Omega Theorems for The Twisted Divisor Function
Kamalakshya Mahatab Anirban Mukhopadhyay




NTNU, Trondheim, Norway
[Kamalakshya Mahatab]accessing.infinity@gmail.com

Institute of Mathematical Sciences, HBNI, 
CIT Campus, Taramani, Chennai 600113, India
[Anirban Mukhopadhyay]anirban@imsc.res.in


For a fixed θ≠ 0,  we define the twisted divisor function

    τ(n, θ):=∑_d| nd^iθ .

In this article we consider the error term Δ(x) in the following asymptotic
formula

    ∑_n≤ x^*|τ(n, θ)|^2=ω_1(θ)xlog x + ω_2(θ)xcos(θlog x)
    +ω_3(θ)x + Δ(x),

where ω_i(θ) for i=1, 2, 3 are constants depending only on θ.
We obtain

    Δ(T)=Ω(T^α(T))  where α(T) =3/8-c/(log T)^1/8 and  c>0,

along with an Ω-bound for the Lebesgue measure of the set of points where the above
estimate holds.



[2010]11M41, 11N37


Omega theorems for the twisted divisor function
    Anirban Mukhopadhyay
    December 30, 2023
===============================================




4mm



§ INTRODUCTION


For an arithmetical function f(n) we write

    ∑_n ≤ xf(n)=M(x)+Δ(x),

where M(x) is the main term and Δ(x) is the error satisfying Δ(x)=o(M(x)).
An Ω-estimate for Δ(x) helps us understand the magnitude of fluctuation of error and thereby 
measures the sharpness of an upper bound for error.
 
In <cit.> and <cit.>, Balasubramanian and Ramachandra introduced
a method to obtain a lower bound for 

    ∫_T^T^𝔟 |Δ(x)|^2/x^2α+1x̣
 
in terms of the second moment of the corresponding Dirichlet series D(s),
for some  𝔟>0 and α>0. A nondecreasing lower bound gives 

    Δ(x)=Ω(x^α-ϵ)   for any ϵ>0 .

In these papers, they considered the error terms in asymptotic formulas for partial sums of certain arithmetic functions such as sum of square-free divisors and counting function for non-isomorphic abelian groups. 
This method requires the Riemann Hypothesis to be assumed in certain cases. Balasubramanian, Ramachandra and Subbarao <cit.> modified this technique to apply on error term in the asymptotic 
formula for the counting function of k-full numbers without assuming Riemann Hypothesis. This method has been used by several authors including
<cit.> and <cit.>.

 
 
For a fixed θ≠ 0,  we consider 

    τ(n, θ)=∑_d| nd^iθ .


Note that 

    ∑_d|n
    a≤log d≤ b^*1=1/2π∫_-∞^∞τ(n, θ)e^-ibθ-e^-iaθ/-iθθ̣,

where * denotes that if e^a|n or e^b|n then their contribution to the sum is . So in principle we can restate questions on distribution of divisors of n in terms of τ(n, θ) and 
can take advantage of the multiplicative structure of τ(n, θ).

This function is used in <cit.> to measure the clustering of divisors. In this paper we will study 
the Dirichlet series of |τ(n, θ)|^2, which can be expressed in terms of the Riemann zeta function as 

    D(s)=∑_n=1^∞|τ(n, θ)|^2/n^s=ζ^2(s)ζ(s+iθ)ζ(s-iθ)/ζ(2s)    for  (s)>1.

In <cit.>, Hall and Tenenbaum proved that

    ∑_n≤ x|τ(n, θ)|^2=ω_1(θ)xlog x + ω_2(θ)xcos(θlog x)
    +ω_3(θ)x + Δ(x),

where ω_i(θ)s are explicit constants depending only on θ and

    Δ(x)=O_θ(x^1/2log^6x).

Here the main term comes from the residues of  D(s) at s=1, 1± iθ.
All other poles of D(s) come from the zeros of ζ(2s). Using a pole on the line (s)=1/4, 
Landau's method gives

    Δ(x)=Ω_±(x^1/4).

In <cit.>, we show that

    μ( _j∩ [T, 2T])=Ω(T^1/2(log T)^-12)    for  j=1, 2,

where

    _1   ={x: Δ(x)>(λ(θ)-ϵ)x^1/4}   
       and   _2   ={ x : Δ(x)<(-λ(θ)+ϵ)x^1/4},

for any ϵ>0 and λ(θ)>0. 
Moreover, under Riemann Hypothesis, we obtained

    μ(_j∩ [T, 2T]) =Ω(T^3/4-ϵ),   for  j=1, 2

and for any ϵ>0.

Adopting the method of Balasubramanian, Ramachandra and Subbarao in case of this twisted divisor function, we derive the following theorem.

For any c>0, there exist constants K(c)>0 and T(c)>0 such that for all T ≥ T(c), we get 

    ∫_T^∞|Δ(x)|^2/x^2α+1e^-2x/yx̣≥ K(c)exp( c(log T)^7/8),

where

    α=α(T)=3/8 -c/(log T)^1/8  and  y=T^𝔟   for 𝔟≥ 80.


In particular, this implies

    Δ(x)=Ω(x^3/8exp(-c(log x)^7/8)  )

for some suitable c>0.

The following localised version of the above theorem is immediate from its proof.

For any c>0 and for all sufficiently large T depending on c,
there exists an 

    X ∈[ T, T^𝔟/2log^2 T]
 
for which we have

    ∫_X^2X|Δ(x)|^2/x^2α+1 dx 
    ≥exp( (c-ϵ)(log X)^7/8),

with α as in Theorem <ref> and for any ϵ>0.

Optimality of the above bound is justified in Proposition <ref>. 
We also prove a measure version  of this result:

For any c>0, let

    α(x)=3/8- c/(log x)^1/8
 and  
={x: |Δ(x)|≫ x^α(x)}. 
Then 

    μ(∩ [X,2X])=Ω(X^2α(X)),   as  X→∞.





§ PREREQUISITES


In order to prove the theorem, we need several lemmas, which form the content of this section. 
We begin with a fixed δ_0 ∈ (0,1/16] for which we would choose a numerical value at the end of this section.


 For T>1, let Z(T) be the set of all γ such that 


  * T≤γ≤ 2T,

  * either ζ(β_1+iγ)=0 for some β_1≥+δ_0/2 

or ζ(β_2+i 2γ)=0 for some β_2≥ +δ_0/2.

Let

    I_γ,k = { T≤ t ≤ 2T: |t-γ| ≤ klog^2 T } for   k=1, 2.

We finally define

    J_k(T)=[T,2T] ∖∪_γ∈ Z(T) I_γ,k.




With the above definition, we have for k=1,2

    μ(J_k(T)) = T +O( T^1-δ_0/8log^3 T ).




We shall use an estimate on the function N(σ, T), which is defined as 

    N(σ, T):=|{σ'+it:σ'≥σ, 0<t≤ T, ζ(σ'+it)=0}|.

Selberg <cit.> proved that

    N(σ, T) ≪ T^1-1/4(σ -)log T,   for  σ>1/2.


Now the lemma follows from the above upper bound on N(σ, t), and the observation that

    μ(∪_γ∈ Z(T) I_γ,k) ≪ N(+ δ_0/2, T)log^2 T.



The next lemma closely follows Theorem 14.2 of  <cit.>, but we are including a proof 
as we could not find a clearly written proof of this version which unlike the original one,
does not use Riemann Hypothesis.

For t∈ J_1(T) and σ= 1/2+δ with δ_0 < δ < 1/4-δ_0/2,  we have

    |ζ(σ+it)|^± 1≪exp(loglog t (log t/δ_0)^1-2δ/1-2δ_0)


and

    |ζ(σ+2it)|^± 1≪exp(loglog t (log t/δ_0)^1-2δ/1-2δ_0).




We provide a proof of the first statement, and the second statement can be similarly proved.

Let 1 <σ' ≤log t. We consider two concentric circles centered at σ'+it,
with radius σ'-1/2-δ_0/2 and σ' -1/2- δ_0. 
Since t∈ J_1(T) and the radius of the circle is ≪log t, we conclude that

    ζ(z)≠ 0   for   |z-σ'-it | ≤σ' -  -  δ_0/2

and also ζ(z) has polynomial growth in this region. 
Thus on the larger circle, log |ζ(z)| ≤ c_5log t, for some constant c_5>0.
By Borel-Caratheodory theorem, 

    |z-σ'-it | ≤σ' -  - δ_0   implies |logζ(z)| ≤c_6σ'/δ_0log t,

for some c_6>0.
Let 1/2+δ_0< σ < 1, and ξ>0 be such that 1+ξ< σ'. 
We consider three concentric circles centered at 
σ'+it with radius r_1=σ'-1-ξ, r_2=σ'-σ and 
r_3=σ'-1/2-δ_0,
and call them 𝒞_1, 𝒞_2 and 𝒞_3 respectively. 
Let

    M_i = sup_z∈𝒞_i |logζ(z)|.

From the above bound on |logζ(z)|, we get 

    M_3 ≤c_6σ'/δ_0log t.

Suitably enlarging c_6, we see that 

    M_1 ≤c_6/ξ.

Hence we can apply the Hadamard's three circle theorem to conclude that

    M_2 ≤ M_1^1-ν M_3^ν,    for  ν=log(r_2/r_1)/log(r_3/r_1).

Thus

    M_2 ≤( c_6/ξ)^1-ν(c_6σ' log t/δ_0)^ν.

It is easy to see that

    ν=2-2σ + 4δ_0(1-σ)/1+2ξ-2δ_0 
    +O(ξ) + O( 1/σ').

Now we put 

    ξ=1/σ'=1/loglog t .

Hence

    M_2 ≤c_6  log^ν t loglog t/δ_0^ν
    =c_7 loglog t/δ_0^ν(log t)^2-2σ+4δ_0(1-σ)/1+2ξ-2δ_0,

for some c_7>0.
We observe that

    2-2σ+4δ_0(1-σ)/1+2ξ-2δ_0 < 
    2-2σ+4δ_0(1-σ)/1-2δ_0 =1-2δ/1-2δ_0.

So we get 

    |logζ(σ +it) | 
    ≤ c_7 loglog t (log t/δ_0)^1-2δ/1-2δ_0,

and hence the lemma.


We put y=T^𝔟, for a constant 𝔟≥ 80. Now suppose that

    ∫_T^∞|Δ(u)|^2/u^2α+1e^-u/yụ≥log^2 T,

for sufficiently large T. Then clearly 

    Δ(u) =Ω( u^α) .
 
Our next result explores the situation when such an inequality does not hold.


Let δ_0<δ<1/4-δ_0/2.
For 1/4+δ/2 < α <1/2, suppose that
 
    ∫_T^∞|Δ(u)|^2/u^2α+1e^-u/yụ≤log^2 T,

for a sufficiently large T.
Then we have

    ∫_J_2(T)|D(α+it)|^2/|α+it|^2ṭ≪ 
    1 + ∫_T^∞|Δ(u)|^2/u^2α+1e^-2u/yụ.



Before embarking on a proof, we need the following lemma which is easy to prove using 
Stirling's formula for Γ-function.


Let z be a complex number with 0≤(z) ≤ 1 and |(z)|≥log^2T. For y as above, we have

    ∫_T^∞ e^-u/yu^-zụ =T^1-z/1-z + O(T^-𝔟')

and

    ∫_T^∞ e^-u/y u^-zlog u ụ =T^1-z/1-zlog T + O(T^-𝔟'),

where 𝔟'>0 depends only on 𝔟.





Under the assumption (<ref>), there exists T_0 with T≤ T_0 ≤ 2T such that

    Δ(T_0)e^-T_0/y/T_0^α≪log^2 T,


    and  1/y∫_T_0^∞Δ(u)e^-u/y/u^αụ≪log T.






The assumption (<ref>) implies that

    log^2T    ≥   ∫_T^2T|Δ(u)|^2/u^2α+1e^-u/yụ 
    = ∫_T^2T|Δ(u)|^2/u^2αe^-2u/ye^u/y/uụ
       ≥   min_T≤ u≤ 2T(|Δ(u)|/u^αe^-u/y)^2,

which proves the first assertion.
To prove the second assertion, we use the previous assertion and Cauchy- Schwartz inequality along with assumption (<ref>) to get

    ( ∫_T_0^∞|Δ(u)|/u^αe^-u/yụ)^2
       ≤   ( ∫_T_0^∞|Δ(u)|^2/u^2α+1e^-u/yụ)
    (  ∫_T_0^∞ u e^-u/yụ) 
       ≪    y^2 log^2 T.

This completes the proof of this lemma.


We now recall a mean value theorem due to Montgomery and Vaughan <cit.>.

 For a real number θ, let θ:=min_n∈ℤ|θ -n|.



Let a_1,⋯, a_N be arbitrary complex numbers, and let λ_1,⋯,λ_N be distinct real numbers such that 

    δ = min_m,n
     m≠ nλ_m-λ_n>0.

Then

    ∫_0^T | ∑_n≤ N a_n exp(iλ_n t) |^2 ṭ 
    =(T +O(1/δ))∑_n≤ N |a_n|^2.





For T≤ T_0≤ 2T and (s)=α, we have

    ∫_T^2T| ∑_n≤ T_0|τ(n,θ)|^2/n^se^-n/y|^2 t^-2 dt ≪ 1.




Using  theorem <ref>, we get

    ∫_T^2T| ∑_n≤ T_0|τ(n,θ)|^2/n^se^-n/y|^2 t^-2 dt 
       ≤   1/T^2( T ∑_n≤ T_0  |b(n)|^2 
    + O( ∑_n≤ T_0 n|b(n)|^2)),

where 

    b(n)=|τ(n,θ)|^2/n^αe^-n/y.

Thus

    ∑_n≤ T_0  |b(n)|^2 ≤∑_n≤ T_0d(n)^4/n^2α≪ T_0^1-2α+ϵ

and 

    ∑_n≤ T_0  n|b(n)|^2 ≤∑_n≤ T_0d(n)^4/n^2α-1≪ T_0^2-2α + ϵ

for any ϵ>0, since the divisor function d(n)≪ n^ϵ. As we have α>0, this completes the proof.




For (s)=α and T≤ T_0 ≤ 2T, we have

    ∫_T^2T| ∑_n≥ 0∫_0^1 Δ(n+x+T_0) 
    e^-(n+x+T_0)/y/(n+x+T_0)^s+1x̣|^2 ṭ≪∫_T^∞|Δ(x)|^2/x^2α+1e^-2x/yx̣.



  
Using Cauchy- Schwarz inequality, we get

    | ∑_n≥ 0∫_0^1  Δ(n+x+T_0)/(n+x+T_0)^s+1
    e^-(n+x+T_0)/yx̣|^2 
    ≤   ∫_0^1 | ∑_n≥ 0Δ(n+x+T_0)/(n+x+T_0)^s+1e^-(n+x+T_0)/y|^2 x̣.

Hence 

    ∫_T^2T| ∫_0^1 ∑_n≥ 0Δ(n+x+T_0) e^-(n+x+T_0)/y/(n+x+T_0)^s+1x̣|^2 ṭ
    ≤   ∫_T^2T∫_0^1 | ∑_n≥ 0Δ(n+x+T_0)/(n+x+T_0)^s+1
    e^-(n+x+T_0)/y|^2 x̣ṭ
    
    =   ∫_0^1 ∫_T^2T| ∑_n≥ 0Δ(n+x+T_0)/(n+x+T_0)^s+1
    e^-(n+x+T_0)/y|^2 ṭx̣.

From Theorem <ref>, we can get

    ∫_T^2T| ∑_n≥ 0Δ(n+x+T_0)/(n+x+T_0)^s+1e^-(n+x+T_0)/y|^2 ṭ
       =    T∑_n≥ 0 |Δ(n+x+T_0)|^2/(n+x+T_0)^2α+2
    e^-2(n+x+T_0)/y 
    + O( ∑_n≥ 0 |Δ(n+x+T_0)|^2/(n+x+T_0)^2α+1e^-2(n+x+T_0)/y)
       ≪   ∑_n≥ 0 |Δ(n+x+T_0)|^2/(n+x+T_0)^2α+1e^-2(n+x+T_0)/y.

Hence

    ∫_T^2T| ∑_n≥ 0∫_0^1 Δ(n+x+T_0) e^-(n+x+T_0)/T/(n+x+T_0)^s+1x̣|^2 ṭ
       ≪   ∫_0^1 ∑_n≥ 0 |Δ(n+x+T_0)|^2/(n+x+T_0)^2α+1e^-2(n+x+T_0)/yx̣≪∫_T^∞|Δ(x)|^2/x^2α+1e^-2x/yx̣,

completing the proof.



For s=α+it with 1/4 +δ  < α < 1/2 and t∈ J_2(T), we have

    ∑_n=1^∞|τ(n,θ)|^2/n^s e^-n/y   =   1/2π i∫_2-i∞^2+i∞
    D(s+w) Γ(w) y^w ẉ
       =   1/2π i∫_2-ilog^2T^2+ilog^2T
    +O( y^2∫_log^2T^∞ |D(s+2+iv)||Γ(2+iv)|ṿ).

The above error term is estimated to be o(1). We move the integral to 

    [1/4+δ/2 -α-ilog^2T, 
     1/4+δ/2 -α+ilog^2 T].

Let δ'=1/4+δ/2 -α.
In the region to the right side of this line, (2s+2w)≥ 1/2+δ.  Writing w=u+iv we observe that t+v ∈ J_1(T) since 
t∈ J_2(T). So we can apply 
Lemma <ref> to 
conclude that

    ζ(2s+2w)≫ T^-1.
 
On the above line, we have (s+w)=1/4+δ/2, Thus

    ζ^2(s+w)ζ(s+w+iθ)ζ(s+w-iθ) ≪ T^3/2-δlog^4 T,

where we use the fact that ζ(z)≪(z)^(1-(z))/2log((z)) if 0≤(z) ≤ 1.
Hence by convexity, we see that ζ^2(s+w)ζ(s+w+iθ)ζ(s+w-iθ) has polynomial growth 
on the horizontal lines of integration.  Therefore the horizontal integrals are o(1) by exponential decay of Γ-function.
Since the only pole inside this contour is at w=0, we get

    ∑_n=1^∞|τ(n,θ)|^2/n^s e^-n/y 
    = D(s) + 1/2π i∫_δ'-ilog^2 T^δ'+ilog^2 T
    D(s+w)Γ(w)y^w ẉ +o(1).

For the integral on the right hand side, we have

    D(s+w) y^w ≪ T^5/2 -δ(𝔟/2 +1)

where the exponent of T is negative by our choice of 𝔟 and δ.
Therefore this integral is also o(1). 

Using T_0 as in Lemma <ref>, we now divide the sum into two parts:

    D(s)= ∑_n≤ T_0|τ(n, θ)|^2/n^s e^-n/y 
    + ∑_n > T_0|τ(n, θ)|^2 /n^s e^-n/y+o(1).

To estimate the second sum, we write

    ∑_n > T_0|τ(n, θ)|^2/n^s e^-n/y   =   ∫_T_0^∞ e^-x/y/x^s(∑_n≤ x|τ(n, θ)|^2 )
       =   ∫_T_0^∞ e^-x/y/x^s(̣(x)+Δ(x))
       =   ∫_T_0^∞ e^-x/y/x^s'(x)x̣ 
       + ∫_T_0^∞ e^-x/y/x^s(̣Δ(x)).

Recall that

    (x)=ω_1(θ)xlog x + ω_2(θ)xcos(θlog x)
    +ω_3(θ)x,

thus 

    '(x)=ω_1(θ)log x + ω_2(θ)cos(θlog x)
    -θω_2(θ)sin(θlog x)+ω_1(θ)+ω_3(θ).

 Observe that

    ∫_T_0^∞ e^-x/y/x^scos(θlog x) x̣
    =∫_T_0^∞ e^-x/y/x^s+iθx̣
    +∫_T_0^∞ e^-x/y/x^s-iθx̣.

Applying Lemma <ref>, we conclude that 

    ∫_T_0^∞ e^-x/y/x^s'(x)x̣ = o(1).

Integrating the second integral by parts:

    ∫_T_0^∞ e^-x/y/x^s(̣Δ(x)) 
       =   e^-T_0/yΔ(T_0)/T_0^s
       +   1/y∫_T_0^∞ e^-x/y/x^sΔ(x)  x̣
    -s∫_T_0^∞ e^-x/y/x^s+1Δ(x) x̣.

Applying Lemma <ref>, we get

    ∑_n > T_0|τ(n, θ)|^2/n^s e^-n/y   =   
    s∫_T_0^∞Δ(x) e^-x/y/x^s+1x̣  + O(log T) 
       =    s∑_n≥ 0∫_0^1Δ(n+x+T_0) e^-(n+x+T_0)/y/(n+x+T_0)^s+1x̣ +O(log T).

Hence we have

    D(s)= ∑_n≤ T_0|τ(n, θ)|^2/n^s e^-n/y 
    +s∑_n≥ 0∫_0^1Δ(n+x+T_0) e^-(n+x+T_0)/y/(n+x+T_0)^s+1x̣ +O(log T) .

Squaring both sides, and then integrating on J_2(T), we get

    ∫_J_2(T)|D(α+it)|^2/|α+it|^2ṭ   ≪∫_T^2T|  ∑_n≤ T_0|τ(n, θ)|^2/n^s 
      e^-n/y|^2 ṭ/t^2
        + ∫_T^2T| ∑_n≥ 0∫_0^1Δ(n+x+T_0) e^-(n+x+T_0)/y/(n+x+T_0)^s+1x̣|^2 ṭ.

The proposition now follows using Lemma <ref> and Lemma <ref>.




§ PROOFS OF THE MAIN THEOREMS



 §.§ Proof of Theorem <ref>


We prove by contradiction. Suppose that (<ref>) does not hold.
Then there exists a constant c>0 such that given any N_0>1, there exists T>N_0 for which

    ∫_T^∞|Δ(x)|^2/x^2α+1e^-2x/yx̣≤exp( c(log T)^7/8).

Note that the above statement is weaker than the contrapositive of the statement of theorem. 
This gives

    ∫_T^∞|Δ(x)|^2/x^2β+1e^-2x/yx̣≪ 1,

where

    β=3/8-c/2(log T)^1/8 .

We apply Proposition <ref> to get 

    ∫_J_2(T)|D(β+it)|^2/|β+it|^2ṭ≪ 1.

Now we compute a lower bound for the last integral over J_2(T).
Write the functional equation for ζ(s) as

    ζ(s) =π^1/2-sΓ((1-s)/2)/Γ(s/2)ζ(1-s).

Using the Stirling's formula for Γ function, we get

    |ζ(s)|=π^1/2-σt^1/2-σ|ζ(1-s)|(1+O(1/T))

for s=σ+it. This implies

    |D(β+it)|=t^2-4β|ζ(1-β+it)^2ζ(1-β-it-iθ)
    ζ(1-β-it+iθ)|/|ζ(2β+i2t)|.

Let δ_0=1/16, and 

    β=3/8 -c/2(log T)^1/8=-δ

with 

    δ=1/8+c/2(log T)^1/8.

Then using Lemma <ref>, we get

    |ζ(1-β+it)| = |ζ(+δ+it)|
    ≫exp(loglog t (log t/δ_0)^1-2δ/1-2δ_0).

For t∈ J_2(T) we observe that t±θ∈ J_1(T), and so the same
bounds hold for ζ(1-β+it+iθ) and ζ(1-β+it -iθ). 
Further

    |ζ(2β+i2t)| = |ζ(+(-2δ)+i2t)|
    ≪exp(loglog t (log t/δ_0)^4δ/1-2δ_0).

Combining these bounds, we get

    |D(β+it)| ≫ t^2-4βexp(-5loglog t (log t/δ_0)^1-2δ/1-2δ_0).

Therefore

    ∫_J_2(T) |D(β+it)|^2 ṭ   ≫    T^4-8βexp(-10loglog T(log T/δ_0)^1-2δ/1-2δ_0)
    μ(J_2(T)) 
       ≫    T^5-8βexp(-10loglog T (log T/δ_0)^1-2δ/1-2δ_0),

where we use Lemma  <ref> to show that μ(J_2(T))≫ T.
Now putting the values of  δ and δ_0 as chosen above, we get

    ∫_J_2(T)|D(β+it)|^2/|β+it|^2 dt 
    ≫exp(3c(log T)^7/8),

since 1-2δ/1-2δ_0< 7/8. This contradicts (<ref>), and hence the theorem follows.




 §.§ Proof of Theorem <ref>


Suppose that the conclusion does not hold, hence

    μ(∩ [X,2X]) ≪ X^2α(X).
 
Thus for every sufficiently large X, we get

    ∫_∩ [X,2X] |Δ(x)|^2/x^2α+1dx 
    ≪ X^2α M(X)/X^2α+1= M(X)/X,

where α=α(X) and M(X)=sup_X≤ x ≤ 2X |Δ(x)|^2.
Using dyadic partition, we can prove

    ∫_∩ [T,y] |Δ(x)|^2/x^2α+1dx ≪M_0(T)/Tlog T,  where  M_0(T) =sup_T≤ x≤ y |Δ(x)|^2

and y=T^𝔟 for some 𝔟>0 and T sufficiently large. 
This gives

    ∫_T^∞|Δ(x)|^2/x^2α+1e^-2x/y dx
    ≪M_0(T)/Tlog T.

Along with (<ref>), this implies

    M_0(T)≫ Texp( c/2(log T)^7/8).

Thus

    |Δ(x)| ≫ x^exp(c/4(log x)^7/8),

for some x∈ [T,y].
This contradicts the fact that |Δ(x)| ≪ x^ (log x)^6.






§ OPTIMALITY OF THE OMEGA BOUND FOR THE SECOND MOMENT 

The following proposition shows the optimality of the omega bound in Corollary <ref>.

 Under Riemann Hypothesis (RH), we have
 
    ∫_X^2XΔ^2(x)x̣≪ X^7/4+ϵ

for any ϵ>0. 


Perron's formula gives

    Δ(x)=1/2π i∫_-T^TD(3/8+it)x^3/8+it/3/8+itṭ + O(x^ϵ),

for any ϵ>0 and for T=X^2 with x∈[X, 2X]. Using this expression for Δ(x), we write its second moment as

    ∫_X^2XΔ^2(x)x̣ 
    = 1/(2π)^2∫_X^2X∫_-T^T∫_-T^TD(3/8+ it_1)D(3/8- it_2)/(3/8+it_1)(3/8- it_2)x^3/4+ i(t_1-t_2)x̣ ṭ_1 ṭ_2 
         
    + O(X^1+ϵ(1+|Δ(x)|))
       ≪ X^7/4∫_-T^T∫_-T^T|D(3/8+ it_1)D(3/8- it_2)/(3/8+it_1)(3/8-it_2)(7/4+ i(t_1-t_2))|ṭ_1 ṭ_2 
    + O(X^3/2+ϵ).

In the above calculation, we have used the fact that Δ(x)≪ x^+ϵ as in (<ref>). Also note that for 
complex numbers a, b, we have |ab|≤(|a|^2 + |b|^2). We use this inequality with

    a=|D(3/8+it_1)|/|3/8+it_1|√(|7/4+i(t_1-t_2)|)  and  b=|D(3/8-it_2)|/|3/8-it_2|√(|7/4+i(t_1-t_2)|),

to get

    ∫_X^2XΔ^2(x)x̣   ≪ X^7/4∫_-T^T∫_-T^T|D(3/8- it_2)/(3/8-it_2)|^2ṭ_1/|7/4+ i(t_1-t_2)|ṭ_2 + O(X^3/2+ϵ)
       ≪ X^7/4log X∫_-T^T|D(3/8- it_2)/(3/8-it_2)|^2 ṭ_2 + O(X^3/2+ϵ).

Under RH, convexity bound gives ζ(σ+it) ≪ t^1/2-σ for 0 ≤σ≤ 1/2,
hence |D(3/8-it_2)|≪ |t_2|^+ϵ. So we have 

    ∫_X^2XΔ^2(x)x̣≪ X^7/4+ϵ  for any  ϵ>0.




§ ACKNOWLEDGEMENT

We thank R. Balasubramanian and K. Srinivas for many pertinent comments.
K. Mahatab is supported by Grant 227768 of the Research Council of Norway, and this work was carried out when he was a research fellow at the Institute of Mathematical Sciences, Chennai.

abbrv


