Published by De Gruyter February 5, 2015

The mean square of the product of the Riemann zeta-function with Dirichlet polynomials

Sandro Bettin , Vorrapan Chandee and Maksym Radziwiłł

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2014-0133

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Abstract

Improving earlier work of Balasubramanian, Conrey and Heath-Brown [1], we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length T12+δ, with δ=0.01515⁢…. As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec [8], obtaining asymptotic estimates in place of bounds. Using the work of Watt [19], we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up to T34 provided that the Dirichlet polynomial assumes a special shape. Finally, we exhibit a conjectural estimate for trilinear sums of Kloosterman fractions which implies the Lindelöf Hypothesis.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1128155

Funding statement: The third author was partially supported by NSF grant DMS-1128155. The corresponding author is Vorrapan Chandee.

A On Conjecture 1

Proposition 4.

Let A,M,N≥1 and let A≪(M⁢N)12+ε. Then

maxα,β,ν⁡|SA,M,N|≫(A⁢M⁢N)12-ε⁢(M+N)12+A⁢(M+N)1-ε

for all ε>0, where the maximum is taken over all choices of coefficients αm,βn,νa≪1.

Proof.

By the reciprocity relation m¯n≡-n¯m+1m⁢n⁢(mod⁢1), we can assume M≥N. Moreover, we can assume N,A≫Mε for some small ε>0 and M arbitrary large, since otherwise the result is easy.

First, we consider the case M1-δ≫N for some δ>0 and we take αm=f⁢(m) for some smooth function f:[M,2⁢M]→[0,1] which is such that f(j)⁢(x)≪jx-j for all j≥0 and ∫ℝf⁢(x)⁢𝑑x=K⁢M, for some K>0. Also, let βn=-γn, where γn is the indicator function of the primes congruent to 1⁢(mod⁢4) in [N,2⁢N], and let νa be the indicator function of the primes congruent to 3⁢(mod⁢4) in [A,2⁢A].

By Poisson summation, we have

∑mf⁢(m)⁢e⁡(a⁢m¯n)=K⁢MN⁢(cn⁢(a)+O⁢(M-100)),

where

cn⁢(a)=∑b=1(b,n)=1ne⁡(b⁢an)=μ⁢(n(n,a))⁢φ⁢(n)φ⁢(n(n,a))

is the Ramanujan sum. It follows that

SA,M,N=K⁢MN⁢∑a∑nβn⁢νa⁢(cn⁢(a)+O⁢(M-100))=K⁢MN⁢∑a∑nγn⁢νa⁢(1+O⁢(M-100))≫(M⁢A)1-ε.

We now prove

maxα,β,ν⁡|SA,M,N|≫M⁢(A⁢N)12-ε,

which then implies the proposition even in the case M1-δ≪N for all δ>0.

By choosing αm appropriately, we have

maxα,β,ν⁡|SA,M,N|≫maxβ,ν⁢∑mFm;β,ν,

where

Fm;β,ν:=|∑a∑(n,m)=1νa⁢βn⁢e⁡(a⁢m¯n)|.

First, notice that we have

maxβ,ν⁢∑mFm;β,ν≥1φ⁢(q)2⁢∑χ1,χ2⁢(mod⁢q)∑mFm;β⁢(χ1),ν⁢(χ2)

with q any prime greater than 4⁢(A4+N4) and where β⁢(χ1),ν⁢(χ2) denotes sequences defined by β⁢(χ1)n=χ1⁢(n) and ν⁢(χ2)a=χ2⁢(a) respectively. Moreover, by Hölder’s inequality,

1φ⁢(q)2⁢∑χ1,χ2⁢(mod⁢q)∑mFm;β⁢(χ1),ν⁢(χ2)2

≤(1φ⁢(q)2⁢∑χ1,χ2⁢(mod⁢q)∑mFm;β⁢(χ1),ν⁢(χ2))23⁢(1φ⁢(q)2⁢∑χ1,χ2⁢(mod⁢q)∑mFm;β⁢(χ1),ν⁢(χ2)4)13.

The left-hand side is

1φ⁢(q)2⁢∑χ1,χ2⁢(mod⁢q)∑mFm;β⁢(χ1),ν⁢(χ2)2=∑m∑a∑(n,m)=11≫M⁢A⁢N,

and we also have

1φ⁢(q)2⁢∑χ1,χ2⁢(mod⁢q)∑mFm;β⁢(χ1),ν⁢(χ2)4=∑m∑a1⁢a2=a3⁢a4∑n1⁢n2=n3⁢n4(m,n1⁢n2)=11≪M⁢(A⁢N)2+ε.

Thus,

1φ⁢(q)2⁢∑χ1,χ2⁢(mod⁢q)Fm;β⁢(χ1),ν⁢(χ2)≫M⁢(A⁢N)12-ε

and the proposition follows. ∎

Acknowledgements

We are very grateful to Brian Conrey for suggesting to us the problem of breaking the 12 barrier in Theorem 1 and to Micah B. Milinovich and Nathan Ng for pointing out the paper of Duke, Friedlander, Iwaniec [9]. We also wish to thank the referee for a very careful reading of the paper and for indicating several inaccuracies and mistakes.

References

[1] R. Balasubramanian, J. B. Conrey and D. R. Heath-Brown, Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial, J. reine angew. Math. 357 (1985), 161–181. 10.1515/crll.1985.357.161Search in Google Scholar

[2] S. Bettin and V. Chandee, Trilinear forms with Kloosterman fractions, preprint (2015), http://arxiv.org/abs/1502.00769. 10.1016/j.aim.2018.01.026Search in Google Scholar

[3] J. Bredberg, Large gaps between consecutive zeros, on the critical line, of the Riemann zeta-function, preprint (2011), http://arxiv.org/abs/1101.3197. Search in Google Scholar

[4] J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. reine angew. Math. 399 (1989), 1–26. 10.1515/crll.1989.399.1Search in Google Scholar

[5] J. B. Conrey, A. Ghosh and S. M. Gonek, Large gaps between zeros of the zeta-function, Mathematika 33 (1986), no. 2, 212–238. 10.1112/S0025579300011219Search in Google Scholar

[6] H. Davenport, Multiplicative number theory, 3rd ed., Grad. Texts in Math. 74, Springer, New York 2000. Search in Google Scholar

[7] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219–288. 10.1007/BF01390728Search in Google Scholar

[8] J.-M. Deshouillers and H. Iwaniec, Power mean-values for Dirichlet’s polynomials and the Riemann zeta-function. II, Acta Arith. 43 (1984), no. 3, 305–312. 10.4064/aa-43-3-305-312Search in Google Scholar

[9] W. Duke, J. Friedlander and H. Iwaniec, Bilinear forms with Kloosterman fractions, Invent. Math. 128 (1997), no. 1, 23–43. 10.1007/s002220050135Search in Google Scholar

[10] W. Duke, J. Friedlander and H. Iwaniec, Representations by the determinant and mean values of L-functions, Sieve methods, exponential sums, and their applications in number theory (Cardiff 1995), London Math. Soc. Lecture Note Ser. 237, Cambridge University Press, Cambridge (1997), 109–115. 10.1017/CBO9780511526091.009Search in Google Scholar

[11] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam 2007. Search in Google Scholar

[12] A. Harper, Sharp conditional bounds for moments of the Riemann zeta function, preprint (2013), http://arxiv.org/abs/1305.4618. Search in Google Scholar

[13] D. R. Heath-Brown, Fractional moments of the Riemann zeta function, J. Lond. Math. Soc. (2) 24 (1981), no. 1, 65–78. 10.1112/jlms/s2-24.1.65Search in Google Scholar

[14] C. P. Hughes and M. P. Young, The twisted fourth moment of the Riemann zeta function, J. reine angew. Math. 641 (2010), 203–236. 10.1515/crelle.2010.034Search in Google Scholar

[15] X. Li and M. Radziwiłł, The Riemann zeta function on vertical arithmetic progressions, Int. Math. Res. Not. IMRN 2015 (2015), no. 2, 325–354. 10.1093/imrn/rnt197Search in Google Scholar

[16] M. Radziwiłl, Limitations to mollifying ζ⁢(s), preprint (2012), http://arxiv.org/abs/1207.6583. Search in Google Scholar

[17] M. Radziwiłl, The 4.36th moment of the Riemann zeta-function, Int. Math. Res. Not. IMRN 2012 (2012), no. 18, 4245–4259. 10.1093/imrn/rnr183Search in Google Scholar

[18] K. Soundararajan, Mean-values of the Riemann zeta-function, Mathematika 42 (1995), no. 1, 158–174. 10.1112/S0025579300011438Search in Google Scholar

[19] N. Watt, Kloosterman sums and a mean value for Dirichlet polynomials, J. Number Theory 53 (1995), no. 1, 179–210. 10.1006/jnth.1995.1086Search in Google Scholar

Received: 2013-8-7

Revised: 2014-11-7

Published Online: 2015-2-5

Published in Print: 2017-8-1