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MODELING CONVOLUTIONS OF LFUNCTIONS
by
RALPH MORRISON
STEVEN J. MILLER, ADVISOR
A thesis submitted in partial ful.llment
of the requirements for the
Degree of Bachelor of Arts with Honors
in Mathematics
WILLIAMS COLLEGE
Williamstown, Massachusetts
May 14, 2010
ABSTRACT
A number of mathematical methods have been shown to model the zeroes of Lfunctions with remarkable success, including the Ratios Conjecture and Random Matrix Theory. In order to understand the structure of convolutions of families of Lfunctions, we investigate how well these methods model the zeros of such functions. Our primary focus is the convolution of the Lfunction associated to Ramanujan’s tau function with the family of quadratic Dirichlet Lfunctions, for which J.B. Conrey and N.C. Snaith computed the Ratios Conjecture’s prediction. Our main result is performing the number theory calculations and verifying these predictions for the onelevel density for suitably restricted test functions up to squareroot error term. Unlike Random Matrix Theory, which only predicts the main term, the Ratios Conjecture detects the arithmetic of the family and makes detailed predictions about their dependence in the lower order terms. Interestingly, while Random Matrix Theory is frequently used to model behavior of Lfunctions (or at least the main terms), there has been little if any work on the analogue of convolving families of Lfunctions by convolving random matrix ensembles. We explore one possibility by considering Kronecker products; unfortunately, it appears that this is not the correct random matrix analogue to convolving families..
ACKNOWLEDGEMENTS
First and foremost, I would like to thank Professor Steven J. Miller for being an outstanding advisor. Without his guidance and support I would never have been able to face down the pagelong equations and daunting theoretical concepts that have arisen over the past year. I would like to thank my second reader Professor Mihai Stoiciu for providing feedback on my work and helping me guide it to its .nal form, and Professor Carston Botts for his notes on the AcceptReject method. I would also like to thank Eduardo Dueñez, Duc Khiem Huynh, and Nina Snaith for their advice and assistance via email over the past two semesters. Finally, I would like to thank my fellow thesis students and the rest of the Williams mathematics department (faculty, students and all) for all their support and for creating and maintaining a fun and intellectually stimulating environment in which to do research.
CONTENTS
1. Introduction 4
1.1. The Ratios Conjecture 6
1.2. Random Matrix Theory 8
2. Quadratic Twists of the Tau Lfunction 9
2.1. The Explicit Formula 10
2.2. Analyzing the Sum Over Zeros 12
2.3. Analyzing the OneLevel Density 16
3. Comparison With the Ratios Conjectures’ Predictions 21
3.1. The Logarithmic Derivatives 22
3.2. The Derivative of B.
23
3.3. The Error Term 29
3.4. Where To Go From Here 34
4. Random Matrix Theory Models of Convolutions 36
4.1. Types of Matrices 36
4.2. Sampling from Distributions 38
4.3. Combinations of Orthogonal and Symplectic Matrices 38
4.4. What Happens When We Include Unitary 39
4.5. Other Methods of Combining Matrices 39
4.6. Random Matrix Conclusions 41 Appendix A. Key Lemmas 43 References 50
1. INTRODUCTION
One of the most important areas in modern number theory is the study of the distribution of the zeroes of Lfunctions, meromorphic functions on the complex plane that are continuations of in.nite series. The simplest is the most wellknown Lfunction, the Riemannzeta function. It is de.ned by
8 1
0
11
.(s):= =1  (1.1)
ns psn=1 p prime
for Re(s) > 1 and extended to a meromorphic function. The extension satis.es a functional equation relating its value at s to its value at 1  s, and trivially vanishes at the negative even integers (which are called the trivial zeros):
1 sp s
.(s) := s(s  1)G2 .(s)= .(1  s). (1.2)
22
The Riemann Hypothesis, often considered the most important open question in mathematics, is the conjecture that all nontrivial zeros of .(s) have real part equal to 1/2. The distribution of the zeros of this and other Lfunctions encode crucial number theoretic information on subjects ranging from the distribution of the primes to properties of class numbers and even mirror the energy levels of neutrons in quantum mechanics, suggesting a deep connection between this branch of mathematics and nuclear physics. As proofs of properties of these zeros are often out of reach of rigorous methods, methods of modeling these zeros are vital in understanding and formulating appropriate conjectures about Lfunctions. A familiarity with the standard properties of Lfunctions is important in understanding the content and results of this thesis, though intuitive interpretations will be offered whenever appropraite. (See [IK, MTB] for background on Lfunctions and [FM, Ha] for the history of the interplay between nuclear physics and zeros of the Riemann zeta function.)
The particular object we will study is the onelevel density of the low lying zeros of a family of Lfunctions, which relates sums of an even Schwartz function f at the zeros of the Lfunction to sums of the Fourier transform f¢at the primes. As f is a Schwartz function, it vanishes rapidly as x.8. Intuitively, this will be the window through which we observe the lowlying zeros. Ideally, we would like to use a delta spike instead of a Schwartz test function to get a perfect picture at a point; however, the delta spike has a Fourier transform of in.nite support, which makes such a function inapplicable as the resulting sums of the Fourier transform cannot be evaluated. Following [ILS], we study the
onelevel density for an Lfunction f, de.ned by
0
.f L
D(f, f) := f ; (1.3)
p
.f
here 1/2+ i.f runs over the nontrivial zeros of the Lfunction (which under the Generalized Riemann Hypothesis all have . . R) and Lp is a scaling factor (de.ned explicitly in equation 2.31) that measures the spacings between zeros near the central point. As each Lfunction only has a bounded number of zeros within this distance of the central point, it is necessary to average the onelevel density over all f in a family F. This allows us to use results from number theory to determine the behavior on average near the central point 1/2. The exact nature of just what constitutes a family is still being determined, but standard examples include Lfunctions attached to Dirichlet characters, cuspidal newforms, and families of elliptic curves.
We assume our family of Lfunctions F can be ordered by conductor, and denote by F(Q) all elements of the family whose conductor is at most Q. The quantity of interest ends up being the limit of
00
1 .L
f (1.4)
F(Q) p
f.F(Q) .
as Q .8. Thus we consider the limiting behavior of the average of the onelevel densities as the conductors grow.
For a “nice” family of Lfunctions, Random Matrix Theory (see [KaSa1, KaSa2]) pre
dicts that the behavior of the zeros as the conductors tend to in.nity agree with the N .8 scaling limits of a classical compact group of N × N matrices, most often either unitary, symplectic, or a type of orthogonal (even, odd or mixed). Given two families of Lfunctions F and G, the RankinSelberg convolution F×G is a new family of Lfunctions built from elements of F and G. This is a natural type of Lfunction family to study, and is likely to be accessible in the simplest nontrivial case of convolving a family of size 1 with another family. An interesting feature of these convolutions was found by Miller and Dueñez in [DM2], namely that for “nice” families of Lfunctions F and G, the underlying symmetry groups of F and G determine the underlying symmetry group of F×G in a simple, multiplicative way. Speci.cally, to each family F is associated a symmetry constant cF (0 for unitary, 1 for symplectic and 1 for orthogonal) and cF×G = cF · cG. Unfortunately this only leads to predictions for the main term of the onelevel density, and it is in the lower order terms that the arithmetic of the families surface.
In this thesis we focus on testing the Ratios Conjecture’s power of modeling the convolution a family of size 1 with another family (Sections §2 and §3). This will allow us to see how the arithmetic of our family enters. Additionally, as Random Matrix Theory has successfully predicted numerous properties of Lfunctions, we try and .nd the random matrix analogue of convolving two families. To our knowledge this has yet to be investigated in the literature. In Section §4 we report on numerical investigations of the Kronecker products of families of random matrices, which is a natural candidate to model convolutions.
1.1. The Ratios Conjecture. The Lfunction Ratios Conjecture of Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] (see also [CS1] for many worked out examples of the conjec
ture’s prediction) are formulas for the averages over families of Lfunctions of ratios of products of shifted Lfunctions. Their “recipe” for performing these calculations starts by using the approximate functional equation, where the error term is discarded, to expand the Lfunctions in the numerator; the Lfunctions in the denominator are expanded via the Mobius function. They then average over the family, and retain only the diagonal pieces. These are restricted sums over integers, but are then completed and extended to sums over all integers; again the error term introduced is ignored. These methods, far simpler to implement than rigorous analysis, have easily predicted the answers to many dif.cult computations, and have shown remarkable accuracy. The resulting formulas make detailed predictions on numerous problems, ranging from moments to spacings between adjacent zeros and values of Lfunctions.
A standard test of the Ratios Conjecture is to compare the Ratios Conjecture’s predictions for the onelevel density of a family of Lfunctions with rigorous calculation. Agreement has been found (for suitably restricted test functions) for families of Dirichlet Lfunctions and cuspidal newforms (see [GJMMNPP, Mil3, Mil5, MilMo]). In addition to strengthen
ing the credibility of the conjecture, these calculations provide insight into the signi.cance of the terms that arise in the number theoretic calculations whose corresponding terms in the Ratios Conjecture’s predictions are more clearly understandable. For example, in [Mil3] the Ratios Conjecture’s prediction allows interpretation of a lower order term in the behavior of the family of quadratic Dirichlet characters as arising from the nontrivial zeros of the Riemann zeta function.
Our primary object of study is the collection of quadratic twists of the Lfunction associated to Ramanujan’s tau function, a family that can be viewed as the convolution of the family of quadratic Dirichlet Lfunctions with the family consisting solely of the tau Lfunction. The Ratios Conjecture’s prediction for this family was computed by Conrey and Snaith in [CS1]. We perform the number theoretic calculations of the zero statistics for the onelevel density for this family, and compare our results to the Ratios Conjecture’s prediction. Our main result is the following.
Theorem 1.1. Consider the family of quadratic twists of the tau Lfunction with even fundamental discriminants d = X; denote the number of such d by X* (which is essential a constant times X). For supp(f¢) . (s, s) with s< 1, the onelevel density equals
00
1 L
g.d
X* p
d=X.d
8
0
1 d G. p. G. p.
= g(.)2log + 6+ i +6  i
2LX*8 2p G L G L
d=X
88
00 (ap 2k + a2pk)) log p 0 log p 0 (ap 2k + a2pk)
+2 + d.
k(1+ 2pi. k(1+ 2pi.
pL ) (p + 1) )
L
pk=1 pk=1 p
+ O(X(1s)/2 log6 X) (1.5)
which agrees with the Ratios Conjecture’s prediction up to an error term of size O(X(1s)/2+e) for any e> 0 (essentially the error term of the expression).
In addition to being of interest in its own right, understanding this family is useful for investigations of elliptic curves. These families are of considerable importance, as they are ideal for viewing effect of multiple zeros on nearby zeros. By work of C. Breuil, B. Conrad,
F. Diamond. R. Taylor and A. Wiles [BCDT, TW, Wi], the Lfunction of an elliptic curve agrees with that of a weight 2 cuspidal newform of level N (where the integer N> 1 is the conductor of the elliptic curve). There are many similarities between these Lfunctions and that associated to the Ramanujan tau function, and two major differences. The .rst is that the tau function is associated to a weight 12 cusp form, and the second is that the level of the tau function is 1 and not N. Both of these effects make the tau function more amenable to analysis and numerical experimentation: the higher weight leads to less discretization in the value of the Lfunction at the central point, and the level being 1 means that there are no bad primes in the explicit formula.
In spite of these differences, the analysis of our family of Lfunctions is comparable to that of the family of quadratic twists of the Lfunction associated to an elliptic curve, for which the Ratios Conjecture’s predictions have not yet been shown to agree with the number theory results. Analysis of the family of quadratic twists of the tau Lfunction provides a useful starting point for the elliptic curvebased family. The .rst lower order term of this family is very important in ongoing investigations of the excess repulsion observed in the .rst zero above the central point (see [DHKMS1, DHKMS2]. The Ratios Conjecture’s prediction for these lower order terms have been inputted in some of these models, but has yet to be veri.ed. The analysis of quadratic twists of the Ramanujan tau function is almost identical to the analysis needed there, the only difference being the effects of the bad primes are not present. Thus this work provides the framework that can be applied to study these elliptic curve families.
1.2. Random Matrix Theory. Random Matrix Theory (see [Co, Dy1, Dy2, KaSa1, KaSa2, Wig1, Wig2, Wig3, Wig4, Wig5, Wis]) has been extraordinarily successful in modeling diverse systems ranging from nuclear physics to statistics to number theory. In this thesis we are interested in its applications to predicting the behavior of Lfunctions. The nlevel correlation between the normalized zeros of the Riemann zetafunction and the normalized eigenvalues of matrices in the Gaussian Unitary Ensemble was .rst noted in the early 1970s by Dyson and Montgomery [Mon], and then extended by many others (see [Hej, Od1, Od2, RS]). While the behavior of zeros far from the central point is universal, he behavior near the central point depends on the family. This is observed in additional statistics such as nlevel densities (see for example [ILS, KaSa1, KaSa2]) and moments (see for example [CFKRS]). Following the success of these investigations, Random Matrix Theory has served as an extremely useful tool for predicting the behavior of Lfunctions.
In an attempt to model the RankinSelberg convolution of families of matrices, we investigate the eigenvalue statistics of Kronecker products of matrices in the Gaussian Unitary Ensemble. Inspired by [DM2], we look at lowest eigenangle statistics of combinations of different types of matrices to see if there seems to be a multiplicative symmetry constant. Qualitative attributes of our computed distributions indicate that this is not the appropriate model for convolving families of Lfunctions. For instance, orthogonal combined with orthogonal looks symplectic on the number theory side; but the distribution of lowest eigenangles for the orthogonal/orthogonal matrix combination features repulsion from zero, while that for symplectic matrices does not. However, the similarities and differences between various combinations suggest that there is a great deal of structure in the eigenangle statistics of these Kronecker products that warrants further investigation.
2. QUADRATIC TWISTS OF THE TAU LFUNCTION
The .rst family of Lfunctions used in our main convolution is
G = {L(s, .d)  d> 0 is an even fundamental discriminant},
where .d is the quadratic Dirichlet character modulo d. A Dirichlet character (modulo d), denoted ., is a type completely multiplicative function on the units of Zwith period d, and .d denotes the unique quadratic Dirichlet character mod d. We let d be a fundamental discriminant, meaning that either d = 1 mod 4 is squarefree or d/4 = 2, 3 mod 4 is squarefree. We further restrict to d even. If .d is the quadratic character associated to the fundamental discriminant d with d> 0, we have .d(1) = 1.
The second family, which consists of one element, arises from Ramanujan’s tau function. The Ramanujan tau function t : N . Z is de.ned by the coef.cients of the Fourier expansion of .(t)24, where . is the Dedekind eta function. That is,
8
0
.(z)24 n
= t (n)q (2.1)
n=1
with q = e2piz. Note that .(z)24 is a scalar multiple of the discriminant modular function, a holomorphic cusp form of weight 12 and level 1. In 1917 Mordell proved that t(mn)= t(m)t(n) if gcd(m, n)=1 (that is, t is a multiplicative function) and that
r+1) 11t(p r1)
t(p = t(p)t(p)r  p (2.2)
for p prime and r a positive integer. In 1974 Deligne proved that t(p)= 2p11/2 for all p prime. (For more on the tau function see [Se].) De.ning t*(n)= t (n)/n11/2 , we have t*(p)= 2 for all p prime. Using equation (2.2), we have
r+1) t(pr+1)
t * (p =
(r+1)(11/2)
p
11t(p
t(p)t (p)r pr1)
= 
(r+1)(11/2) (r+1)(11/2)
pp
r1)
t(p) t(p)r t(p
=+
11/2 pr(11/2) (r1)(11/2)
pp
r1)
= t * (p)t * (p r)  t * (p (2.3)
for p prime and r a positive integer. Since t* is a multiplicative function, we may consider the Lfunction
0
8t*(n) t*(p)1 1
L(s, t * )= =1  + (2.4)
ns ps p2s n=1 p prime
for Re(s) > 1.
We consider the Lfunction families G = {L(s, .d) .d is a quadratic character } and H = {L(s, t*)} (noting that H has only one element). Convolving these families, we have (by the work of DueñezMiller [DM2]) the orthogonal family F = G×H of quadratic twists of the Lfunction L(s, t*), denoted by L.(s, .d). The Ratios Conjecture’s calculations for this family of Lfunctions were performed by Conrey and Snaith in [CS1]. To test the power of the Ratios Conjecture as it applies to the convolution F, we perform the number theory computations and determine the onelevel density of the zeros for suitably restricted test functions. This comprises the remainder of Section §2. We then compare this to the Ratios Conjecture’s predictions in Section §3, and see that they agree up to O(X(1s)/2+e), where the support of the transform of our test function is contained in (s, s), where s< 1 (i.e., supp(f¢) . (s, s)).
2.1. The Explicit Formula. In this subsection we derive the explicit formula, which connects sums of our test function evaluated at the zeros of our family to sums of the Fourier transform of our test function evaluated at the logarithms of the primes; the onelevel density is just a scaled version of this. We follow the arguments in [RS].
Let L.(s, .d) .F. The essence of our strategy is to consider a contour integral of the logarithmic derivative L.(s, .d) and then shift this integral, picking up contributions from the zeros of L.(s, .d). As L.(s, .d) appears in the denominator of this logarithmic derivative, the contour shift of this integral picks up those zeros as poles, giving us information about their distribution. We analyze the resulting expression for a .xed d and then take the limit of the average over all d = X (as we cannot average over an in.nite number).
For the purposes of averaging, we de.ne X* = d=X 1 where d is an even fundamental discriminant. By Lemma A.1 we have
3
X * = p2 X + O(X1/2), (2.5)
and thus X* is of the same order of magnitude as X. For all subsequent sums over d, this will be the range of d (i.e., we always assume d to be an even fundamental discriminant at most X).
First we establish some key formulas. Written as an Euler sum and an Euler product, we have
8.d(n)t*(n) t*(p).d(p) .d(p2) 1
0
L.(s, .d)= =1  +
ns ps p2s n=1 p
1 1
ap.d(p) ap.d(p)
=1  1  (2.6)
ps ps p
where ap,ap are the roots of the quadratic (in 1/ps) equation 1t*(p).d(p)/ps+.d(p2)/p2s ,
meaning they are t *(p).d(p) ±(t*(p).d(p))2  4.d(p2) /.d(p2). Given that
t * (p).d(p))2  4.d(p 2) = 0,
these roots are either the same (and real) or are distinct and complex conjugates of one another. In both cases, we have that they are complex conjugates (justifying our notation), and that they satisfy ap · ap =1 and ap + ap = t*(p). Since both have multiplicative inverse equal to complex conjugate, they are both of norm 1. We now wish to extend our function to the entire complex plane. For d> 0, our Lfunction has the functional equation
d s
..(s, .d) := G(s + 11/2)L.(s, .d)= ..(1  s, .d) (2.7)
2p
(see, for instance, [CS1, IK]).
We integrate the logarithmic derivative of ..(s, .d) weighted by a Schwartz function to ensure suf.cient decay rate. We assume the Generalized Riemann Hypothesis (GRH), so that if 12 + i. is a zero of .(s, .d) then . . R. Let f be an even Schwartz function where its Fourier transform
8
¢2pix.dx
f(.)= f(x)e (2.8)
8
has .nite support; that is, supp(f¢) . (s, s) for some .nite s. Extend f(x) to the whole complex plane via
s  1 H(s)= f 2 . (2.9)
i Note that H(s) is scaled so that if s = 21 + i.d is a zero of .(s, .d), H(s)= f(.d). Set
1 ..(s, .d)
I = H(s)ds (2.10)
2pi ..(s, .d)
Re(s)=3/2
0
1 ..(s, .d)
I = f(.d)+ H(s)ds (2.11)
2pi ..(s, .d)
Re(s)=1/2
.d
where .d is the imaginary part of a nontrivial zero, and the sum is over all such values. Recall from equation (2.7) that ..(s, .d)= ..(1  s, .d); it follows that ..(s, .d)= ..(1  s, .d). Combined with equation (2.11), this gives us
0
1 ..(1  s, .d)
I = f(.d)  H(s)ds. (2.12)
2pi ..(1  s, .d)
Re(s)=1/2
.d
Performing the change of variables s . 1  s, we obtain
0
1 ..(s, .d)
I = f(.d)  H(1  s)ds. (2.13)
2pi ..(s, .d)
Re(s)=3/2
.d
Subtracting equation (2.13) from (2.10) proves
Theorem 2.1.
0
1 ..(s, .d)
f(.d)= [H(s)+ H(1  s)]ds. (2.14)
2pi ..(s, .d)
Re(s)=3/2
.d
This result, when properly averaged over a .nite subset of the family F, will give us the onelevel density.
2.2. Analyzing the Sum Over Zeros. Having found an expression for . f(.) for a .xed d, we wish to manipulate it into a more informative form before averaging over d to obtain the onelevel density for our family. First we .nd more a more useful way to express the logarithmic derivative of ..(s, .d). Taking the logarithmic derivative of equation (2.7), we have
..(s, .d) d G(s + 11/2) L.(s, .d)
=log+ + . (2.15)
..(s, .d)2p G(s + 11/2) L.(s, .d)
It will also be useful to have the logarithmic derivative of equation (2.6), which is
ap.d(p) ap.d(p)L.(s, .d) 0 ps ps
=  log p +
L.(s, .d)1  ap.d(p) 1  ap.d(p)
pps ps
8
00 (apk + apk).dk(p) =  log p sk . (2.16)
p
k=1 p
Using equation (2.15) we expand the logarithmic derivative in (2.14) and shift the con
.(s,.d)
tours of all terms except the L.term to Re(s)= 1 . This gives us
L.(s,.d)2
0
f(.d)= I1 + I2 (2.17)
.d
where
1 d G
I1 = log +(s + 11/2) [H(s)+ H(1  s)]ds (2.18)
2pi 2p G
Re(s)=1/2
and
1 L.(s, .d)
I2 =[H(s)+ H(1  s)]ds. (2.19)
2pi L.(s, .d)
Re(s)=3/2
The integral in (2.18) with s = 21 + iy is
8
1 d G1 11
I1 = log++ iy +2f(y)idy
2pi 2p G2 2
8
8
1 d G
= 2log +2 (6+ iy) f(y)dy
2p 2p G
8
8
1 d GG
= 2log + (6+ iy)+ (6  iy) f(y)dy. (2.20)
2p 2p GG
8
We now analyze I2. Combining equations 2.16 and 2.19, we have
0
8
2pi pks
Re(s)=3/2
k=1 p
0
(akp + akp).kd(p) log p
1
[H(s)+ H(1  s)]ds.
(2.21)
I2 =

We wish to switch the order of integration and summation (over k and p). To justify this, we will prove
Lemma 2.2.
0
0
8
Re(s)=3/2
k=1 p
(apk + akp).kd(p) log p
pks
[H(s)+ H(1  s)]
ds < 8.
(2.22)
Proof. Note that
00 00
8(ak + ak).k(p) log p 8log p ·ak + ak·.k(p)
ppdppd
=
pks pk(3/2+iy)k=1 pk=1 p
00
82 · 1 · log p
=
p3k/2 ·eiyk log pk=1 p
00
82 log p
=
k)3/2
(p
k=1 p
0
82 log n
=
3/2
n
n=1
=  2. (3/2), (2.23)
a constant independent of s and d. Thus we have
8
00 (apk + apk).k(p) log pd[H(s)+ H(1  s)] ds
pks
Re(s)=3/2
k=1 p
00
8(apk + akp).kd(p) log p
= H(s)+ H(1  s) ds
pks
Re(s)=3/2
k=1 p
= C H(s)+ H(1  s)ds
Re(s)=3/2
= C H(s)ds + H(1  s)ds . (2.24)
Re(s)=3/2 Re(s)=3/2
To see that both integrals in equation (2.24) are convergent, note that
8
¢2pix.d.
f(x)= f(.)e
8 8
¢2pi(x+iy).d.
f(x + iy)= f(.)e
8 8
H(x + iy)
=
8
f¢(.)e
2p(x
1
2
). 2piy.d..
· e
(2.25)
2p(x
For .xed x, H(x+iy) is the Fourier transform of a Schwartz function (namely f¢(.)e
1
2
)),
meaning that it itself is Schwartz. This means that it decays faster than 1/yk for any k . N, implying that both integrals converge. The claim (equation (2.22)) follows. .
By the FubiniTonelli Theorem, we may switch summation and integration in (2.22) as the absolute value leads to a .nite integral in the product measure. Doing so, pulling out terms constant with respect to s, and noting that 1/pks = eks log p, we may rewrite equation
(2.21) as
8
00
1
I2 =  (apk + apk).dk(p) log p [H(s)+ H(1  s)]e ks log pds.
2pi
Re(s)=3/2
k=1 p
(2.26)
We wish to shift our contour to Re(s)= 12 . Consider the integral
ks log pds
[H(s)+ H(1  s)]e (2.27)
C
where C is the rectangle de.ned by the points 3 + iM, 3 iM, 1 +iM, and 1 iM, where
222 2
M> 0. As there are no poles of our integrand, this integral equals 0. (Note the original integrand did have poles from the zeros of the Lfunction; however, by switching the order of summation and integration and considering the integral for a .xed prime, we need only consider integrals of analytic functions.) As M .8, the horizontal components of the rectangle become negligible (since H(x + iy) decays rapidly as y increases), meaning that in the limit the two vertical components must cancel each other. It follows that
ks log pds
[H(s)+ H(1  s)]e =[H(s)+ H(1  s)]e ks log pds. Re(s)=3/2 Re(s)=1/2
(2.28)
Shifting contours as described above and changing variables by s = 12 + iy, we have
8
00 8
1 k(1/2+iy) log pidy
I2 =  (apk + apk).dk(p) log p 2f(y)e
2pi
8
k=1 p
0
2 80 (ak + ak).k(p) log p 82piy log pk
2p
=  p pk/2 df(y)e dy
2pp
8
k=1 p
00 (ak + ak).kk
2 8p pd(p) log p log p
¢
=  k/2 f. (2.29)
2pp2p
k=1 p
Combining equations (2.20) and (2.29), we have
0 8
1 d GG
f(.d) = 2log + (6+ iy)+ (6  iy) f(y)dy
2p 2p GG
8
.d
8
k
2 00 (apk + akp).kd(p) log p log p
 k/2 f¢. (2.30)
2pi p2p
k=1 p
To rewrite equation (2.30), we sum over twists d and scale the zeros by the mean density of zeros arising from even fundamental discriminants at most X. One could instead consider the related quantities where each Lfunction’s zeros are scaled by the logarithm of its conductor, a local instead of a global rescaling. Similar behavior is seen; see for example [GM, Mil1] for such investigations.
We set
X
L = log (2.31)
2p (which is essentially the average logconductor) and replace f(y) with
g(.)= f (y) (2.32)
where . = y · Lp . It is a straightforward calculation that if F (x)= G(cx) (where c 0)
=
¢
and F¢(.) is the Fourier transform of F (x), then 1 c F.cis the Fourier transform of G(cx).
It follows that .ˆ(.)= Lp gˆLp .. Summing over quadratic twists with even fundamental discriminant d = X and dividing by X*, the number of terms in the sum (which is proportional to X), we have proven a tractable explicit formula for the onelevel density.
Theorem 2.3 (Expansion for the onelevel density). The onelevel density for the family of twists of the Ramanujan tau function by even fundamental discriminants at most X satis.es
1 0 0 L
X* g .d p
d=X .d
1 8 0 d G p. G p.
= 2LX* 8 g(.) d=X 2 log 2p + G 6 + i L + G 6  i L d.
 2 2LX* 0 d=X 80 k=1 0 p (ak p + ak p).k d(p) log p pk/2 ˆg log pk 2L , (2.33)
where f is an even Schwartz function such that supp(f¢) is contained in a bounded interval.
2.3. Analyzing the OneLevel Density. We analyze the term
8
2 000 (akp + akp).kd(p) log p log pk S =  k/2 gˆ. (2.34)
2LX* p2L
d=Xk=1 p
by splitting it into two sums: S = Seven + Sodd (2.35)
Speci.cally,
8
k
1 000 (a2k + a2k).2(p) log p log p
p pd
Seven =  gˆ(2.36)
X* pkLL
d=Xk=1 p
and
8(a2k+1 + a2k+1
1 000 pp ).d(p) log p log p2k+1
Sodd =  gˆ. (2.37)
X* (2k+1)/2L
pL
d=Xk=0 p
No higher powers of .d(p) appear because .d is a quadratic character, implying that .2dk(p)= .2 d(p) and .2dk+1(p)= .d(p) for any positive integer k.
Note that
.
.1 if p . d,
.2 d(p)= (2.38)
.
0 if pd.
This allows us to split Seven into
Seven = Seven,1 + Seven,2 (2.39)
where
0 + a2k
1 0 8(a2pk p ) log p log pk Seven,1 =  ¢g (2.40)
Lpk L
pk=1
and
8
k
1 000 (a2k + a2k) log p log p
Seven,2 = pp ¢g (2.41)
X* pkLL
d=Xk=1 pd
(there is no 1/X* in Seven,1 as that was canceled by the X* from the dsum). We will analyze these two terms separately.
Consider Seven,1. Changing back to g by writing ¢g as the Fourier transform of g, and switching summation and integration (which is justi.ed because of the rapid decay), we have
8 0 8(a2k
1 0 p + ap 2k)) log pSeven,1 =  g(.) d.. (2.42)
(k+1)(1+ 2pi. )
L
L 8 p p
k=1
We will see in Corollary 3.3 that this contribution exactly agrees with some of the terms in the expansion from the Ratios Conjecture’s prediction.
Consider Seven,2. Changing the order of summation, we may write
8
1 000 (ap 2k + a2pk) log p log pk Seven,2 = g¢
X* pkLL
d=Xk=1 pd 8
1 00 (a2pk + a2pk) log p log pk 0
= g¢1. (2.43)
LX* pk L
pk=1 d=X pd
By Lemma A.1 (proven in Appendix A), we have
0 X* 1= + O(X1/2). (2.44)
p +1
d=X pd
Plugging (2.44) into (2.43) yields
00 (a2k + a2kk
1 8pp ) log p log pSeven,2 = ¢g + O(X1/2), (2.45)
Lpk(p + 1) L
pk=1
where we used Lemma A.1 to note that X* =3X/p2 + O(X1/2). To see that the error term is P (X1/2), note the error term is bounded by 0 8a2k + a2kk
O(X1/2) 0 pp log p log p
¢g. (2.46)
LX* pk L
pk=1
As remarked, by Lemma A.1 we have O(X1/2/X*)= O(X1/2). We next note that the sum over k = 2 is trivially seen to be O(1), and by Mertens’ theorem (which states
p=Xs log p/p = log Xs + O(1)), the contribution from k =1 divided by L (which is of size log X) is also O(1). This completes the proof of the size of the error term in (2.45) for
Seven,2. We now turn to the analysis of the main term of Seven,2 in (2.45). Note that
k 8 k 8
log p2pi log p 2pi. ¢= g(.)e L d. = g(.)p L kd.. (2.47)
g L
8 8
Combining (2.45) and (2.47), we have
0
1 0 8(a2k + a2k) log p 8 2pi.
L
Seven,2 = pp g(.)p kd. + O(X1/2)
Lpk(p + 1)
8
pk=1
00 (a2k + a2k 8 = g(.)p kd. + O(X1/2)
1 8pp ) log p  2pi.
L
Lpk(p + 1)
8
pk=1
8
1 8 0 log p 0 (ap 2k + ap 2k)
= g(.) d. + O(X1/2). (2.48)
L 8 (p + 1) pk(1+ 2pi. )
L
pk=1
We will now bound Sodd. The following lemma and the proof thereof were modi.ed with permission from [Mil3].
2
Lemma 2.4. For supp(ˆg) . (s, s), we have Sodd = O(X 1s log6 X).
Proof. Jutila’s bound (see equation (3.4) of [Ju3]) is
2
00
.d(n) « NX log10 N (2.49)
1 0, this is a larger error term than the O(X1/2) we have from Seven, and thus absorbs that error term. Taking all these pieces together, we .nd that the number theoretic calculations of the onelevel density give us
00
1 L
g.d
X* p
d=X.d
8 0
1 d G p. G p.
= g(.) 2log + 6+ i +6  i d.
2LX* 8 2p G L G L
d=X
8 00 (a2k + a2k
1 8pp )) log p
 g(.) d.
k(1+ 2pi. )
L
L
8 p
pk=1
8 00 (a2k + a2k
1 log p 8pp )
2
+ g(.) d. + O(X 1s log6 X)
L (p + 1) k(1+ 2pi. )
L
8 p
pk=1
8 0
1 d G p. G p.
= g(.) 2log + 6+ i +6  i
2LX* 8 2p G L G L
d=X 88
00 (a2k + a2k)) log p 0 log p 0 (a2k + a2k)
pp pp
+2  + d.
pk(1+ 2pi. ) (p + 1) pk(1+ 2pi. )
LL
pk=1 pk=1
+ O(X 1s
2
log6 X). (2.53)
To have a power savings in the error term, we require s< 1. Thus we have proven the .rst part of Theorem 1.1, namely that equation (2.53) is the onelevel density for the family of quadratic twists of the tau Lfunction for suitably restricted test functions.
Having calculated the onelevel density on the number theory side, we now compare it to the Ratios Conjecture’s predictions for the onelevel density.
3. COMPARISON WITH THE RATIOS CONJECTURES’ PREDICTIONS
Using the Ratios Conjecture, Conrey and Snaith [CS1] compute the onelevel density for our family to be
00 8 0
1 d GG
f (.d)= f(y) 2log + (6+ iy)+ (6  iy)
2p 2p GG
8
d=X.d d=X
+2  . (1 + 2iy)+ L. (sym2 , 1+2iy)+ B.(iy; iy)
.L.
d 2iy G(6  iy) .(1 + 2iy)L.(sym2 , 1  2iy)
 B.(iy, iy) dy
2p G(6 + iy) L.(sym2 , 1)
+ O(X1/2+.)
(3.1)
where
1 1 1
a2 a2
1
L. sym 2 ,s =1  p 1  1  p , (3.2)
ps ps ps p
8
p 0 t*(p2m)
B.(a; .)= 1+
pm(1+2a)
p +1
pm=1
88
t*(p) 0 t*(p2m+1) 0 t*(p2m)
1
 +
1+a+. pm(1+2a) 1+2. pm(1+2a)
pp
m=0 m=0
1  t*(p2) t*(p2)1 1 1+2a 2+4a 3+6a 1+2.
+  1  × pppp, (3.3)
t*(p2) t*(p2)1  1+a+. + 2+2a+2.  3+31 a+3. 1  1+1 a+.
pppp
and where the derivative of B. is with respect to a. Again setting f(y)= g(.) and dividing by X*, this equation becomes
00
1 L
g.d =
X* p
d=X.d
8 0
1 d G p. G p.
g(.) 2log + 6+ i +6  i
2LX* 8 2p G L G L
d=X
. p. p. p. p.
L.2
+2  1+2i + sym , 1+2i + B. i ; i
. LL. L LL
2i p.
6  ip. 2 , 1  2ip.
L
G . 1+2ip. sym
d LL L. L p. p.
 B. i ,i d.
2p G 6+ ip. L.(sym2 , 1) LL
L
+ O(X1/2+.).
(3.4)
We will show that equations (2.53) and (3.4) agree up O(X(1s)/2+e), a power savings error term when s< 1. As the two expressions agree in their general form and in the 2 log (d/2p)+ GG (6 + i.)+ GG (6  i.) term, we need only analyze the four remaining terms of equation (3.4), showing the .rst three are equal to the remaining terms in (2.53) and bounding the last one as our error term. We will frequently use the variable y (equal to p./L) for notational convenience.
3.1. The Logarithmic Derivatives. First we show that
8
0
0
(ap 2k + a2pk)) log p
.L.
(1 + 2iy) + (sym2
,
(3.5)

, 1+2iy)=

k(1+2iy)
.
L.
p
pk=1
which gives the same contribution as the Seven,1 term.
Lemma 3.1. The logarithmic derivative of L.(sym2,s) is
8
0
,s)=  log p p skp . (3.6)
0
a2k
+ a2k +1
L.
(sym2
L.
p
pk=1
Proof. Taking the logarithmic derivative of equation (3.2) and using the geometric series expansion, we .nd
..
22
a
a
1 p
0
(sym2 ,s)=  .
p log p
ps ps
log p
log p
ps
1  p
L.
L.
.
+
+
1
22
aa
1 
p ps
1 
ps p ps
kk
k
8
0
pp pp
=  log p ++
ps ps ps ps ps ps
pk=1
0
a2 a2 a2 a2
11
k+1 k+1 k+1
88
0
=  log p p ++ p
ps ps ps pk=0
0
0
a2 a2
1
0
a2k + a2k
+1
pp . (3.7)
sk
= 
p
pk=1
Lemma 3.2. The logarithmic derivative of .(s) is
0
.
.
8
0
(s)=  log p
1 sk . (3.8)
p
pk=1
22
Proof. This can be veri.ed through explicit calculation:
. d 1 1
. (s) = ds log p 1  ps
0 1 log p =  ps 1
1 
ps p
0
=  log p
ps ps pk=0
8
k
0
1
1
0
=  log p
8
0
1
. (3.9)
psk pk=1
8
Putting these results together, we get
Corollary 3.3. We have
0
0
(ap 2k + a2pk)) log p
.L.
(1 + 2iy) + (sym2
(3.10)

, 1+2iy)=

,
pk(1+2iy)
.
L.
pk=1
which gives the same contribution as the Seven,1 piece. Proof. Applying the previous two lemmas, we have
8
0
, 1+2iy) = log p
8
0
k=1
0
0
a2k + a2k
+1
pp
.L.
1
(sym2


(1 + 2iy)+
log p
k(1+2iy) k(1+2iy)
.
L.
p
p
k=1
p p
8
0
k(1+2iy)
p
pk=1
0
(a2pk + a2pk +1  1) log p
= 
8
0
pk(1+2iy)
pk=1
0
(a2k + a2k) log p
pp (3.11)
= 
8
as claimed. . This corollary constitutes the .rst pairing of terms in equations (2.53) and (3.4).
3.2. The Derivative of B.. Next we will consider the B.(iy; iy) term, and show that it equals
0
0
log p
a2m + a2m
pp . (3.12)
pm(1+2i.)
p +1
pm=1
Recall from [CS1] that B.(r; r)=1 (as can be veri.ed by direct substitution). For notational convenience, we de.ne
0
2m+1) 2m)
p 88t*(p8t*(ppm(1+2a) p1+a+. pm(1+2a) p1+2. pm(1+2a)
p +1
m=1 m=0 m=0
(3.13)
0
0
t*(p2m) t*(p)
1
f1(a; .) =1+

+
t*(p2) t*(p2)1
f2(a; .) =1  +  (3.14)
p1+2a p2+4a p3+6a
1
f3(a; .) =1  (3.15)
p1+2.
t*(p2) t*(p2)1
f4(a; .) =1  +  (3.16)
1+a+. 2+2a+2. 3+3a+3.
ppp
1
f5(a; .) =1  , (3.17)
p1+a+.
giving us
f2(a; .)f3(a; .)
B.(a; .)= f1(a; .) · . (3.18)
f4(a; .)f5(a; .)
p
Lemma 3.4.
.B
(r; r)=
.a
880
0
pp pp
log p 
pm(1+2r) pm(1+2r)
p +1
pm=1 m=1 t*(p2)  2·t*(p2)3
+
1+2r 2+4r 3+6r
1
ppp
++ .
2) t*(p2) 1+2r
1  t*(p+  1 1  p
p1+2r p2+4r p3+6r
(3.19)
Proof. By taking the logarithmic derivative, we reduce the product of our .ve functions to a sum, allowing us to more easily compute it piece by piece. Taking the logarithmic partial
0
a2m + a2m a2m + a2m
1
derivative of (3.18) with respect to a, we have .B (a; .) .f2(a; .)f3(a; .)
.a = log f1(a; .) ·
B(a, .) .a f4(a; .)f5(a; .)
p
0
.
f2(a; .)f3(a; .)
log f1(a; .) ·
f4(a; .)f5(a; .)
=
.a
p
0
.
= (log(f1(a; .)) + log(f2(a; .)) + log(f3(a; .))
.a
p
 log(f4(a; .))  log(f5(a; .)))
0
f1(a; .) f2(a; .) f3(a; .) f4(a; .) f5(a; .)
++  , (3.20)
=
f1(a; .) f2(a; .) f3(a; .) f4(a; .) f5(a; .)
p
where the derivatives fi are with respect to a. Since B(r; r)=1, we have
0
.B
f1(r; r) f2(r; r) f3(r; r) f4(r; r) f5(r; r)
++  . (3.21)
(r; r)=
.a
f1(r; r) f2(r; r) f3(r; r) f4(r; r) f5(r; r)
p
We shall evaluate each of these logarithmic derivatives at a = . = r.
Note that f1(r; r)=1. Taking the derivative of f1(a; .) with respect to a, we have
880
0
0p 8
pm(1+2a) pm(1+2a)+1+a+.
p +1
m=1 m=0
2m log p · t*(p2m)
2m+1)t*(p)
(2m + 1) log p · t*(p

f1(a; .)=
2m log p · t*(p2m)
1
+
1+2.
pm(1+2a)
p
m=0
880
0
pm(1+2a) pm(1+2a)+1+a+.
p +1
m=1 m=0
2mt*(p2m)
2m+1)t*(p)
(2m + 1)t*(p
p log p


=
0
8
1+2. pm(1+2a)
p
m=0
Plugging in a = . = r, we have
2mt*(p2m)
1
(3.22)
+
.
8880
0
0
pm(1+2r)(m+1)(1+2r)(m+1)(1+2r)
p +1 pp
m=1 m=0 m=0
2mt*(p2m)
2m+1)t*(p)
(2m + 1)t*(p
2mt*(p
2m)
p log p
(r; r)= 

f1
+
p log p
= 
2m2)
2mt*(p2m) + (2m  1)t*(p2m1)t*(p) + (2m  2)t *(p
0
8
pm(1+2r)
p +1
m=1
(3.23)
25
.
Using the fact that t*(p2m1)t *(p)= t *(p2m)  t*(p2m2), this expression becomes
8
pm(1+2r)
p +1
m=1
0
2m2)
2mt*(p2m) + (2m  1)(t *(p2m)  t*(p2m2)) + (2m  2)t*(p
p log p
f1(r; r)= 
8
pm(1+2r)
p +1
m=1
8
00
t*(p2m)  t *(p2m2)
p log p
= 
a2m + a2m
pp
p log p
(3.24)
= 
.
pm(1+2r)
p +1
m=1
Finally, noting that  p = 1+ 1 , we have
p+1 p+1
8
00
8
 log p pp . (3.25)
a2m + a2m
pp
a2m + a2m
log p
f1(r; r)=
pm(1+2r) pm(1+2r)
p +1
m=1
m=1
We now move on to f2 and f2. Plugging in, we have
t*(p2) t*(p2)1
f2(r; r)=1  +  . (3.26)
1+2r 2+4r 3+6r
ppp
Taking the derivative with respect to a and evaluating at a = . = r, we have
2 · t*(p2)4 · t*(p2)6
f2(r; r) = log p  + (3.27)
1+2r 2+4r 3+6r
ppp
Similar calculations give us
f4(r; r)= f2(r, r) (3.28)
and
t*(p2)2 · t *(p2)3
f4(r; r) = log p  + . (3.29)
1+2r 2+4r 3+6r
ppp
It follows that
t*(p2)  2·t*(p2)
1+2r 2+4r 3+6r
f2(r; r) f4(r; r) pp+ p3
 = log p. (3.30)
f2(r : r) f4(r : r)1  t*(p2) t*(p2)1
+ 
1+2r 2+4r 3+6r
ppp
Noting that f3(r; r) is a constant with respect to a, we have f3(r; r)=0. Finally, we have
f5(r; r) log p · p1 log p
= 1+2r = . (3.31)
1 p1+2r  1
f5(r; r)1  p
1+2r
Combining all these expressions, we have
26
.B
(r; r)=
.a
pp pp
log p 
pm(1+2r) pm(1+2r)
p +1
pm=1 m=1
t*(p2)2·t*(p2)3 1+2r 2+4r 3+6r
+
+ ppp+1 (3.32)
1  t*(p2) t*(p2) 1 1+2r
+  1  p
1+2r 2+4r 3+6r
ppp
as claimed. .
880
Evaluating this derivative at r = iy, we have
0
0
a2m + a2m a2m + a2m
1
880
0
pp pp
B (iy; iy) = log p 
pm(1+2iy) pm(1+2iy)
p +1
pm=1 m=1
t*(p2)  2·t*(p2)3 p1+2iy p2+4iy + p3+6iy 1
++ . (3.33)
1  t*(p2) t*(p2) 1 1+2iy
+  1  p
1+2iy 2+4iy 3+6iy
ppp
Equation (3.33) contains a term present in the number theory calculations, as well as an algebraically messy term after it. The following vital lemma eliminates the extra term, greatly simplifying our expression and giving us perfect equality between B (iy; iy) and equation 3.12. Without it, our correspondence of terms between 2.53 and 3.4 would not work out as desired.
Lemma 3.5. We have
0
a2m + a2m a2m + a2m
1
t*(p2)  2·t*(p2)
0
8
m=1 3
a2m + a2m
pp
+
1
1+2iy
2+4iy
pp
3+6iy
p
0.
(3.34)

+
+=
1  t*(p2) t*(p2)
+
pm(1+2iy)
1+2iy
 1 1  p
1+2iy
pp
2+4iy 3+6iy
p
Proof. Set
t*(p2)  2·t*(p2)
0
8
pp
M =  +
3
a2m + a2m
+
1
1+2iy
2+4iy
pp
3+6iy
p
.
(3.35)
+
1  t*(p2) t*(p2)
+
pm(1+2iy)
1+2iy
 1 1  p
m=1
3+6iy
p
a2 a2
pp
We rewrite the series that is the .rst term of M. Setting A1 = pm(1+2iy) and A2 = pm(1+2iy) and noting that both have absolute value less than 1, we have
ppp p
=+
pm(1+2iy) pm(1+2iy) pm(1+2iy) m=1 m=1 m=1
8880
0
0
a2m + a2m a2m a2m
Am Am
=+
12 m=1 m=1
A1 A1
=+
1  A1 1  A2
880
A1 + A2  2A1A2
=
1  A1  A2 + A1A2 a2+a2
pp 2

1+2iy 2+4iy
0
= pp. (3.36)
a2+a2
pp 11  1+2iy + 2+4iy
pp
Using the fact that ap 2 + ap 2 = t*(p2)  1, we have
t*(p2)1
0
8
.
pm(1+2iy) 1  t*(p2)11
+
m=1 p1+2iy p2+4iy
Note that
1/p1+2iy
1
2
a2m + a2m
pp

p1+2iy p2+4iy
(3.37)
(3.38)
=
= 
1+2iy
1  (1/p1+2iy)
1  p
Letting t denote t*(p) and q denote 1/p1+2iy, we have
t*(p2)  2·t*(p2)
0
8
pp
M =  +
3
a2m + a2m
+
1
1+2iy
2+4iy
pp
3+6iy
p
+
1  t*(p2) t*(p2)
+
pm(1+2iy)
1+2iy
 1 1  p
m=1
1+2iy
pp
2+4iy 3+6iy
p
t*(p2)1 t*(p2)  2·t*(p2)
23
 +1/p1+2iy
1+2iy 2+4iy 1+2iy 2+4iy 3+6iy
ppppp
=  + 
1  t*(p2)11 1  t*(p2) t*(p2)1 1  (1/p1+2iy)
++ 
1+2iy 2+4iy 1+2iy 2+4iy 3+6iy
ppppp
(1  t)q +2q2 tq  2tq2 +3q3 q
=+  . (3.39)
1  (t  1)q + q2 1  tq + tq2  q3 1  q
Noting that 1  tq + tq2  q3 = (1  (t  1)q + q2)(1  q), we have
(1  q)((1  t)q +2q2)+ tq  2tq2 +3q3  (1  (t  1)q + q2)q
M =
1  tq + tq2  q3 q  qt +2q2  q2 + tq2  2q3 + tq  2tq2 +3q3  q + tq2  q2  q3
=
1  tq + tq2  q3 (1  t + t  1)q + (2  1+ t  2t + t  1)q2 +(2+3  1)q3
=
1  tq + tq2  q3 =0. (3.40)
Combined with equation (3.33), this lemma immediately implies
Corollary 3.6.
00 a2m + a2m
log p 8pp
B (iy; iy)= . (3.41)
pm(1+2iy)
p +1
pm=1
This gives us our second correspondence of terms.
3.3. The Error Term. From Corollary 3.3 and Proposition 3.6, we have that equations
(2.53) and (3.4) are in agreement save for the term
2i p.
8 0 6  ip.
L
1 d G LR(g; X)=  g(.)
LX* 8 2p G 6+ ip.
d=XL
2 , 1  2ip.
. 1+2ip. L. symp. p.
× LL B. i ,i d..
L.(sym2 , 1) LL
(3.42)
Our general technique will be to perform a contour shift and bound all the terms from G onward in the expression above by a polynomial in ., then use the rapid decay of g(.) to show the integral over . converges so that we need only worry about X terms.
First we will show that B.(iy; iy) converges and will continue to do so for contour shifts of y up to a cutoff point.
Proposition 3.7. Let 12 + t = w = 14  t . If we shift from y to y  iw, then we have that B. (i(y  iw); i(y  iw)) is Ow(1).
Proof. We have
0
88t*(p8*(p2m)
2m+1)
t pm(12iy) pm(12iy) p1+2iy pm(12iy)
p +1 p
pm=1 m=0 m=0
1  t*(p2) t*(p2)1 1
12iy 24iy 36iy 1+2iy
+  1  × pppp. (3.43)
1  t*(p2) t*(p2)1
+  1  1
23
pppp
0
0
t*(p2m) t*(p)
1
p
B.(iy; iy)= 1+

+
Letting y = y  iw (our shift), we will show that
0
0
88t*(p8t*(p2m)
2m+1)
p
pm(12iy ) pm(12iy ) 1+2iy pm(12iy )
p +1 pp
m=1 m=0 m=0 1  t*(p2) t*(p2)1 1
+  1 
12iy 24iy 36iy 1+2iy
pppp
×
1  t*(p2) t*(p2)1
+  1  1
23
0
pppp
11
=1+ O + O, (3.44)
24w. 2+2w.
pp
t*(p2)  t*(p2)
implying that the product converges as the error is O(1/p1+. ). Noting that + 1
pp2 p3
t *(p2m) t*(p)
1
T(p)= 1+

+
<
1 and p 1 < 1, we may rewrite T(p) as
0
0
0
2m+1)
p 88t*(p8t*(p2m) pm(12iy ) pm(12iy ) 1+2iy pm(12iy )
p +1 pp
m=1 m=0 m=0 t *(p2) t*(p2)1 1
× 1  +  1 
12iy 24iy 36iy 1+2iy
pppp
t*(p2m) t *(p)
1
T(p)= 1+

+
0
0
88
pp2 p3 pn n=0 n=0
t*(p2) t*(p2)1 n
1
. (3.45)

×
+
We now rewrite some of the in.nite sums of (3.45) as a main term plus an error term. By Lemma A.3 in the appendix, for any t> 0 we may truncate the terms of (3.45) as follows, preserving multiple error terms depending on the direction in which y has been shifted:
pt*(p2) t *(p)2 11 1
T(p)= 1+ +++ O + O
12iy 2 1+2iy 24w. 2.
p +1 pppppt*(p2)1 1
× 1  + O( )1 
12iy 24w 1+2iy
ppp
t *(p2)1 11
× 1+ + O 1+ + O
pp2 pp2 t*(p2) t*(p2)+1 1 1 t *(p2) t*(p2)+1 1
= 1+  +  +
12iy 1+2iy 12iy 1+2iy
pppp +1 ppp11
+O + O
24w. 2.
pp
t*(p2)1 t *(p2)+1 1
× 1  ++ O
12iy 1+2iy 24w
ppppt*(p2) t*(p2)+1 11 1 1
= 1+  ++ O + O + O
p12iy p1+2iy p24w. p2+2w. p2.
p t*(p2)1 t *(p2)+1 1
× 1  ++ O
p12iy p1+2iy p24w
p t*(p2)1 t*(p2)+1 t*(p2) t *(p2)+1 1
=1  ++  +
p12iy p1+2iy p12iy p1+2iy
pp
11
+ O + O
24w. 2+2w.
pp
11
=1+ O + O. (3.46)
24w. 2+2w.
pp
If we have shifted in the positive direction, the larger error term is O (1/p24w.), meaning we may shift 2t close to 1/4 and have O(1/p1+d) where d> 0. Similarly, if we have shifted in the negative direction, the larger error term is O (1/p2+2w.), meaning we may shift 2t close to 1/2 and have O(p1+d) where d> 0. In both cases we have convergence with the product Ow(1). .
Thus the B. term is Ow(1) for .xed w and any y (as the bound is independent of the imaginary part of our variable). To bound . and L., we use the standard fact (see for example [IK]) that both grow polynomially in y as y.8 (where y is our imaginary part).
We now consider G(6iy) . Our shifting restrictions from B. allow us to only consider w
G(6+iy)
with 1/2 0 small). Then
p 4
1
0 log(d/2p) 2piz
2piz 2piz
log(X/2p)
e = X * e 1  + O(X2.). (A.10)
log(X/2p)
d=X
Proof. We may rewrite our sum as
00
log(d/2p) (log dlog 2p)
2piz 2piz
e L = e L
d=Xd=X
0
2piz log 2p
2piz log d
LL
= ee
d=X
0
2piz log 2p
d2piz/L
L
= e. (A.11)
d=X
Recall the integral version of partial summation, namely that if h(x) is a continuously
differentiable function and A(x)= n=x an, then
0 x
anh(n)= A(x)h(x)  A(u)h (u)du. (A.12)
1
n=x
2piz/L and an
Considering h(x)= x=1 for n an even fundamental discriminant and an =0 otherwise, by Lemma A.1 we have
0
3u
A(u)= 1= p2 + O(u 1/2). (A.13)
d=u
This allows us to rewrite equation (A.11) as
0
2piz log 2p
d2piz/L
e
L
d=X
X
2piz log 2p 3X 3u 2piz
X2piz/L  1/2) 2piz/L1
L
=e + O(X1/2)+ O(u · uu
p2 p2 L
1
(A.14)
2pi(.iwL/p)
Note that we may rewrite 2Lpiz = L = 2w + id, where d . R. As order of magnitude depends only on the real part of the exponent, we have O(X1/2)X2piz/L =
2piz/L1
)(X1/22w)= O(X2.) due to our choice of w. Similarly, we may write O(u1/2)uu
1/22w1)
O(u= O(u1+2.), meaning the integral (from 1 to X) of that term ends up as 46
=
du
2piz log 2p
L
O(X2.). As multiplying these terms by the eonly decreases their order of magni
tude in X, we have
0
2piz log 2p
d2piz/L
e
L
d=X
X
2piz log 2p 3X 3u 2piz
X2piz/L  2piz/L1
L
=e · uu du + O(X2.)
p2 p2 L
1 3 X
2piz log 2p 3 · 2piz
X12piz/L p2 2piz/Ldu
= e L + u + O(X2.)
p2 L 1
3 X12piz/L
2piz log 2p 3 · 2piz
X12piz/L p2
L
= e ++ O(X2.)
p2 L 1  2piz/L
8
p2 L 2piz
k=0
0 1 LL
3  log 2p 2piz
Xe2piz(log X )
=1  + O(X2.)
p2 L
2piz
= X * e 1  2piz 1 + O(X2.).
k
3 2piz log 2p 2piz
L
L
X12piz/L
+ O(X2.)
1+
=
e
(A.15)
log(X/2p)
0
Lemma A.3. Let y = y  iw be as in Proposition 3.7. Then for all t> 0,
i.)
8
t*(p2m) t *(p2)1
=+ O (A.16)
pm(12iy ) p12iy p24w.
0
m=1
ii.)
8
t*(p)
2m+1)
t*(pt*(p)2 1
=+ O (A.17)
pm(12iy ) p2 p22w.
p
m=0
iii.)
0
8
p1+2iy pm(12iy ) p1+2iy p2. m=0
t*(p2m)
1
11
=+ O. (A.18)
Proof. Fix t, and pick C such that t*(n)= Cn. for all n (such a C exists by Deligne’s theorem, which implies t(n)= O(n11/2+.) for all t> 0; see [Se]). Also, recall that for real numbers a, x, y, we have ax+iy = ax.
i.) The .rst claim follows from
8
m=2
0
t*(p2m)
pm(12iy )
=
8
m=2
0
2m).
C · (p
pm(12iy )
8
= C
p12iy 2.m
m=2
0
1
8
= C
p12w2iy2.m
m=2
0
1
8
= C
p12w2.
m=2
0
1 m
0
8
= C
12w2. 12w2.
pp
m=0
2
11
= C ·
12w2. 1
p1  p
12w1.
11
= C = O. (A.19)
24w4. 24w4.
pp
ii.) The second claim follows from
2
m
1
1
t*(p)
p
0
8
m=1
0
0 8
8
m=1
t*(p2m+1) t*(p)
=
2m+1).
C · (p
pm(12iy ) pm(12iy )
1
p
Ct*(p)
p1.
=
p12iy 2.m
1
0
m=1
8
Ct*(p)
1.
=
12w2iy2.m
p
0
p
m=1
8
Ct*(p)
1.
m
1
=
12w2.
pp
m=1
0
8
p12w2. p12w2. m=0
Ct*(p) 1
p1.
m
1
=
Ct*(p) 1
=
22w3. 1
p1  p
12w1.
= C 1 p22w3. = O 1 p22w3. . (A.20)
48
iii.) The third claim follows from the fact that t*(1) = 1 and from the following bound (where several steps are omitted due to similarity of the bound for the .rst claim):
2m).
88C · (p
0
0
t *(p2m)
1
1
=
1+2iy pm(12iy ) p1+2iy  m=1 pm(12iy )
p
m=1
8
=
p1+2w p12iy 2.m m=1
0
C
1
8
=
1+2w 12w2. 12w2.
ppp
m=0
C 11
0
= ·
1+2w 12w2. 1
pp1  p
12w1.
11
= C = O. (A.21)
24. 24.
pp
m
C 1
1
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