Rankin–Selberg coefficients in arithmetic progressions

DefinitionNormalized Fourier coefficients of a cusp form#✓fully proved

Let \(f\) be a cusp form of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with \(q\)-expansion \(f(z) = \sum_{n \ge 0} a_f(n) e(nz)\), where \(e(z) = e^{2\pi i z}\). For each natural number \(n\), the normalized Fourier coefficient of \(f\) is the complex number \(\lambda_f(n) = a_f(n) / n^{(k-1)/2}\), where \(n^{(k-1)/2}\) is the real power.

At \(n = 0\) the real power follows \(0^0 = 1\) and \(0^s = 0\) for \(s \ne 0\), and division by \(0\) gives \(0\).

public noncomputable def cuspFormNormalizedCoeff {k : ℤ} (f : CuspForm 𝒮ℒ k) (n : ℕ) : ℂ :=
  (qExpansion 1 f).coeff n / (((n : ℝ) ^ (((k : ℝ) - 1) / 2) : ℝ) : ℂ)

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Section 1 (Fourier expansion of f ∈ H_k^*).

Used by 8

Used byNormalized cusp form coefficient at one · Normalized Hecke eigenform · Hecke relation at the square of a prime · Rankin–Selberg coefficients · Rankin–Selberg coefficients as a sum over square divisors · Rankin–Selberg coefficient at one · Rankin–Selberg coefficients at primes · Sums of squared normalized coefficients at perfect powers in arithmetic progressions

TheoremNormalized cusp form coefficient at one#✓fully proved

Let \(f\) be a cusp form of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with \(q\)-expansion \(f(z) = \sum_{n \ge 0} a_f(n) e^{2\pi i n z}\), and let \(\lambda_f\) be its normalized Fourier coefficients. Then \(\lambda_f(1) = a_f(1)\), the coefficient of \(e^{2\pi i z}\) in the \(q\)-expansion of \(f\).

public theorem cuspFormNormalizedCoeff_one {k : ℤ} (f : CuspForm 𝒮ℒ k) :
    cuspFormNormalizedCoeff f 1 = (qExpansion 1 f).coeff 1

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Section 1.

Uses 1

Statement usesNormalized Fourier coefficients of a cusp form

DefinitionNormalized Hecke eigenform#✓fully proved

Let \(f\) be a cusp form of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with normalized Fourier coefficients \(\lambda_f\). Then \(f\) is a normalized Hecke eigenform when \(\lambda_f(1) = 1\) and, for all positive integers \(m\) and \(n\), \(\lambda_f(m) \lambda_f(n) = \sum_{d \mid \gcd(m, n)} \lambda_f(mn/d^2)\), the sum running over the positive divisors \(d\) of \(\gcd(m, n)\). The set of these forms is denoted \(H_k^*\).

At level one this is equivalent to \(f\) being a simultaneous eigenform of all Hecke operators normalized by \(a_f(1) = 1\), where \(f(z) = \sum_{n \ge 0} a_f(n) e^{2\pi i n z}\) is the \(q\)-expansion of \(f\).

public def IsNormalizedHeckeEigenform {k : ℤ} (f : CuspForm 𝒮ℒ k) : Prop :=
  cuspFormNormalizedCoeff f 1 = 1 ∧
    ∀ m n : ℕ, 0 < m → 0 < n →
      cuspFormNormalizedCoeff f m * cuspFormNormalizedCoeff f n =
        ∑ d ∈ (Nat.gcd m n).divisors, cuspFormNormalizedCoeff f (m * n / d ^ 2)

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Section 1, Equation (1).

Uses 1 · Used by 8

Statement usesNormalized Fourier coefficients of a cusp form

Used byHecke relation at the square of a prime · Sums of squared normalized coefficients at perfect powers in arithmetic progressions · Sums of Rankin–Selberg convolution powers in arithmetic progressions · Sums of Rankin–Selberg convolution powers twisted by a primitive character · Sums of Rankin–Selberg convolution powers twisted by the principal character · Hua's theorem on squared Rankin–Selberg coefficients in arithmetic progressions · Sums of squared Rankin–Selberg coefficients twisted by a primitive character · Sums of squared Rankin–Selberg coefficients twisted by the principal character

TheoremHecke relation at the square of a prime#✓fully proved

Let \(f\) be a normalized Hecke eigenform of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with normalized Fourier coefficients \(\lambda_f\). Then \(\lambda_f(p)^2 = \lambda_f(p^2) + 1\) for every prime \(p\).

public theorem IsNormalizedHeckeEigenform.normalizedCoeff_prime_sq {k : ℤ}
    {f : CuspForm 𝒮ℒ k} (hf : IsNormalizedHeckeEigenform f) {p : ℕ} (hp : p.Prime) :
    cuspFormNormalizedCoeff f p ^ 2 = cuspFormNormalizedCoeff f (p ^ 2) + 1

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Equation (1) with m = n = p.

Uses 2

Statement usesNormalized Hecke eigenform · Normalized Fourier coefficients of a cusp form

DefinitionRankin–Selberg coefficients#✓fully proved

Let \(f\) be a cusp form of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with normalized Fourier coefficients \(\lambda_f\). The Rankin–Selberg coefficients of \(f\) form the arithmetic function \(\lambda_{f \times f}\) with \(\lambda_{f \times f}(0) = 0\) and, for \(n \ge 1\), \(\lambda_{f \times f}(n) = \sum_{l^2 m = n} \lambda_f(m)^2\), the sum running over the pairs of positive integers \((l, m)\) with \(l^2 m = n\); equivalently, \(\lambda_{f \times f}(n)\) is the sum of \(\lambda_f(n/d)^2\) over the positive divisors \(d\) of \(n\) that are perfect squares.

These are the Dirichlet coefficients of \(L(f \times f, s) = \zeta(2s) \sum_{n \ge 1} \lambda_f(n)^2 n^{-s}\). The \(w\)-th power of \(\lambda_{f \times f}\) under Dirichlet convolution is denoted \(\lambda_{w, f \times f}\).

public noncomputable def rankinSelbergCoeff {k : ℤ} (f : CuspForm 𝒮ℒ k) : ArithmeticFunction ℂ :=
  ⟨fun n => ∑ d ∈ n.divisors with IsSquare d, cuspFormNormalizedCoeff f (n / d) ^ 2, by simp⟩

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Section 1 (definition of λ_{f×f} and λ_{w,f×f}).

Uses 1 · Used by 9

Statement usesNormalized Fourier coefficients of a cusp form

Used byRankin–Selberg coefficients as a sum over square divisors · Rankin–Selberg coefficient at one · Rankin–Selberg coefficients at primes · Sums of Rankin–Selberg convolution powers in arithmetic progressions · Sums of Rankin–Selberg convolution powers twisted by a primitive character · Sums of Rankin–Selberg convolution powers twisted by the principal character · Hua's theorem on squared Rankin–Selberg coefficients in arithmetic progressions · Sums of squared Rankin–Selberg coefficients twisted by a primitive character · Sums of squared Rankin–Selberg coefficients twisted by the principal character

TheoremRankin–Selberg coefficients as a sum over square divisors#✓fully proved

Let \(f\) be a cusp form of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with normalized Fourier coefficients \(\lambda_f\) and Rankin–Selberg coefficients \(\lambda_{f \times f}\). For every natural number \(n\), \(\lambda_{f \times f}(n) = \sum_d \lambda_f(n/d)^2\), where \(d\) runs over the positive divisors of \(n\) that are perfect squares (so the sum is empty when \(n = 0\)).

public theorem rankinSelbergCoeff_apply {k : ℤ} (f : CuspForm 𝒮ℒ k) (n : ℕ) :
    rankinSelbergCoeff f n =
      ∑ d ∈ n.divisors with IsSquare d, cuspFormNormalizedCoeff f (n / d) ^ 2

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Section 1.

Uses 2 · Used by 2

Statement usesNormalized Fourier coefficients of a cusp form · Rankin–Selberg coefficients

Used byRankin–Selberg coefficient at one · Rankin–Selberg coefficients at primes

TheoremRankin–Selberg coefficient at one#✓fully proved

Let \(f\) be a cusp form of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with normalized Fourier coefficients \(\lambda_f\) and Rankin–Selberg coefficients \(\lambda_{f \times f}\). Then \(\lambda_{f \times f}(1) = \lambda_f(1)^2\).

public theorem rankinSelbergCoeff_one {k : ℤ} (f : CuspForm 𝒮ℒ k) :
    rankinSelbergCoeff f 1 = cuspFormNormalizedCoeff f 1 ^ 2

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Section 1.

Uses 3

Statement usesNormalized Fourier coefficients of a cusp form · Rankin–Selberg coefficients

Proof usesRankin–Selberg coefficients as a sum over square divisors

TheoremRankin–Selberg coefficients at primes#✓fully proved

Let \(f\) be a cusp form of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with normalized Fourier coefficients \(\lambda_f\) and Rankin–Selberg coefficients \(\lambda_{f \times f}\). Then \(\lambda_{f \times f}(p) = \lambda_f(p)^2\) for every prime \(p\).

public theorem rankinSelbergCoeff_prime {k : ℤ} (f : CuspForm 𝒮ℒ k) {p : ℕ} (hp : p.Prime) :
    rankinSelbergCoeff f p = cuspFormNormalizedCoeff f p ^ 2

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Section 1.

Uses 3

Statement usesNormalized Fourier coefficients of a cusp form · Rankin–Selberg coefficients

Proof usesRankin–Selberg coefficients as a sum over square divisors

TheoremSums of squared normalized coefficients at perfect powers in arithmetic progressions#●ready to prove

Let \(f\) be a normalized Hecke eigenform of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with normalized Fourier coefficients \(\lambda_f\), let \(j \ge 2\) be an integer, and put \(\delta = 92/597\) if \(j = 2\) and \(\delta = 92/(69(j-1)(j+3) + 247)\) if \(j \ge 3\). Then there are real numbers \(c_q > 0\), one for each prime \(q\), with the following property: for every \(\varepsilon > 0\) there is a real constant \(C\) such that for every prime \(q\), every natural number \(l\) coprime to \(q\) and every real \(x \ge 1\) with \(q \le x^{3\delta/4}\), \(\bigl|\sum_{n} \lambda_f(n^j)^2 - c_q x / \varphi(q)\bigr| \le C q x^{1 - 3\delta/2 + \varepsilon}\), where \(n\) runs over the integers \(1 \le n \le x\) with \(n \equiv l \pmod q\) and \(\varphi\) is Euler's totient function.

Hua quotes this result from Zou et al. [52, Theorem 1]. The constant \(c_q\) is allowed to depend on \(q\), a weaker reading of the printed main term, whose constant does not depend on \(q\); the constant \(C\) depends only on \(f\), \(j\) and \(\varepsilon\).

public theorem_wanted sum_cuspFormNormalizedCoeff_pow_sq_arithmeticProgression {k : ℤ}
    {f : CuspForm 𝒮ℒ k} (hf : IsNormalizedHeckeEigenform f) {j : ℕ} (hj : 2 ≤ j) :
    let δ : ℝ := if j = 2 then 92 / 597 else 92 / (69 * ((j : ℝ) - 1) * ((j : ℝ) + 3) + 247)
    ∃ c : ℕ → ℝ, (∀ q : ℕ, q.Prime → 0 < c q) ∧
      ∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ q : ℕ, q.Prime → ∀ l : ℕ, Nat.Coprime l q →
        ∀ x : ℝ, 1 ≤ x → (q : ℝ) ≤ x ^ (3 / 4 * δ) →
          ‖(∑ n ∈ Finset.Icc 1 ⌊x⌋₊ with n ≡ l [MOD q], cuspFormNormalizedCoeff f (n ^ j) ^ 2)
              - ((c q * x / (q.totient : ℝ) : ℝ) : ℂ)‖ ≤
            C * q * x ^ (1 - 3 / 2 * δ + ε)

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Theorem 1.1.

Uses 2

Statement usesNormalized Hecke eigenform · Normalized Fourier coefficients of a cusp form

TheoremHua's theorem on squared Rankin–Selberg coefficients in arithmetic progressions#●ready to prove

Let \(f\) be a level-one cusp form of weight \(k\) that is a normalized Hecke eigenform, let \(\lambda_{f\times f}\) be its Rankin–Selberg coefficients, let \(\varphi\) be Euler's totient function, and put \(\eta = 23/261\). There are real polynomials \(P_q\), one for each prime \(q\), each of degree \(1\) with positive leading coefficient, such that for every \(\varepsilon > 0\) there is a constant \(C\) with

\[ \Bigl|\sum_{\substack{1 \le n \le x \\ n \equiv l \ (\mathrm{mod}\ q)}} \lambda_{f\times f}(n)^2 - \frac{x}{\varphi(q)}\,P_q(\log x)\Bigr| \le C\,q\,x^{1 - 3\eta/2 + \varepsilon} \]

for every prime \(q\), every natural number \(l\) coprime to \(q\) and every real \(x \ge 1\) with \(q \le x^{\eta}\).

public theorem_wanted sum_rankinSelbergCoeff_sq_arithmeticProgression {k : ℤ}
    {f : CuspForm 𝒮ℒ k} (hf : IsNormalizedHeckeEigenform f) :
    ∃ P : ℕ → Polynomial ℝ, (∀ q : ℕ, q.Prime → (P q).natDegree = 1 ∧ 0 < (P q).leadingCoeff) ∧
      ∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ q : ℕ, q.Prime → ∀ l : ℕ, Nat.Coprime l q →
        ∀ x : ℝ, 1 ≤ x → (q : ℝ) ≤ x ^ ((23 : ℝ) / 261) →
          ‖(∑ n ∈ Finset.Icc 1 ⌊x⌋₊ with n ≡ l [MOD q], rankinSelbergCoeff f n ^ 2)
              - ((x / (q.totient : ℝ) * (P q).eval (Real.log x) : ℝ) : ℂ)‖ ≤
            C * q * x ^ (1 - 3 / 2 * ((23 : ℝ) / 261) + ε)

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Theorem 1.2.

Uses 2

Statement usesNormalized Hecke eigenform · Rankin–Selberg coefficients

TheoremSums of Rankin–Selberg convolution powers in arithmetic progressions#●ready to prove

Let \(f\) be a normalized Hecke eigenform of weight \(k \in \mathbb{Z}\) for \(\mathrm{SL}_2(\mathbb{Z})\), with Rankin–Selberg coefficients \(\lambda_{f \times f}\), let \(w \ge 2\) be an integer, write \(\lambda_{w, f \times f}\) for the \(w\)-fold Dirichlet convolution power of \(\lambda_{f \times f}\), and put \(\vartheta_w = 92/(247w + 10)\). Then there are real polynomials \(P_q\), one for each prime \(q\), each of degree \(w - 1\) with positive leading coefficient, with the following property: for every \(\varepsilon > 0\) there is a real constant \(C\) such that for every prime \(q\), every natural number \(l\) coprime to \(q\) and every real \(x \ge 1\) with \(q \le x^{\vartheta_w}\), \(\bigl|\sum_{n} \lambda_{w, f \times f}(n) - (x / \varphi(q)) P_q(\log x)\bigr| \le C q x^{1 - 3\vartheta_w/2 + \varepsilon}\), where \(n\) runs over the integers \(1 \le n \le x\) with \(n \equiv l \pmod q\), \(\varphi\) is Euler's totient function and \(\log\) is the natural logarithm.

The printed hypothesis \(q \ll x^{\vartheta_w}\) is read as \(q \le x^{\vartheta_w}\), and the polynomial is allowed to depend on \(q\); the constant \(C\) depends only on \(f\), \(w\) and \(\varepsilon\).

public theorem_wanted sum_rankinSelbergCoeff_pow_arithmeticProgression {k : ℤ}
    {f : CuspForm 𝒮ℒ k} (hf : IsNormalizedHeckeEigenform f) {w : ℕ} (hw : 2 ≤ w) :
    ∃ P : ℕ → Polynomial ℝ,
      (∀ q : ℕ, q.Prime → (P q).natDegree = w - 1 ∧ 0 < (P q).leadingCoeff) ∧
      ∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ q : ℕ, q.Prime → ∀ l : ℕ, Nat.Coprime l q →
        ∀ x : ℝ, 1 ≤ x → (q : ℝ) ≤ x ^ ((92 : ℝ) / (247 * w + 10)) →
          ‖(∑ n ∈ Finset.Icc 1 ⌊x⌋₊ with n ≡ l [MOD q], (rankinSelbergCoeff f ^ w) n)
              - ((x / (q.totient : ℝ) * (P q).eval (Real.log x) : ℝ) : ℂ)‖ ≤
            C * q * x ^ (1 - 3 / 2 * ((92 : ℝ) / (247 * w + 10)) + ε)

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Theorem 1.3.

Uses 2

Statement usesNormalized Hecke eigenform · Rankin–Selberg coefficients

TheoremSums of squared Rankin–Selberg coefficients twisted by a primitive character#●ready to prove

Let \(f\) be a level-one cusp form of weight \(k\) that is a normalized Hecke eigenform, let \(\lambda_{f\times f}\) be its Rankin–Selberg coefficients, and put \(\eta = 23/261\). For every \(\varepsilon > 0\) there is a constant \(C\) such that for every prime \(q\), every primitive Dirichlet character \(\chi\) modulo \(q\) and every real \(x \ge 1\) with \(q \le x^{\eta}\),

\[ \Bigl|\sum_{1 \le n \le x} \lambda_{f\times f}(n)^2\,\chi(n)\Bigr| \le C\,q\,x^{1 - 3\eta/2 + \varepsilon}. \]

public theorem_wanted sum_rankinSelbergCoeff_sq_mul_primitive_character {k : ℤ}
    {f : CuspForm 𝒮ℒ k} (hf : IsNormalizedHeckeEigenform f) {ε : ℝ} (hε : 0 < ε) :
    ∃ C : ℝ, ∀ q : ℕ, q.Prime → ∀ χ : DirichletCharacter ℂ q, χ.IsPrimitive →
      ∀ x : ℝ, 1 ≤ x → (q : ℝ) ≤ x ^ ((23 : ℝ) / 261) →
        ‖∑ n ∈ Finset.Icc 1 ⌊x⌋₊, rankinSelbergCoeff f n ^ 2 * χ n‖ ≤
          C * q * x ^ (1 - 3 / 2 * ((23 : ℝ) / 261) + ε)

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Proposition 3.1.

Uses 2

Statement usesNormalized Hecke eigenform · Rankin–Selberg coefficients

TheoremSums of squared Rankin–Selberg coefficients twisted by the principal character#●ready to prove

Let \(f\) be a level-one cusp form of weight \(k\) that is a normalized Hecke eigenform, let \(\lambda_{f\times f}\) be its Rankin–Selberg coefficients, and for a prime \(q\) let \(\chi_0\) be the principal Dirichlet character modulo \(q\). There are real polynomials \(P_q\), one for each prime \(q\), each of degree \(1\) with positive leading coefficient, such that for every \(\varepsilon > 0\) there is a constant \(C\) with

\[ \Bigl|\sum_{1 \le n \le x} \lambda_{f\times f}(n)^2\,\chi_0(n) - x\,P_q(\log x)\Bigr| \le C\,x^{179/209 + \varepsilon} \]

for every prime \(q\) and every real \(x \ge q\).

public theorem_wanted sum_rankinSelbergCoeff_sq_mul_principal_character {k : ℤ}
    {f : CuspForm 𝒮ℒ k} (hf : IsNormalizedHeckeEigenform f) :
    ∃ P : ℕ → Polynomial ℝ, (∀ q : ℕ, q.Prime → (P q).natDegree = 1 ∧ 0 < (P q).leadingCoeff) ∧
      ∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ q : ℕ, q.Prime → ∀ x : ℝ, (q : ℝ) ≤ x →
        ‖(∑ n ∈ Finset.Icc 1 ⌊x⌋₊, rankinSelbergCoeff f n ^ 2 * (1 : DirichletCharacter ℂ q) n)
            - ((x * (P q).eval (Real.log x) : ℝ) : ℂ)‖ ≤
          C * x ^ ((179 : ℝ) / 209 + ε)

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Proposition 3.2.

Uses 2

Statement usesNormalized Hecke eigenform · Rankin–Selberg coefficients

TheoremSums of Rankin–Selberg convolution powers twisted by a primitive character#●ready to prove

Let \(f\) be a level-one cusp form of weight \(k\) that is a normalized Hecke eigenform, let \(\lambda_{f\times f}\) be its Rankin–Selberg coefficients, let \(w \ge 2\), let \(\lambda_{w,f\times f}\) be the \(w\)-th power of \(\lambda_{f\times f}\) under Dirichlet convolution, and put \(\vartheta_w = 92/(247w + 10)\). For every \(\varepsilon > 0\) there is a constant \(C\) such that for every prime \(q\), every primitive Dirichlet character \(\chi\) modulo \(q\) and every real \(x \ge 1\) with \(q \le x^{\vartheta_w}\),

\[ \Bigl|\sum_{1 \le n \le x} \lambda_{w,f\times f}(n)\,\chi(n)\Bigr| \le C\,q\,x^{1 - 3\vartheta_w/2 + \varepsilon}. \]

public theorem_wanted sum_rankinSelbergCoeff_pow_mul_primitive_character {k : ℤ}
    {f : CuspForm 𝒮ℒ k} (hf : IsNormalizedHeckeEigenform f) {w : ℕ} (hw : 2 ≤ w)
    {ε : ℝ} (hε : 0 < ε) :
    ∃ C : ℝ, ∀ q : ℕ, q.Prime → ∀ χ : DirichletCharacter ℂ q, χ.IsPrimitive →
      ∀ x : ℝ, 1 ≤ x → (q : ℝ) ≤ x ^ ((92 : ℝ) / (247 * w + 10)) →
        ‖∑ n ∈ Finset.Icc 1 ⌊x⌋₊, (rankinSelbergCoeff f ^ w) n * χ n‖ ≤
          C * q * x ^ (1 - 3 / 2 * ((92 : ℝ) / (247 * w + 10)) + ε)

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Proposition 4.1.

Uses 2

Statement usesNormalized Hecke eigenform · Rankin–Selberg coefficients

TheoremSums of Rankin–Selberg convolution powers twisted by the principal character#●ready to prove

Let \(f\) be a level-one cusp form of weight \(k\) that is a normalized Hecke eigenform, let \(\lambda_{f\times f}\) be its Rankin–Selberg coefficients, let \(w \ge 2\), let \(\lambda_{w,f\times f}\) be the \(w\)-th power of \(\lambda_{f\times f}\) under Dirichlet convolution, and for a prime \(q\) let \(\chi_0\) be the principal Dirichlet character modulo \(q\). There are real polynomials \(P_q\), one for each prime \(q\), each of degree \(w - 1\), such that for every \(\varepsilon > 0\) there is a constant \(C\) with

\[ \Bigl|\sum_{1 \le n \le x} \lambda_{w,f\times f}(n)\,\chi_0(n) - x\,P_q(\log x)\Bigr| \le C\,x^{1 - 210/(317w + 115) + \varepsilon} \]

for every prime \(q\) and every real \(x \ge q\).

public theorem_wanted sum_rankinSelbergCoeff_pow_mul_principal_character {k : ℤ}
    {f : CuspForm 𝒮ℒ k} (hf : IsNormalizedHeckeEigenform f) {w : ℕ} (hw : 2 ≤ w) :
    ∃ P : ℕ → Polynomial ℝ, (∀ q : ℕ, q.Prime → (P q).natDegree = w - 1) ∧
      ∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ q : ℕ, q.Prime → ∀ x : ℝ, (q : ℝ) ≤ x →
        ‖(∑ n ∈ Finset.Icc 1 ⌊x⌋₊, (rankinSelbergCoeff f ^ w) n * (1 : DirichletCharacter ℂ q) n)
            - ((x * (P q).eval (Real.log x) : ℝ) : ℂ)‖ ≤
          C * x ^ (1 - (210 : ℝ) / (317 * w + 115) + ε)

Sources

• Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Proposition 4.2.

Uses 2

Statement usesNormalized Hecke eigenform · Rankin–Selberg coefficients

8 fully proved7 ready to prove

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