [16] and [17, §2.10].

3. An upper bound for N

In this section we prove the following lemma which gives an upper bound on N , the number of zeros of V (s) inside the contour .

Lemma 3.1. Let N be as in the previous lemma. Then the following inequality holds N ≤ U L

4π log J U



+ O(U ), where

J = 1

|d^∗|²

M

X

j,l=0

˜b

X

r,m=˜a

i^a+2j(−i)^a+2lcjc_l˜cr˜cm

4^a+j+l

× Z _{T +U}

T

B_jψ_rB_lψ_m+ χ^∗χ^∗D_jψ_rD_lψ_m+ B_jψ_rχ^∗D_lψ_m+ χ^∗D_jψ_rB_lψ_m
dt, where

d^∗= i 2

a

c^∗ 6= 0, (3.1)

˜

cu ∈ R for all ˜a ≤ u ≤ ˜b, and

B_k(s) = X

n≤

qT 2π



1 −log n L

  1 + πi

2L −2 log n L

k

n^−s for all a ≤ k ≤ b,

and

D_k(s) = X

n≤

qT 2π

log n L

 2 log n L + πi

2L − 1

k

n^s−1 for all a ≤ k ≤ b ; χ^∗(t) = eeⁱ(^π₄^{−t log}(_2πe^t )),

with

ψu(s) =X

n≤y

µ(n) n^1/L

 log y/n log y

u

n^−s for all ˜a ≤ u ≤ ˜b.

Proof. Notice that N, the number of zeros of V (s) inside the contour R is the same as the number of zeros of _d∗¹V (s)ψ(s) therein where ψ(s) is a mollifying function which on average approximates the behavior of inverse of the function F_~c,a,T(s). This mollifier is defined in the following way

ψ(s) =X

n≤y

µ(n) n^1/L

˜b

X

r=˜a

˜

c_r log y/n log y

r

n^−s=:

˜b

X

r=˜a

˜ c_rψ_r(s), where ψr(s) is given by

ψr(s) =X

n≤y

µ(n) n^1/L

 log y/n log y

r

n^−s=:X

n≤y

ar(n)n^−s.

Therefore to bound N , we bound the number of zeros of _d∗¹V (s)ψ(s). For this we shall apply Littlewood’s lemma [10] to _d∗¹V (s)ψ(s) on the rectangular contour Ω = [σ_a+ iT, σ₁+ iT, σ₁+ i(T + U ), σa+ i(T + U )], where σ1= log L/ log 2, σa= 1/2 − 1/L. Littlewood’s lemma gives

2πi X

ρ=β+iγ

(β − σ_a) = − I

Ω

log 1

d^∗ψ(s)V (s)

 ds, (3.2)

where the summation is performed over all the zeros ρ of V (s)ψ(s) inside Ω and on its upper side.

Using estimates of the integral I

Ω

log 2 L

k

ψ_k(s)V_k(s) ds

from [11, §3] we get approximations for our integral in (3.2) along the right and horizontal sides of the contour Ω. In doing so, we note that

1

d^∗V (s)ψ(s) − 1    

= O 1 L



on the right side of the contour. This implies that the change in argument of 1

d^∗V (s)ψ(s) is bounded by π in absolute value on this side. Now the number of zeros of the product V (s)ψ(s) in a larger domain Ω is greater than or equal to the number of zeros of V (s) in a smaller domain R, the imaginary part of the left hand side of (3.2) is at least N

L. Putting all these facts together, we conclude

N ≤ L 2π

Z _{T +U}

T

log    

1

d^∗ψ(σ_a+ it)V (σ_a+ it)    

dt + O (U ) .

Finally using Jensen’s inequality, we get an expression for the number of zeros in terms of an integral as

N ≤ U L

2π log I U



+ O (U ) , (3.3)

where

I = 1

|d^∗| Z _{T +U}

T

|ψ(σ_a+ it)V (σ_a+ it) dt.

We first write V (σa+ it) as

V (σa+ it) = B(σa+ it) + χ(σa+ it)D(σa+ it), where

B(σ_a+ it) =

M

X

j=0

c_ji^a+2j L^a+2j

X

n≤√

t 2π

 1

2log σ_a+ it 2π



− log n + O 1

|t|

a+2j

1 −log n L



n^−σa−it,

and D(σa+ it) is the sum

M

X

j=0

cji^a+2j L^a+2j

X

n≤√

t 2π



log n −1

2log 1 − σ_a− it 2π

 + O

 1

|1 − t|

a+2j log n L



n^σa+it−1.

Here

χ(s) = H(1 − s) H(s) .

A slight simplification of B(σ_a+ it) using definition of the logarithm of a complex number yields B(σa+ it) =

M

X

j=0

c_ji^a+2j L^a+2j

 X

n≤√

t 2π

 1 2log

 t 2π

 + iπ

4 − log n

+ O 1 t

 a+2j

1 −log n L



n^−σa−it+ O



jT^−3/4 L 2

a+2j

.

Next we break the sum over n in the expressions for B(σa+ it) and D(σa+ it) into three parts and use the approximations from [11, §3] to estimate each of them. Extracting ¹₂log_2π^t out of the sum over n and using the fact that L = log_2π^T , we get,

B(σ_a+ it) =

M

X

j=0

cji^a+2j 2^a+2j

 X

n≤

qT 2π

 1 + πi

2L− 2 log n L

a+2j

1 −log n L



n^−σa−it



+

M

X

j=0

c_ji^a+2j 2^a+2j

 X

n≤

qT 2π

  log(t/2π)

L + πi

2L −2 log n L

a+2j

−

 1 + πi

2L −2 log n L

a+2j

×



1 −log n L



n^−σa−it



+

M

X

j=0

cji^a+2j 2^a+2j



X

qT 2π≤n≤√

t 2π

 log(t/2π)

L + πi

2L −2 log n L

a+2j

1 −log n L



n^−σa−it



+ O

 T^−3/4

M

X

j=0

jcji^a+2j 2^a+2j

 . (3.4)

Let Bj(σa+ it) denote the first sum over n, that is B_j(σ_a+ it) = X

n≤

qT 2π



1 −log n L

  1 + πi

2L −2 log n L

a+2j

n^−σa−it= X

n≤

qT 2π

b_j(n)n^−σa−it.

The square of the integrals of the latter two inner sums over n inside B(σ_a+ it) in [11, §3] can be estimated as

Z T +U T

X

n≤

qT 2π

  log(t/2π)

L + πi

2L −2 log n L

a+2j

−

 1 + πi

2L −2 log n L

a+2j

×



1 −log n L



n^−σa−it    

2

dt  U L¹⁹,

and

Z T +U T

X

qT 2π≤n≤√

t 2π

 log(t/2π)

L + πi

2L −2 log n L

a+2j

1 −log n L



n^−σa−it    

2

dt  U L¹⁰. Also the square of the integral of the mollifier satisfies

Z T +U T

|ψ_j(σa+ it)|² dt  U L.

Hence, on applying the Cauchy-Schwarz inequality for the latter two sums in (3.4) we obtain Z T +U

T

ψ(σa+ it)B(σa+ it) dt =

˜b

X

r=˜a

˜ cr

M

X

j=0

c_ji^a+2j 2^a+2j

Z T +U T

ψr(σa+ it)Bj(σa+ it) dt

+ O U L⁹

 + O

 U L^9/2

 . (3.5)

Now for the other term in V (σ_a+ it), we first estimate the function χ(σ_a+ it) by adding and subtracting to it the function

χ^∗(t) = eeⁱ(^π₄^{−t log}(_2πe^t )).

This gives,

|χ(σ_a+ it)| ≤ |χ^∗(t)| + |χ(σ_a+ it) − χ^∗(t)| ≤ |χ^∗(t)| + O L⁻¹¹
. We perform an exercise similar to B(σ_a+ it) for D(σ_a+ it) and consequently obtain,

Z T +U T

χ^∗(t)ψ(σa+ it)D(σa+ it) dt =

˜b

X

r=˜a

˜ cr

M

X

j=0

cji^a+2j 2^a+2j

Z T +U T

χ^∗(t)ψr(σa+ it)Dj(σa+ it) dt

+ O U L⁹

 + O

 U L^9/2

 , (3.6)

where

Dj(σa+ it) = X

n≤

qT 2π

log n L

 2 log n L + πi

2L − 1

a+2j

n^σa+it−1= X

n≤

qT 2π

dj(n)n^σa+it−1.

Equations (3.5) and (3.6) together give I = 1

|d^∗| Z T +U

T

|ψ(σ_a+ it)V (σ_a+ it)| dt

= 1

|d^∗| Z T +U

T

˜b

X

r=˜a

˜ c_r

M

X

j=0

c_j

2^a+2j(B_j(σ_a+ it) + χ^∗(t)D_j(σ_a+ it))ψ_r(σ_a+ it)    

dt + O

 U L^9/2

 . By denoting the double sum by J , we obtain

J U = 1

|d^∗|²

b

X

j,l=a

c_jc_li^a+2j(−i)^a+2l 4^a+j+l A_j,l, (3.7)

where

A_j,l= 1 U

Z T +U T



(B_j+ χ^∗D_j)

˜b

X

r=˜a

˜

c_rψ_r(σ_a+ it)



B_l+ χ^∗D_l

˜b

X

m=˜a

˜

c_mψ_m(σ_a+ it)

 dt

=

˜b

X

r,m=˜a

˜

cr˜cmE_j,l,r,m, (3.8)

and

Ej,l,r,m= 1 U

Z T +U T

(BjψrBlψm+ χ^∗χ^∗DjψrDlψm+ Bjψrχ^∗Dlψm+ χ^∗DjψrBlψm) dt.

(3.9)

Employing the Cauchy-Schwarz inequality with identity as one of the functions and using the above notations we get,

I ≤

√

J U + O

 U L^9/2

 .

Thus using the above inequality and (3.3), the number of zeros is bounded by N ≤ U L

4π log J U



+ O (U ) ,

which proves the lemma. 

4. Proportion of zeros on the critical line

We now apply the results of the previous two sections to obtain the proportion of zeros of F_~c,a,T(s) on Re(s) = ¹₂. As we shall see this proportion tends to 1, at a speed independent of the vector ~c.

First, we consider the following integral appearing in J in Lemma 3.1.

1 U

Z T +U T

BjψrBlψm(σa+ it) dt = X

n1,n2,n3,n4

n1,n2≤√

T /2π n3,n4≤y

b_j(n₁)b_l(n₂)a_r(n₃)a_m(n₄) (n1n2n3n4)^σa

Z T +U T

 n₂n₄ n1n3

it

dt

= X

n1,n2,n3,n4

n2n4=n1n3

n1,n2≤√

T /2π n3,n4≤y

bj(n1)b_l(n2)ar(n3)am(n4) (n₂n₄)^2σa

+ 1 U

X

n1,n2,n3,n4

n2n46=n₁n3

n1,n2≤√

T /2π n3,n4≤y

bj(n1)bl(n2)ar(n3)am(n4) (n₁n₂n₃n₄)^σa

(ⁿ_n²ⁿ⁴

1n3)^{i(T +U )}− (ⁿ_n²ⁿ⁴

1n3)^iT logⁿ_n²ⁿ⁴

1n3

 (4.1)

Note that, bj(n1), b_l(n2), ar(n3), am(n4)  1, and σa= 1/2 − 1/L. This gives that the second sum above is less than

1 U

X

n1,n2,n3,n4

n2n46=n₁n3

n1,n2≤√

T /2π n3,n4≤y

1 (n1n2n3n4)^1/2

1

| logⁿ_n²ⁿ⁴

1n3|.

Rewriting n1n3 = m and n2n4= n, we obtain 1

U

X

n1,n2,n3,n4

n2n46=n1n3

n1,n2≤√

T /2π n3,n4≤y

1 (n1n2n3n4)^1/2

1

| logⁿ_n²ⁿ⁴

1n3|  1 U

X

m,n≤y qT

2π

m6=n

d(m)d(n) (mn)^1/2

1

| log(m/n)|.

Also d(m) = O (m^) = O (T^), d(n) = O (n^) = O (T^). Let h = n − m. One may assume without loss of generality that m < n. This gives,

1 U

X

m,n≤y qT

2π

m6=n

d(m)d(n)

(mn)^1/2| log(m/n)|  T^ U

X

1≤m<n≤T^1/2+θ

1

(mn)^1/2| log(m/n)|

 T^ U

X

2≤n≤T^1/2+θ 1≤h≤n−1

1

(n(n − h))^1/2| log(1 − ^h_n)|

 T^ U

X

2≤n≤T^1/2+θ 1≤h≤n−1

n^1/2 h(n − h)^1/2

 T^ U

X

1≤h≤T^1/2+θ

1 h

X

1+h≤n≤T^1/2+θ

1 1 −^h_n
1/2

 T^

U log²(T )T^1/2+θ T^log¹²(T )T^1/2+θ

T  T^−1/2+2+θ 1

T^0. The first sum in the second equality in (4.1) can be written as

X

n1,n2,n3,n4

n2n4=n1n3=n n1,n2≤√

T /2π n3,n4≤y

bj(n1)bl(n2)ar(n3)am(n4)

(n2n4)^2σa = X

n≤y qT

2π

1 n^2σa



X

k1|n

√n T 2π

≤k₁≤y

ar(k1)bj

 n k1



×



X

k2|n

√n 2πT

≤k₂≤y

a_m(k₂)b_l n k₂



. (4.2)

We now reverse the order of summation so that the partial sums of bj(x)bl(x) are in the inner sum for which estimates are easier to handle. In order to do so, we let d = gcd(k1, k2) and n = ^k1_d^k2w, where

w ≤

qT 2πd max(k1, k2).

Using the above relations and interchanging the sums over k1, k2 and n, the right-hand side of equation (4.2) can be rewritten as

X

k1,k2≤y

ar(k1)am(k2)

(k₁k₂)^2σa d^2σa X

w≤

√T 2πd max(k1,k2)

bj

 k2

dw

 b_l k1

dw

 w^−2σa.

Therefore, 1

U

Z T +U T

B_jψ_rB_lψ_m(σ_a+ it) dt = X

k1,k2≤y

a_r(k₁)a_m(k₂)

(k1k2)^2σa d^2σa X

w≤

√T 2πd max(k1,k2)

b_j k₂ dw

 b_l k₁

dw

 w^−2σa

+ O

 1 T^0

 . (4.3)

Next, following previous arguments, we can estimate the following term from J in Lemma 3.1 as follows

1 U

Z T +U T

χ^∗χ^∗D_jψ_rD_lψ_m(σ_a+ it) dt = e² X

r1,r2≤y

a_r(r₁)a_m(r₂) r1r2

D^2−2σa

× X

w≤

√T 2πD max(r1,r2)

d_jr₂ Dw

r_lr₁ Dw

w^2σa−2+ O

 1 T^0

 , (4.4)

as well as 1 U

Z T +U T

χ^∗DjψrB_lψm(σa+ it) dt = e² X

n3,n4≤y n3≥n4

ar(n3)am(n4)m^∗2−2σa n₃n₄

× X

qT 2π

m∗

n4≤n^∗≤q

T 2π

m∗

n3

d_j(ⁿ_m^∗n∗⁴)d_l ⁿ_m^∗n∗³

n^∗2−2σa + O 1 L⁷

 , (4.5)

and 1 U

Z T +U T

χ^∗B_jψ_rD_lψ_m(σ_a+ it) dt = e² X

n3,n4≤y n3≤n₄

a_r(n₃)a_m(n₄)m^∗2−2σa n3n4

× X

qT 2π

m∗

n3≤n^∗≤ qT

2π m∗

n4

dj(ⁿ_m^∗n∗⁴)dl n^∗n3

m^∗

n^∗2−2σa + O 1 L⁷

 . (4.6)

Combining (4.5) and (4.6), we obtain 1

U Z T +U

T

χ^∗DjψrB_lψm+ χ^∗BjψrD_lψm
(σ_a+ it) dt = e² X

n3,n4≤y

ar(n3)am(n4)m^∗2−2σa n₃n₄

× X

qT 2π

m∗

max (n3,n4)≤n^∗≤ qT

2π m∗

min (n3,n4)

dj(ⁿ_m^∗n∗⁴)dl n^∗n3

m^∗

n^∗2−2σa + O 1 L⁷

 . (4.7)

The three integrals in (4.3), (4.4) and (4.7) upon substitution in (3.9) and using dk T

2πx
= b_k(x) amount to

E_j,l,r,m = X

n3,n4≤y

a_r(n₃)a_m(n₄) (n3n4)^2σa m^∗2σa

Z ^T

2π m∗2 n3n4

1

b_jxn₄ m^∗



b_lxn₃ m^∗



x^−2σa dx

+ O

 1 T^0



+ O 1 L

 . (4.8)

Integration by parts gives that for a polynomial P (u) of degree N in the variable u Z B

A

P (log u)u^α du =

N

X

n=1

(−1)ⁿP⁽ⁿ⁾(log B)

(1 + α)ⁿ⁺¹B^1+α−

N

X

n=1

(−1)ⁿP⁽ⁿ⁾(log A) (1 + α)ⁿ⁺¹A^1+α. For 0 ≤ k ≤ M , denote

φ_k(x) := (1 − x)



1 − 2x + πi 2L

a+2k

,

then

b_k(x) = φ_k log x L



is a polynomial in log x. On applying the above identity on φj(^{log x}_L ) and φ_l(^{log x}_L ), we express the integral in (4.8) as the sum

E_j,l,r,m = X

n3,n4≤y

ar(n3)am(n4) (n₃n₄)^2σa m^∗2σa

2a+2j+2l+2

X

n=0

(−1)ⁿ 2ⁿ

 n

X

v=0

n v



φj(v) 1

Llog xn4

m^∗



× φ_l^(n−v) 1

Llog xn₃ m^∗



x^−2a

   

T 2π

m∗2 n3n4

1

+ O

 1 T^0



+ O 1 L

 ,

Using the Taylor series expansion of φ^(v)_j (x) and φ^(n−v)_l (x) at x = 1 and at x = 0, and generalizations of Lemmas 18 and 20 from [11], we obtain

Ej,l,r,m= 1 2

j+l+2

X

n=0

(−1)ⁿ 2ⁿ



e²(φj(x)φl(x))⁽ⁿ⁾    x=1

− (φ_j(x)φl(x))⁽ⁿ⁾    x=0



×

 1

2(r + m + 1)+ 2rm r + m − 1



+1 2

j+l+2

X

n=0



e²(φj(x)φ_l⁰(x) + φj0(x)φ_l(x))⁽ⁿ⁾    x=1

− (φ_j(x)φ_l⁰(x) + φj0(x)φ_l(x))⁽ⁿ⁾    x=0



× 1

2(r + m + 1) +1

2

j+l+2

X

n=0

(−1)ⁿ 2ⁿ



e²(φj0(x)φ⁰_l(x))⁽ⁿ⁾    x=1

− (φ_j⁰(x)φ_l⁰(x))⁽ⁿ⁾    x=0

 1

2(r + m + 1) + U + O(L⁻¹log⁵L) + O

 1 T^0



+ O 1 L

 , where U is defined by

U := e² 2

j+l+2

X

n=0

(−1)ⁿ 2ⁿ ×

×



(φ_j(x))φ_l(x))⁽ⁿ⁾    x=1

+ (φ_j(x))φ_l⁰(x))⁽ⁿ⁾    x=1

r

r + m + (φ_j⁰(x))φ_l(x))⁽ⁿ⁾    x=1

m r + m



+1 2

j+l+2

X

n=0

(−1)ⁿ 2ⁿ

×



(φ_j(x))φ_l(x))⁽ⁿ⁾    _x=0

+ (φ_j(x))φ_l⁰(x))⁽ⁿ⁾    _x=0

r

r + m + (φ_j⁰(x))φ_l(x))⁽ⁿ⁾    _x=0

m r + m

 . (4.9)

Using Lemma 28 from [11] in the above expression for E_j,l,r,m, we have E_j,l,r,m =

 1

2(r + m + 1)+ 2rm r + m − 1

 Z 1 0

e^2xφ_j(x)φ_l(x) dx + 1 2(r + m + 1)

Z 1 0

e^2xφ_j⁰(x)φ_l⁰(x) dx

+ 1

2(r + m + 1) Z 1

0

e^2x(φ_j(x)φ⁰_l(x) + φ_j⁰(x)φ_l(x)) dx + U + O log⁵L L

 + O

 1 T^0

 . (4.10)

The third integral above can be simplified in terms of the first one as Z 1

0

e^2x(φ_j(x)φ⁰_l(x) + φ_j⁰(x)φ_l(x)) dx = −1 − 2 Z 1

0

e^2xφ_j(x)φ_l(x) dx + O(L⁻¹).

Therefore (4.10) becomes Ej,l,r,m =

 2rm

r + m − 1 − 1 2(r + m + 1)

 Z 1 0

e^2xφj(x)φl(x) dx + 1 2(r + m + 1)

Z 1 0

e^2xφj0

(x)φl0

(x) dx

− 1

2(r + m + 1) + U + O

 1 T^0



+ O 1 L

 . (4.11)

By substituting the values of the function φk in the integral, and using binomial expansions, we estimate the first two integrals in (4.11). For the first integral, we have

Z 1 0

e^2xφj(x)φ_l(x) dx = Z 1

0

e^2x(1 − x)²(1 − 2x)^2a+2j+2l dx + O(L⁻¹).

We write a formula for the integral on the right hand side using repeated integration by parts and obtain,

Z 1 0

e^2x(1 − x)²(1 − 2x)^2a+2j+2l dx

= 1

2(2a + 2j + 2l)− 1

2(2a + 2j + 2l)² + (e²+ 1)

4(2a + 2j + 2l)³ − 9e²− 3 4(2a + 2j + 2l)⁴ + O a⁻⁵
,

(4.12)

For the second integral in (4.11), we have Z 1

0

e^2xφ_j⁰(x)φ_l⁰(x) dx

= Z 1

0

e^2x(1 − 2x)^2a+2j+2l−2((1 + 2a + 4j) − (2a + 4j + 2)x)((1 + 2a + 4l) − (2a + 4l + 2)x) dx

= 2(a + 2j)(a + 2l)

2a + 2j + 2l + 2(a + 2j)(a + 2l) (2a + 2j + 2l)² + 1 + (a + 2j)(a + 2l)

(2a + 2j + 2l)² + 1 +(a + 2j)(a + 2l)(e²− 1) (2a + 2j + 2l)²

 1

2a + 2j + 2l

+ −2(3e²+ 1)(a + 2j)(a + 2l)

(2a + 2j + 2l)² + e²− 1 + 4(a + 2j)(a + 2l) (2a + 2j + 2l)²

 1

2(2a + 2j + 2l)² + O(a⁻³).

(4.13)

We also expand the terms involving r and m in (4.11), and substitute (4.12), (4.13) in (4.11) to obtain

E_j,l,r,m= U +(rm + (a + 2j)(a + 2l))

(2a + 2j + 2l)(r + m) + (rm − (a + 2j)(a + 2l)) (2a + 2j + 2l)(r + m)

 1

r + m − 1

2a + 2j + 2l



+ rm

(2a + 2j + 2l)(r + m)³ − rm

(2a + 2j + 2l)²(r + m)² + (e²+ 1)rm 2(2a + 2j + 2l)³(r + m)

− 1

4(2a + 2j + 2l)(r + m)+ (a + 2j)(a + 2l) (2a + 2j + 2l)(r + m)³ −

 2jl

(2a + 2j + 2l)² + 1

 1

2(r + m)² + 2(a + 2j)(a + 2l)

(2a + 2j + 2l)² + 1 + (a + 2j)(a + 2l)(e²− 1) (2a + 2j + 2l)²

 1

2(2a + 2j + 2l)(r + m)