The sixth moment of the Riemann zeta function and ternary additive divisor sums

@article{Ng2016TheSM,
  title={The sixth moment of the Riemann zeta function and ternary additive divisor sums},
  author={Nathan Ng},
  journal={arXiv: Number Theory},
  year={2016}
}
  • N. Ng
  • Published 17 October 2016
  • Mathematics
  • arXiv: Number Theory
Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary… 

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References

SHOWING 1-10 OF 58 REFERENCES

Some remarks on the mean value of the Riemann zetafunction and other Dirichlet series. III

This is a sequel (Part II) to an earlier article with the same title. There are reasons to expect that the estimates proved in Part I without the factor $(\log\log H)^{-C}$ represent the real truth,

Moments of zeta and correlations of divisor-sums: IV

  • B. ConreyJ. Keating
  • Mathematics
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2015
This work examines the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations and identifies terms that are missed in the standard application of these methods.

On the Fourth Power Moment of the Riemann Zeta-Function

Abstract The function E 2 ( R ) is used to denote the error term in the asymptotic formula for the fourth power moment of the Riemann zeta-function on the half-line. In this paper we prove several

Lectures on mean values of the Riemann zeta function

This is an advanced text on the Riemann zeta function, a continuation of the author's earlier book. It presents the most recent results on mean values. An especially detailed discussion is given of

The Theory of the Riemann Zeta-Function

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects

High moments of the Riemann zeta-function

The authors describe a general approach which, in principal, should produce the correct (conjectural) formula for every even integer moment of the Riemann zeta function. They carry it out for the

On the cubic moment of quadratic dirichlet L-functions

where χD is the quadratic Dirichlet character associated to Q( √ D) as defined in [11]. Besides its own interest, this mean value problem also plays a crucial role in the studies of such as the

The Sixth Power Moment of Dirichlet L-Functions

We prove a formula, with power savings, for the sixth moment of Dirichlet L- functions averaged over all primitive characters χ (mod q) with q ≤ Q, and over the critical line. Our formula agrees

The Riemann Zeta-Function

In order to make progress in number theory, it is sometimes necessary to use techniques from other areas of mathematics, such as algebra, analysis or geometry. In this chapter we give some

Mean-values of the Riemann zeta-function

Let Asymptotic formulae for I k ( T ) have been established for the cases k =1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of I k ( T ) remains
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