Finite Ramanujan expansions and shifted convolution

Finite Ramanujan expansions and shifted convolution sums (joint work with M.Ram Murty & B. Saha)

Giovanni Coppola University of Salerno

1

We define the shifted convolution sum (also, correlation) of any couple f, g : N → C as def X

Cf,g (N, a) =

f (n)g(n + a).

n≤N

The integer variable a > 0 is the shift. There’s a lack of asymptotic/explicit formulæ, for correlations of interesting f, g (esp., case f = g = Λ, the von-Mangoldt function, with even a = 2k ≥ 2, involves 2k−twin primes!), too difficult (apart special cases) to achieve, even for one single, fixed shift a > 0.

2

def def 0 0 For f = f ∗ µ and g = g ∗ µ M¨ obius inversion

⇒ f (n) =

X

f 0 (d)

and

g(m) =

d|n

X

g 0(q)

q|m

so : vital remark is that inside

(1)

Cf,g (N, a) =

X

f 0(d)

d

=

X

f 0 (d)

d≤N

X

q≤N +a

X

X

g 0(q)

q

1

n≤N n≡0 mod d n≡−a mod q

g 0(q)

X

1,

n≤N n≡0 mod d n≡−a mod q

our f (n), g(m) become truncated divisor sums X

d|n,d≤N

f 0(d),

X

g 0(q)

q|m,q≤N +a

3

(depending on both variables, N and shift a); the condition d|n can be expressed as 1X 1X ed(jn) = cq (n), 1d|n = d j≤d d q|d involving Ramanujan sums def

cq (n) =

X

eq (jn),

j≤q,(j,q)=1

after g.c.d. rearrangement, from orthogonality def of additive characters eq (m) = e2πim/q .

We immediately get any arithmetic functions f, g : N → C have (inside Cf,g ) following finite Ramanujan expansions (exchanging sums now) f (n) =

X

0

f (d)1d|n =

q≤N

d≤N

g(m) =

X

d≤N +a

X

g 0(d)1d|m =

fb(q)cq (n),

X

q≤N +a

gb(q)cq (m), 4

(finite expansions depending on N , a again) with Ramanujan coefficients def

fb(q) =

X

d≤N d≡0 mod q

f 0 (d) def , gb(q) = d

X

d≤N+a d≡0 mod q

g 0(d) . d

Thus heuristic formula for f and g correlation (2)

Cf,g (N, a) ∼ Sf,g (a)N,

with a ≥ 1, defining the f and g singular series: ∞ def X b Sf,g (a) = f (q)gb(q)cq (a). q=1

This has been proved in our first work (with Murty & Saha, see JNT) for particular f, g.

5

Actually, it is the singular sum (after N , fb = 0) Sf,g (a) =

X

q≤N

fb(q)gb(q)cq (a).

On the other hand, it depends on N . But, this variable is implicit in f, g. Aficionados of Hardy-Littlewood method will say: these are only partial sums of singular series!

6

Heuristic (2) inspired the definition: (3) Cf,g (N, a) =

∞ X

`=1

Cd f,g (N, `)c` (a),

∀a ∈ N

which is the shift-Ramanujan expansion of our correlation. Notice: Hildebrand’s Theorem ensures pointwise convergence! (For all arithmetic functions, here shift a is the argument) (Big!) Problem is to find the shift-Ramanujan coefficients Cd f,g (N, `).

Now, we don’t know, if (3) is a finite sum! For this, Carmichael formula Cd f,g (N, `) =

1 1 X lim Cf,g (N, m)c`(m) x→∞ ϕ(`) x m≤x

is useful. Two pbs: 1) when? & 2) how? Both questions need two new concepts: the purity, of a Ramanujan expansion, and the fair correlations. 7

We say a Ramanujan expansion is “pure”, iff coefficients & their supports do not depend on outer variable. In other words, the variable we expand appears only in Ramanujan sums. In above (3) outer variable’s the shift a. Purity is a strong requirement: finite & pure Ramanujan exp.s are truncated divisor sums! (Hildebrand Th.m expands any f (n) into finite Ramanujan exp. ⇒ not pure: n−dependence)

Very similar is the definition: Cf,g (N, a) is fair def

⇐⇒ a−dependence is only inside g argument (n + a). Equivalently, f (n) and gb(q) do not depend on a, neither in supports, in following (4)

Cf,g (N, a) =

X q

gb(q)

X

f (n)cq (n + a).

n≤N

This formula comes easily from the g finite Ramanujan expansion. 8

Our 2nd paper (C-Murty) proves the following. Abbreviate Ramanujan expansion (3) as s.R.e. P

Theorem 1. Assume g(m) = q|m,q≤Q g 0(q), Q independent of a and Cf,g (N, a) is fair. Then F.A.E. • s.R.e. is pure & uniformly convergent; • s.R.e. coefficients from Carmichael formula; • s.R.e. has Ramanujan exact explicit formula: Cf,g (N, a) =

X g b(`) X

ϕ(`) n≤N `≤Q

f (n)c` (n)c`(a)∀a ∈ N

• s.R.e. is pure & finite. Definition: such a s.R.e. is regular. Remark: Once found the Reef, we’d find the treasure (our’s to prove (2) above) ! 9

This is not a joke, but (for reasonable f , g) a consequence: Corollary 1. Same hypotheses of Theorem 1 P log D give, for f (n) = d|n,d≤D f 0(d), log N < 1 − δ, with regular s.R.e., whenever f, g satisfy the Ramanujan Conjecture, Cf,g (N, a) = Sf,g (a)N + O(N 1−δ ). Notice “gain”, δ > 0, in remainder’s exponent depends on f .

10

The case f = g = Λ is not covered now but try calculating a−primes (a = 2k ≥ 2) correlation, from X

n≤N

Λ(n)c` (n) ≈

X

(log p)c`(p) ∼ µ(`)N

p≤N

(∀`, apart “few cases”, by PNT), in the Reef b of a−twin primes. (Btw, Λ(q) =? See JNT)

(Twin primes regularity gives H-L asymptotic!) Since regularity is “hard”, to prove, we come to “soft”, say, hypotheses: work in progress.

11

Natural question: what can we prove only from “g is of range Q and fair correlation”? These (in Th.m 1), we call “basic hypotheses”, give Cf,g (N, a) =

X

`≤Q

Cd f,g (N, Q, `)c` (a) +

X

0 Cf,g (d),

d|a d>Q

by M¨ obius inversion & d|a formula, with pure “truncated Ramanujan coeff.s”: def

Cd f,g (N, Q, `) =

X

d≤Q d≡0 mod `

0 (d) Cf,g

d

similar to Wintner-Delange formula (for infinite expansions, under hypotheses). Defined Eratosthenes transform of correlation as: def 0 0 Cf,g (d) = Cf,g (N, d) =

X

Cf,g (N, t)µ(d/t).

t|d

12

Then, (4) above, fairness and Carmichael for truncated Cd f,g (from purity) give ∀q ∈ N ˆ g (q) X d f (n)cq (n) − L(q), Cf,g (N, Q, q) = ϕ(q) n≤N

abbreviating ∀q ∈ N X 1 1 X 0 (d)c (m), L(q) = lim Cf,g q ϕ(q) x x m≤x d|m,d>Q def

notice always exists ∈ C and vanishes (as 0−0) on q > Q. In all, Cf,g (N, a) = 



X ˆ g(q) X 0  f (n)cq (n)−L(q) cq (a)+ Cf,g (d) ϕ(q) d|a q≤Q n≤N X



d>Q

comes from ”g of range Q and fair correlation”. 13

Well, L(q) is a kind of “grey box”, since: X 1 X 0 Cf,g (d)cq (m) x m≤x d|m,d>Q X

1 X 0 = Cf,g (d) · cq (dK) x K≤ x QQ

d

f (n)c` (n)c` (a)

c`(a).

`|d `>Q 15

See, if we may exchange `, d sums into ∞ X C 0 (d) X f,g

d

d=1 `|d

c`(a) =

∞ X

X

`=1 d≡0 mod `

0 (d) Cf,g

d

c`(a)

P 1 then on LHS detecting 1`|d = d `|d c`(a) gives ∞ C 0 (d) X X f,g

d=1

d

c`(a) =

`|d

X

0 (d) = C (N, a), Cf,g f,g

d|a

with on RHS the Wintner-Delange coefficients X

d≡0 mod `

0 (d) Cf,g

d

ˆ g(`) X = f (n)c` (n), ∀` ∈ N ϕ(`) n≤N

thus giving, again, the Reef. Under which hypotheses ? For example, Delange Hypothesis:

DH

∞ 2ω(d) |C 0 (d)| X f,g

d=1

d