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| Titchmarsh divisor problem | |
|---|---|
| Name | Titchmarsh divisor problem |
| Field | Analytic number theory |
| Notable proponents | E. C. Titchmarsh; D. R. Heath-Brown; A. Ivic; Y. Motohashi; H. Davenport; J. E. Littlewood |
| First published | 1930s |
| Main objects | Divisor function, Dirichlet series, Riemann zeta function |
| Status | Active research topic |
Titchmarsh divisor problem The Titchmarsh divisor problem is a classical question in Analytic number theory concerning the asymptotic behavior of correlations of the divisor function and its shifted values; it sits at the intersection of studies of the Riemann zeta function, mean values of arithmetic functions, and the distribution of primes. The problem originated in work by Edward Charles Titchmarsh and has driven developments involving spectral theory, trace formulas, and exponential sum estimates in the traditions of G. H. Hardy and John Edensor Littlewood.
The core question asks for an asymptotic formula for sums of the form Σ_{n ≤ x} τ(n)τ(n + h) for fixed nonzero integers h, where τ denotes the divisor function studied by Srinivasa Ramanujan and formalized in classical texts by Tom M. Apostol and Harold Davenport. Early work relating shifted convolution sums to mean square estimates of the Riemann zeta function was developed in the milieu of Atle Selberg and later refined using spectral methods from the theory of Maass forms and the Kuznetsov trace formula associated with researchers such as Evgeny Kuznetsov and Henryk Iwaniec.
Titchmarsh formulated the problem while studying second moments of multiplicative functions, influenced by techniques from G. H. Hardy and J. E. Littlewood; the original analytic approach connected the shifted convolution to properties of the Dirichlet series of τ and products of Riemann zeta function values. Subsequent progress by Atle Selberg and Harald Bohr introduced orthogonality methods, and mid-20th-century work by H. Davenport and A. Selberg clarified error terms and main-term heuristics. The modern precise statement asks for the asymptotic Σ_{n ≤ x} τ(n)τ(n + h) = C(h) x log x + D(h) x + o(x) with explicit constants C(h), D(h) related to convolution of divisor sums and local factors studied by G. H. Hardy-style multiplicative theory.
Attack routes include the use of the Dirichlet hyperbola method favored in classical texts by Tom M. Apostol and Harold Davenport, spectral decompositions via the theory of Maass forms as developed by Atle Selberg and Henryk Iwaniec, and the deployment of the Kuznetsov trace formula introduced by Evgeny Kuznetsov. Exponential sum machinery influenced by I. M. Vinogradov and S. S. Pillai appears through estimates akin to the Weyl differencing paradigm of Hermann Weyl and the bilinear forms method of D. A. Burgess. Mean-value bounds for the Riemann zeta function from investigations by A. A. Karatsuba and K. Ramachandra also play critical roles.
Early bounds giving the main term up to polylogarithmic errors were due to E. C. Titchmarsh and refinements by H. L. Montgomery and R. C. Vaughan reduced error estimates using mean-value theorems and large-sieve ideas from P. X. Gallagher. A breakthrough establishing asymptotics with power-saving error terms in specific averaged settings was achieved through the work of Yoichi Motohashi and Aleksandar Ivić, who combined spectral and trace-formula techniques reminiscent of Atle Selberg and Henryk Iwaniec; later advances by D. R. Heath-Brown and Roger Heath-Brown (alternate name usage) extended spectral refinements and delta-method innovations derived from Duke, Friedlander, and Iwaniec.
The problem connects tightly to correlations studied in the Goldbach problem context by Ivan Matveevich Vinogradov and to shifted convolution problems for cusp forms examined by P. Sarnak and W. Luo. It relates to divisor correlation questions that echo themes in the Hardy–Littlewood prime k-tuples conjecture and to mean-square estimates of L-functions in the spirit of Atle Selberg and Andrew Wiles-adjacent L-function studies. Techniques overlap with bounds for exponential sums used in work by Enrico Bombieri and Harald Helfgott.
The modern proof strategy begins by expressing Σ τ(n)τ(n + h) via a shifted Dirichlet convolution and applying a spectral decomposition of automorphic kernels developed in the Selberg trace formula framework of Atle Selberg. One inserts the Kuznetsov trace formula with weight functions chosen as in work by Yoichi Motohashi and Aleksandar Ivić to convert sums to spectral sums over Maass forms and Eisenstein series parameters studied by Henryk Iwaniec. Bounding auxiliary sums uses the spectral large-sieve inequalities pioneered by J. D. Vaaler and P. Sarnak, together with exponential-sum inputs from I. M. Vinogradov-type estimates and the delta-method as used by Duke, Friedlander, and Iwaniec. The main term emerges from the diagonal contribution tied to classical divisor convolution results of G. H. Hardy and J. E. Littlewood; off-diagonal contributions are controlled by delicate spectral mean-value bounds as refined by D. R. Heath-Brown.
Open directions include improving uniformity in the shift h beyond current ranges studied by Yoichi Motohashi and Aleksandar Ivić, obtaining pointwise power-saving error terms akin to conjectures influenced by G. H. Hardy-style heuristics, and exploring higher correlation analogues connected to the k-fold divisor problem investigated by P. T. Bateman and E. Wright. Researchers like K. Soundararajan, Peter Sarnak, and Rizwanur Khan continue to explore links with subconvexity bounds for L-functions and quantum unique ergodicity questions initiated by Andrew Booker and Akshay Venkatesh.