The fourth power mean of Dirichlet L-functions in $${\mathbb {F}}_q [T]$$

Andrade, Júlio; Yiasemides, Michael

doi:10.1007/s13163-020-00350-2

Cited by 10 publications

(15 citation statements)

References 9 publications

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“…We have a(0) = 1, a(1) = 1, a(2) = 1 ζ(2) = 6 π 2 , and we have an understanding of a(k) for higher values of k. The factor f (k) is more elusive. Clearly, from the results described above, we have f (0) = 1, f (1) = 1, f (2) = 1 12 . It has been conjectured via number-theoretic means that f (3) = 42 9!…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 79%

The Hybrid Euler-Hadamard Product Formula for Dirichlet $L$-functions in $\mathbb{F}_q [T]$

Yiasemides

2021

Preprint

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For Dirichlet L-functions in Fq[T ] we obtain a hybrid Euler-Hadamard product formula. We make a splitting conjecture, namely that the 2k-th moment of the Dirichlet L-functions at 1 2 , averaged over primitive characters of modulus R, is asymptotic to (as deg R −→ ∞) the 2k-th moment of the Euler product multiplied by the 2k-th moment of the Hadamard product. We explicitly obtain the main term of the 2k-th moment of the Euler product, and we conjecture via random matrix theory the main term of the 2k-th moment of the Hadamard product. With the splitting conjecture, this directly leads to a conjecture for the 2k-th moment of Dirichlet Lfunctions. Finally, we lend support for the splitting conjecture by proving the cases k = 1, 2. This work is the function field analogue of the work of Bui and Keating. A notable difference in the function field setting is that the Euler-Hadamard product formula is exact, in that there is no error term.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 79%

The Hybrid Euler-Hadamard Product Formula for Dirichlet $L$-functions in $\mathbb{F}_q [T]$

Yiasemides

2021

Preprint

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“…However, in her proof she claims that d (N ) deg N , which is not the case. Addressing this is non-trivial and was done in [3], as stated above.…”

Section: Remark 75mentioning

confidence: 99%

“…The proof of Lemma 7.8 is similar to the proof of Lemma 7.7. From [3], we use Lemma 7.10 instead of Lemma 7.9.…”

Section: Lemma 76 Let P Degmentioning

confidence: 99%

“…as deg Q −→ ∞ with Q being prime. Andrade and Yiasemides [3] generalised this to the full analogue of Soundararajan's result, by removing the restriction that Q be prime. As we will soon see, this current paper considers another generalisation of Tamam's result.…”

Section: Introductionmentioning

confidence: 98%

“…Here, the sum is over all primitive characters of modulus q; φ * (q) is the number of primitive characters of modulus q; φ is the totient function; and L(s, χ) is the Dirichlet L-function associated to the character χ. Extending the work of Heath-Brown [15], Soundararajan [24] proved that 3 1 + p −1 (log q) 4 .…”

Section: Introductionmentioning

confidence: 99%

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The fourth moment of derivatives of Dirichlet L-functions in function fields

Andrade

Yiasemides

2021

Math. Z.

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We obtain the asymptotic main term of moments of arbitrary derivatives of L-functions in the function field setting. Specifically, we obtain the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus $$Q \in {\mathbb {F}}_q [T]$$ Q ∈ F q [ T ] , and the asymptotic limit is as $${{\,\mathrm{deg}\,}}Q \longrightarrow \infty $$ deg Q ⟶ ∞ . This extends the work of Tamam who obtained the asymptotic main term of low moments of L-functions, without derivatives, in the function field setting. It is also the function field q-analogue of the work of Conrey, who obtained the fourth moment of derivatives of the Riemann zeta-function.

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The variance and correlations of the divisor function in $${\mathbb {F}}_q [T]$$, and Hankel matrices

Yiasemides

2024

Res Math Sci

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We prove an exact formula for the variance of the divisor function over short intervals in $${\mathcal {A}}:= {\mathbb {F}}_q [T]$$ A : = F q [ T ] , where q is a prime power; and for correlations of the form $$d(A) d(A+B)$$ d ( A ) d ( A + B ) , where we average both A and B over certain intervals in $${\mathcal {A}}$$ A . We also obtain an exact formula for correlations of the form $$d(KQ+N) d (N)$$ d ( K Q + N ) d ( N ) , where Q is prime and K and N are averaged over certain intervals with $${{\,\textrm{deg}\,}}N \le {{\,\textrm{deg}\,}}Q -1 \le {{\,\textrm{deg}\,}}K$$ deg N ≤ deg Q - 1 ≤ deg K ; and we demonstrate that $$d(KQ+N)$$ d ( K Q + N ) and d(N) are uncorrelated. We generalize our results to $$\sigma _z$$ σ z defined by $$\sigma _z (A):= \sum _{E \mid A} |A |^z$$ σ z ( A ) : = ∑ E ∣ A | A | z for all monics $$A \in {\mathcal {A}}$$ A ∈ A . Our approach is to use the orthogonality relations of additive characters on $${\mathbb {F}}_q$$ F q to translate the problems to ones involving the ranks of Hankel matrices over $${\mathbb {F}}_q$$ F q . We prove several results regarding the rank and kernel structure of these matrices, thus demonstrating their number-theoretic properties. We also discuss extending our method to other divisor sums, such as those involving $$d_k$$ d k .

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The fourth power mean of Dirichlet L-functions in $${\mathbb {F}}_q [T]$$

Cited by 10 publications

References 9 publications

The Hybrid Euler-Hadamard Product Formula for Dirichlet $L$-functions in $\mathbb{F}_q [T]$

The Hybrid Euler-Hadamard Product Formula for Dirichlet $L$-functions in $\mathbb{F}_q [T]$

The fourth moment of derivatives of Dirichlet L-functions in function fields

The variance and correlations of the divisor function in $${\mathbb {F}}_q [T]$$, and Hankel matrices

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