The effect of repeated differentiation on $L$-functions
Jos Gunns Christopher Hughes




School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW
jos.gunns@bristol.ac.uk



Department of Mathematics, University of York, York, YO10 5DD, United Kingdom
christopher.hughes@york.ac.uk




[2010]11M41


We show that under repeated differentiation, the zeros of the Selberg Ξ-function become more evenly spaced out, but with some scaling towards the origin. We do this by showing the high derivatives of the Ξ-function converge to the cosine function, and this is achieved by expressing a product of Gamma functions as a single Fourier transform.




The effect of repeated differentiation on L-functions
    Christopher Hughes
    1 May 2018
=====================================================






§ INTRODUCTION




In 2006 Haseo Ki <cit.> proved a conjecture of Farmer and Rhoades <cit.>, that differentiating the Riemann Ξ-function evens out the zero spacing. Specifically Ki showed that there exists sequences A_n and C_n with  C_n → 0 slowly such that

    lim_n →∞A_nΞ^(2n)(C_nz)=cos(z),


In this paper we extend Ki's result to the entire Selberg Class of L-functions, showing that there exists sequences A_n and C_n (which depend on the properties of L-function under consideration) and constants M' and θ, such that

    lim_n →∞A_nΞ_F^(2n)(C_n z-M'/Λ)= cos(z+θ).

where Ξ_F is the Xi-function for the L-function F, an element of the Selberg Class. This result is stated more precisely in Theorem <ref>.


In <cit.>, Selberg proposed an axiomatic definition of an L-function, now known as the Selberg Class.

A function F(s) is an element of the Selberg Class if:


  * It has a Dirichlet series of the form

    F(s) = ∑_n=1^∞a_n/n^s

which is absolutely convergent for (s)>1.


  * It is a meromorphic function such that (s-1)^m F(s) is an entire function of order 1, for some integer m≥ 0.


  * It has a functional equation of the form Φ(s) = Φ(1-s), where

    Φ(s) =  ϵ Q^s F(s) ∏_j=1^k Γ(λ_j s +μ_j)

with ϵ, Q, λ_j and μ_j all constants, and subject to |ϵ|=1, Q>0, λ_j>0 and (μ_j)≥ 0.


  * The coefficients in the Dirichlet series satisfy a_1=1 and a_n = O(n^δ) for some fixed positive δ.


  * It has an Euler product in the sense that

    log F(s) = ∑_nb_n/n^s

with b_n=0 unless when n=p^r for some prime p and r a positive integer, and b_n = O(n^θ) for some θ<1/2.



Kaczorowski and Perelli <cit.> define an Extended Selberg Class, essentially by dropping the requirement for the function to satisfy an Euler product. Our results apply equally to elements of this extended class of L-functions.

A function F(s) is an element of the Extended Selberg Class if it satisfies axioms (1)–(3) above.



The degree of an L-function is 2Λ, where

    Λ = ∑_j=1^k λ_j .

It is conjectured that the degree is always an integer. However, this is only known for L-functions of degree 2 or less <cit.>. More specifically, it is believed that, using the duplication formula, the gamma functions can be transformed so that λ_j=1/2 for all j (and in such a case, the L-function has degree k).




Let F be an element of the Selberg Class, and set

    ξ_F(s) = s^m (1-s)^m ϵ Q^s ∏_j=1^k Γ(λ_j s +μ_j) F(s).



Note that by assumption of F being in the Selberg Class, ξ_F(s) is an entire function of order 1, with the functional equation ξ_F(s) = ξ_F(1-s).


Set Ξ_F(z) = ξ_F(12+ z).



From the functional equation Ξ_F(z) is a real function for z∈. If the Dirichlet coefficients of F are real, then Ξ(z) is an even function.


Ki proved his result for the Riemann Ξ-function by starting with the integral representation of the Gamma function to show that

    Ξ_ζ(z)=∫_-∞^∞φ(x)e^ x z x,

where

    φ(x)=2 ∑_n=1^∞(2n^4 π^2 e^9x/2 - 3n^2π e^5x/2) e^-n^2π e^2 x.

Note that the functional equation yields the fact that φ(x) = φ(-x).

After a suitable change of variables, this yields

    Ξ_ζ(z)=2π^2 ∫_0^∞ e^-a e^x e^b x(1+O(e^-x)) (e^ x z/2 + e^- x z/2) x,

with a=π and b=9/4. By differentiating such integrals, Ki was able to explicitly show the existence of sequences A_n and C_n such that (<ref>) held. His method also holds for Hecke L-functions, since the functional equation, analogously to the Riemann Xi-function, can be written with a single Gamma function. However, the Selberg Class of L-functions generally includes a product of disparate Gamma functions, which cannot be simplified down to a single one by the multiplication formula of the Gamma function.


In sections <ref> and <ref>, we find the Fourier transform for the analogous Ξ-function for an element of the (extended) Selberg Class of L-functions, showing it can be written as

    Ξ_F(z)=B ∫_-∞^∞φ(x)e^Λ z x x,

where φ(x) = e^-a e^x e^b x(1+O(e^-x)) as x→∞, and where Λ = ∑λ_j.

In section <ref>, we start from that result to demonstrate the existence of sequences A_n and C_n such that

    lim_n →∞ A_n Ξ_F^(2n)(C_n z-M'/Λ)=cos(z+θ)

where θ=(B) and  M'=∑_j=1^k μ_j. We utilize a similar argument to that used by Ki.


The rates of convergence are considered in section <ref>, demonstrated by numerical examples.



 §.§ Acknowledgements


This research forms part of the first author's PhD thesis from the University of York <cit.>.






§ EXPRESSING THE Ξ-FUNCTION AS A FOURIER TRANSFORM



Let F be an element of the Selberg Class, with data m, k, ε, Q, λ_j, and μ_j. The Fourier transform of the Xi-function related to F is

    Ξ_F(x)    = ∫_-∞^∞Ξ_F(z) e^- x z z
       = B̂exp(- â e^x/Λ+ b̂ x ) (1+O(e^-x/Λ))

where

    â =  Λ Q^-1/Λ∏_j=1^k λ_j^-λ_j / Λ

and

    b̂= 2m+M+12Λ/Λ

and

    B̂ =  (-1)^m ε Q^-(M+2m)/Λ  (2π)^(k+1)/2Λ^2m-1/2∏_j=1^k λ_j^-1/2 + μ_j + λ_j (  - M -2m)/Λ

where

    Λ= ∑_j=1^k λ_j

and

    M = ∑_j=1^k μ_j - 12(k-1).




Note that Λ and M are invariant under the Gamma multiplication formulae.


Recall that

    Ξ_F(z)   =ξ_F(1/2+iz) 
       =ε Q^1/2+iz(14+z^2)^m F(12+ z) ∏_j=1^kΓ(λ_jz+μ_j+12λ_j)

is an entire function.
We wish to find its Fourier transform

    Ξ_F(x) = ∫_-∞^∞Ξ_F(z) e^- x z z.

Shifting the contour so that F(s) can be represented by its Dirichlet series, swapping the order of summation and integration and shifting the contour back, we find that

    Ξ_F(x) = ε Q^1/2∑_n=1^∞a_n/n^1/2∫_-∞^∞(14+z^2)^m ∏_j=1^kΓ(λ_jz+μ_j+12λ_j) (n e^x/Q)^- z z.

Thus the Fourier transform can be found by convolutions and differentiations of the Fourier transform of the Gamma function.




Let λ_1,…,λ_k>0 and let α_1,…,α_k be such that their real parts are all positive. Then for large T,

    ∫_-∞^∞(∏_j=1^k Γ(α_j + λ_j z) ) e^- T z z 
    
    =
    C_k exp(-Λ e^T/Λ∏_j=1^k λ_j^-λ_j / Λ + T(A-(k-1)/2)/Λ) (1+O(e^-T/Λ))

where Λ= ∑_j=1^k λ_j and A = ∑_j=1^k α_j and

    C_k = (2π)^(k+1)/2/√(Λ)∏_j=1^k λ_j^-1/2 + α_j + λ_j ( 1/2 (k-1) - A )/Λ.




Booker stated a similar result in the case when λ_j=1/2 for all j, in section 5.2 of <cit.>.



We prove this theorem by induction. The base case, when k=1 says that for λ>0 and (α)>0,

    ∫_-∞^∞Γ(λ z+α) e^- T z z =  2π/λexp(-e^T/λ+ Tα/λ).

This is simply the Fourier transform of one gamma function, a classical result.

With our choice of Fourier constants the convolution theorem is

    ∫_-∞^∞ f(z) g(z) e^- T z z = 1/2π∫_-∞^∞f(x) g(T-x)  x

where f and g are the Fourier transforms of f and g respectively. The Fourier transform of k+1 gamma functions will be the convolution of the Fourier transform of k gamma functions with the Fourier transform of one gamma function, both of which are known by the inductive hypothesis. That is,

    ∫_-∞^∞(∏_j=1^k+1Γ(α_j + λ_j z) ) e^- T z z 
    
     =
    C_k/λ_k+1∫_-∞^∞exp(-Λ e^x/Λ∏_j=1^k λ_j^-λ_j / Λ + x(A-(k-1)/2)/Λ) (1+O(e^-x/Λ) ) 
    ×exp(-e^(T-x)/λ_k+1+ (T-x)α_k+1/λ_k+1)  x

where we have set Λ= ∑_j=1^k λ_j and A = ∑_j=1^k α_j. Later in the proof, we will also set Λ'= ∑_j=1^k+1λ_j and A' = ∑_j=1^k+1α_j.

We will asymptotically evaluate this integral. Note that the exponential in the integrand is dominated by

    -Λ e^x/Λ∏_j=1^k λ_j^-λ_j / Λ -e^(T-x)/λ_k+1

and this has a maximum at x=x_0 where x_0 is such that

    - e^x_0/Λ∏_j=1^k λ_j^-λ_j / Λ + 1/λ_k+1 e^(T-x_0)/λ_k+1 = 0

that is

    x_0 =  T Λ/Λ' + λ_k+1Λ/Λ'ln(1/λ_k+1∏_j=1^k λ_j^λ_j / Λ)

where Λ' = Λ+λ_k+1 = ∑_j=1^k+1λ_j.

Thus, expanding around x=x_0+ϵ for small ϵ, we have (after a fair amount of straightforward algebraic simplification, and using the identity Λ' = Λ + λ_k+1)

    - Λ e^x/Λ∏_j=1^k λ_j^-λ_j / Λ -e^(T-x)/λ_k+1  = -e^T/Λ'∏_j=1^k+1λ_j^-λ_j / Λ'( Λ e^ϵ / Λ + λ_k+1 e^-ϵ/λ_k+1) 
    
    = - Λ' e^T/Λ'∏_j=1^k+1λ_j^-λ_j / Λ'(1 + 1/2λ_k+1Λϵ^2  + B_1 ϵ^3 + O( ϵ^4) )

where B_1 is an inconsequential constant that depends upon Λ and λ_k+1. (We remark that it is no surprise the coefficient of the ϵ term is zero, as this is the expansion around the maximum of the LHS).

Substituting x=x_0+ϵ in the two other terms in the exponent of the integrand in (<ref>) and letting A' = A + α_k+1 = ∑_j=1^k+1α_j we have

    x(A-12(k-1))/Λ + (T-x)α_k+1/λ_k+1 = T (A'-12(k-1))/Λ'
    
    + λ_k+1 (A-12(k-1)) - α_k+1Λ/Λ'ln(1/λ_k+1∏_j=1^k λ_j^λ_j / Λ)
    + B_2 ϵ

where B_2 = A-12(k-1)/Λ - α_k+1/λ_k+1 is another inconsequential constant.

Substituting both these expansions back into (<ref>) we see that the Fourier transform of the k+1 Gamma functions is asymptotically

    C exp(- Λ' e^T/Λ'∏_j=1^k+1λ_j^-λ_j / Λ' + T (A'-12(k-1))/Λ') 
    ×∫exp( - ϵ^2Λ'/2λ_k+1Λ e^T/Λ'∏_j=1^k+1λ_j^-λ_j / Λ'(1 + B_1 ϵ + O( ϵ^2)) + B_2 ϵ) ϵ

where

    C =  C_k /λ_k+1(1/λ_k+1∏_j=1^k λ_j^λ_j / Λ)^λ_k+1 (A-1/2(k-1)) - α_k+1Λ/Λ'.


We utilise here the normal methods of asymptotic analysis, where the range of the ϵ integral is thought of as being small (so O(ϵ) terms are small), but ϵ^2 e^T/Λ' is large, so the Gaussian integral can be extended to the whole real line with trivially small error. To be concrete, truncate the ϵ integral to be over [-e^-T/3Λ' , e^-T/3Λ'] and let Q=Λ'/2λ_k+1Λ∏_j=1^k+1λ_j^-λ_j / Λ', so we have

    ∫_-e^-T/3Λ'^e^-T/3Λ' e^- ϵ^2 Q e^T/Λ'(1 + B_1 ϵ + O( ϵ^2)) + B_2 ϵϵ
    
    = ∫_-e^-T/3Λ'^e^-T/3Λ' e^ - ϵ^2 Q e^T/Λ'(1 - B_1 Q e^T/Λ'ϵ^3  + B_2 ϵ + O(e^2T/Λ'ϵ^6) ) ϵ.

We can extend the integral to be over all  with a tiny error, of size O( e^-Q e^T/3Λ'). Note that due to the symmetry of the integral, the odd terms in ϵ vanish, and note that the big-O term in the integrand contributes O(e^-3T/2Λ') to the integral. Therefore, the above integral equals

    ∫_-∞^∞ e^ - ϵ^2 Q e^T/Λ'(1 + O(e^2T/Λ'ϵ^6) ) ϵ + O( e^-Q e^T/3Λ')
    
    =√(π/Q) e^-T/2Λ'(1 + O(e^-T/Λ')).


It is easy to see the contribution to (<ref>) from outside the range

    [x_0-e^-T/3Λ' , x_0+e^-T/3Λ']

contributes a tiny amount, dominated by the error term above, and so

    ∫_-∞^∞(∏_j=1^k+1Γ(α_j + λ_j z) ) e^- T z z  = √(2πλ_k+1Λ/Λ')∏_j=1^k+1λ_j^λ_j / (2Λ') C ×
    ×exp(- Λ' e^T/Λ'∏_j=1^k+1λ_j^-λ_j / Λ' + T (A'-12 k)/Λ') (1+O(e^-T/Λ') ).

In order to simplify the constant, recall the definitions of C given in (<ref>) and C_k given in (<ref>). After some rearranging, we see that

    √(2πλ_k+1Λ/Λ')∏_j=1^k+1λ_j^λ_j / (2Λ') C
       = (2π)^(k+2)/2/√(Λ')∏_j=1^k+1λ_j^-1/2 + α_k + λ_j (k/2 - A' )/Λ'
       =C_k+1

which is the  required form for k+1 Gamma functions, thus completing the proof.



Let λ_1,…,λ_k>0 and let α_1,…,α_k be such that their real parts are all positive. Then for large T,

    ∫_-∞^∞(14 + z^2)^m (∏_j=1^k Γ(α_j + λ_j z) ) e^- T z z 
    
    =
    C_k,mexp(-Λ e^T/Λ∏_j=1^k λ_j^-λ_j / Λ + T(2m+A-(k-1)/2)/Λ) (1+O(e^-T/Λ))

where Λ= ∑_j=1^k λ_j and A = ∑_j=1^k α_j and

    C_k,m = (-1)^m (2π)^(k+1)/2Λ^2m-1/2∏_j=1^k λ_j^-1/2 + α_j + λ_j ( 1/2 (k-1) - A -2m)/Λ.




The new term (14 + z^2)^m requires the first 2m derivatives of the RHS to be calculated. The big-O term is differentiable, and note that it dominates all the derivatives other than the 2mth derivative. The result then follows immediately.




First note that from the above Corollary, the contribution to (<ref>) for the terms with n>1 are exponentially smaller than the error term in n=1 term, for large x. Since a_1=1 for an element of the Selberg Class, we have that for large x,

    Ξ_F(x) = (-1)^m ε Q^-(M+2m)/Λ  (2π)^(k+1)/2Λ^2m-1/2∏_j=1^k λ_j^-1/2 + μ_j - λ_j ( M +2m)/Λ
    ×exp(-Λ Q^-1/Λ e^x/Λ∏_j=1^k λ_j^-λ_j / Λ + (2m+M+12Λ)x/Λ) (1+O(e^-x/Λ)),

where we have used the Corollary above, with α_j = μ_j+12λ_j, T=x-log Q and we set M = ∑_j=1^k μ_j - 12(k-1). This is the theorem, with the constants B̂, â and b̂ given explicitly.



The proof made essential use of only the first three assumptions arising from F(s) being an element of the Selberg class. Therefore this result holds for F an element of the Extended Selberg Class (with B̂ being trivially changed if a_1≠1).







§ THE Ξ-FUNCTION UNDER REPEATED DIFFERENTIATION



Note that with our choice of constants, the inverse Fourier transform is

    Ξ_F(z) = 1/2π∫_-∞^∞Ξ_F(x) e^ x z x.



Note that the μ_j, part of the data of the L-function F, could be complex. If we define

    M'=∑_j=1^k μ_j,

and rescale z we have

    Ξ_F(z-M'/Λ)    =Λ/2π∫_-∞^∞Ξ_F(x Λ) e^- x M' e^ x z  x 
       = B ∫_-∞^∞φ(x)e^ixz x

where by Theorem <ref>

    φ(x) = e^-a e^x e^b x(1+O(e^-x)),

with

    a =  Λ Q^-1/Λ∏_j=1^k λ_j^-λ_j / Λ,


    b= 2m+12Λ-12(k-1) + ∑_j=1^k μ_j

and B = B̂Λ / 2π. (Note that a,b ∈ and, in the notation of Theorem <ref>,   a=â and b = Λb̂ -  M').


Let Ξ_F(z) be the Xi-function for the L-function F, an element of the Selberg Class. Let w_n be defined as the solution to

    a w_n e^w_n = b w_n +2n

where a and b are given by (<ref>) and (<ref>) respectively. Then uniformly on compact subsets of ,

    lim_n→∞ A_n Ξ_F^(2n)(C_n z-M'/Λ) = cos(z+(B)),

where  Λ,  M', and B are given in Theorem <ref>, and the sequences A_n and C_n are given by

    A_n = (-1)^n exp(a e^w_n - b w_n)  √(n)/2|B|Λ^2n w_n^2n+1/2√(π)

and

    C_n = 1/Λ w_n.





One can see that, for large n, the w_n defined in the theorem satisfies

    w_n∼log(2n/a)-loglog(2n/a) .




From the functional equation for the L-function we have that

    Ξ_F(z-M'/Λ)=Ξ_F(z-M'/Λ)

so

    B ∫_-∞^∞φ(x)e^ixz x   =B∫_-∞^∞φ(x)e^-ixz x
       =B∫_-∞^∞φ(-x)e^ixz x,

and since this holds for any z ∈ℂ we have

    Bφ(x)=Bφ(-x).

Therefore

    Ξ_F(z-M'/Λ)=∫_0^∞φ(x)(B e^ixz+Be^-ixz)  x.

We can now consider just the integral

    f(z)=∫_0^∞φ(x)e^ixz x

as the second integral will behave in much the same way. Differentiating this, we have that

    f^(2n)(z)=(-1)^n ∫_0^∞φ(x)x^2ne^ixz x.


Haseo Ki <cit.> proved that uniformly on compact subsets of ,

    lim_n →∞∫_0^∞v_nφ(w_nx)x^2ne^ixz x =e^iz,

where φ(x) is of the form given in (<ref>), and w_n is defined such that

    a w_n e^w_n = b w_n +2n

and

    v_n = √(n w_n/π)e^ ae^w_n e^-b w_n .



Therefore, we have that

    f^(2n)(z / w_n)   =(-1)^n∫_0^∞φ(x)x^2ne^ x z / w_n x
       =(-1)^n w_n^2n+1∫_0^∞φ(w_nx)x^2ne^ixz x

and using Ki's work (and including the error term) we have

    f^(2n)(z/w_n)=√(π/nw_n)(-1)^ne^-ae^w_ne^bw_nw_n^2n+1e^iz(1+𝒪(w_n^-2)).

From (<ref>) we see that

    1/Λ^2nΞ_F^(2n)(z-M'/Λ) = B  f^(2n)(z) + Bf^(2n)(-z)

so setting C_n = 1/Λ w_n,

    (-1)^n e^a e^w_n - b w_n w_n^-2n-1√(n w_n/π)1/|B| Λ^2n   Ξ_F^(2n)(C_n z-M'/Λ) 
       =(B/|B| e^iz+B/|B|e^-iz)(1+𝒪(w_n^-2)) 
       =2cos(z+(B))(1+𝒪(w_n^-2))

and after taking the limit, the proof Theorem <ref> is complete.









§ NUMERICAL DEMONSTRATIONS


In this section we briefly discuss how the L-function's data affects the  convergence to the cosine function. Recall that the error term is O(w_n^-2)
where

    w_n ∼log(2n/a),

with

    a = Λ Q^-1/Λ∏_j=1^k λ_j^-λ_j / Λ.


Therefore L-functions with larger conductor converge slightly more quickly, and high degree L-functions converge more slowly. This fact is clearer if one assumes that one can transform the L-function so its data has λ_j=1/2 for all j, since then a = k Q^-2/k.


The sequence C_n effectively scales the density of the zeros of the (2n)th derivative. We have that

    C_n=1/Λ w_n→ 0.

which means that the zeros of the unscaled (2n)th derivative have moved towards the origin. Compare, for example, the Riemann Xi-function before any derivatives have been taken and  after 100 derivatives have been taken.

These figures also demonstrate the convergence to the cosine function.

Finally, the A_n term dictates how large the derivatives of the L-functions get. From

    A_n=√(n)(-1)^ne^ae^w_ne^-bw_n/2w_n^2n+1/2√(π)|B| Λ^2n

and using the defining equation for w_n, a w_n e^w_n = b w_n + 2n, we have that

    log| A_n | = 2n(1-logΛ - log w_n) - a e^w_n ( w_n-1 ) + 12log n - 12log w_n + O(1)

and so since w_n ∼log(2n/a), as n →∞ we have that A_n → 0, which means that the size of the (2n)th derivative gets large as n increases, although for L-functions of small degree where logΛ < 1 the size of the derivatives can initially decrease, before eventually increasing.







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