HEXAGONbeta

THE CROSS-ENERGY OF DIRICHLET PARTIAL SUMS AND THE ZEROS OF THE RIEMANN ZETA FUNCTION

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Pedro Caceres ↗

Version 1 / Oct 09, 2026 / CC BY 4.0

Abstract

Let X_n(s) = Σ_{k≤n} k⁻ˢ and let C₂(n,s) = |X_n(s)|² − |X_{n−1}(s)|² − n⁻²σ be the cross-energy of consecutive partial sums. At every point of the critical strip the limit of C₂(n,s) is 1/(¼ + t²), 0 or +∞ according as σ = ½, σ > ½ or σ < ½; this trichotomy is the trace of the pole at s = 1. On the critical line the associated helix has a natural orientation whose signed radius is Hardy’s Z(t), and the zeros of the truncated functions spiral into the zeros of ζ at the rate n⁻ᵝ. We prove that the mean energy of the normalized error E(u) in the prime number theorem equals Σ_ρ lim_{n→∞} C₂(n,ρ) in [0,∞]; it is at least 2 + γ₀ − log(4π), with equality if and only if the Riemann hypothesis (RH) holds and all zeros are simple, and an unconditional local version expresses the energy in a Gaussian window as a double sum over zeros. We then study the pointwise inequality S(σ,t) = ∂σ log|ξ(σ + it)| > 0, which is equivalent to RH: an off-line pair lowers S exactly inside a disc, the failure set at abscissa σ up to height T has measure at most N(σ,T + 1), and S is Weil’s functional at a Poisson kernel, whose truncation is an exact finite expression in the primes up to eᴸ. Every criterion is tested against the Davenport–Heilbronn function, which has zeros off the critical line, and against its Euler-product twin L(s,χ) mod 5: the former violates each criterion in the predicted way, the latter behaves like ζ. For ζ, the primes up to 10⁸ give mean energy 0.0453 against 0.0462. On the circle of the Euler factor of a prime p the zeros lean toward the angle π, and toward π + argχ(p) for L(s,χ). We do not prove the Riemann hypothesis.

Provenance statement

The research questions, the cross-energy C₂ (introduced by the author in 2017) and the overall program are the author's. The author used Claude (Anthropic), an AI assistant, in the Cowork environment, as a research and writing aid: to derive and check proofs, run and verify the numerical computations in Python, test claims against counterexamples, search the literature, draft and edit the LaTeX text and figures, and review earlier drafts critically. Other AI assistants (e.g. ChatGPT) were used by the author for exploratory discussion; their outputs were checked and only verified statements were retained. All results were checked by the author, who takes full responsibility for the content. No AI system is an author.

Tools used

Anthropic
Claude Sonnet
OpenAI
ChatGPT
Python
PythonVersion 3.13

References

  1. L. Báez-Duarte, M. Balazard, B. Landreau, E. Saias, Notes sur la fonction $ $ de Riemann, 3, Adv. Math. 149 (2000), 130--144.
  2. J.-B. Bost, A. Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), 411--457.
  3. L. Báez-Duarte, A strengthening of the Nyman--Beurling criterion for the Riemann hypothesis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 14 (2003), 5--11.
  4. A. Beurling, A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 312--314.
  5. H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichlet--schen Reihen an/ns a_n/n^s, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. (1913), 441--488.
  6. E. Bombieri, Remarks on Weil's quadratic functional in the theory of prime numbers I, Rend. Mat. Acc. Lincei (9) 11 (2000), 183--233.
  7. E. Bombieri, Counting points on curves over finite fields (d'après S. A. Stepanov), Séminaire Bourbaki 430 (1973).
  8. J.-F. Burnol, A lower bound in an approximation problem involving the zeros of the Riemann zeta function, Adv. Math. 170 (2002), 56--70.
  9. A. A. Buchstab, Asymptotic estimates of a general number-theoretic function, Mat. Sb. 2(44) (1937), 1239--1246.
  10. P. Caceres, Cross-energy of Dirichlet partial sums and the zeros of the Riemann zeta function: critical-line detection, a Fresnel transition law, and structural limits, preprint, Zenodo, 2026, ọi 10.5281/zenodo.23066932 .DOI
  11. P. Caceres, Arithmetic cross-energy of prime partial sums: positivity, phase-locking, and the pole barrier (Part II), preprint, Zenodo, 2026, ọi 10.5281/zenodo.23070165 .DOI
  12. P. Caceres, Cross-energy of Dirichlet partial sums III: monotonicity windows, the derivative at zeros, and criteria for rigidity, preprint, Zenodo, 2026, ọi 10.5281/zenodo.23070322 .DOI
  13. P. Caceres, Cross-energy of Dirichlet partial sums IV: a Jensen formula for the C2C_2 wave and the statement equivalent to the Riemann hypothesis, preprint, Zenodo, 2026, ọi 10.5281/zenodo.23070393 .DOI
  14. P. Caceres, Cross-energy of Dirichlet partial sums V: anatomy of a zero, Euler-product interference and the Davenport--Heilbronn control, preprint, Zenodo, 2026, ọi 10.5281/zenodo.23070452 .DOI
  15. P. Caceres, Cross-energy of Dirichlet partial sums VI: the Euler product inside the critical strip, preprint, Zenodo, 2026, ọi 10.5281/zenodo.23070519 .DOI
  16. P. Caceres, Cross-energy of Dirichlet partial sums VII: Weil positivity as a prime cross-energy, preprint, Zenodo, 2026, ọi 10.5281/zenodo.23070556 .DOI
  17. P. Caceres, Cross-energy of Dirichlet partial sums VIII: circles for the critical line and a pointwise C2C_2 energy inequality, preprint, Zenodo, 2026, ọi 10.5281/zenodo.23070589 .DOI
  18. A. Connes, C. Consani, Weil positivity and trace formula, the archimedean place, Selecta Math. 27 (2021), 77.
  19. K. Conrad, Partial Euler products on the critical line, Canad. J. Math. 57 (2005), 267--297.
  20. 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.
  21. H. Cramér, Ein Mittelwertsatz in der Primzahltheorie, Math. Z. 12 (1922), 147--153.
  22. G. Csordas, R. S. Varga, Necessary and sufficient conditions and the Riemann hypothesis, Adv. in Appl. Math. 11 (1990), 328--357.
  23. H. Davenport, Multiplicative Number Theory , 3rd ed., Graduate Texts in Mathematics 74, Springer, 2000.
  24. P. Deligne, La conjecture de Weil. I, Publ. Math. IHÉS 43 (1974), 273--307.
  25. H. Davenport, H. Heilbronn, On the zeros of certain Dirichlet series I, II, J. London Math. Soc. 11 (1936), 181--185 and 307--312.
  26. P. Caceres, Ellipse symmetry in Dirichlet series: a duality between analytic functions (Riemann zeta) and geometric forms, preprint, 16 September 2024.
  27. P. Caceres, Erratum and addendum to ``Ellipse symmetry in Dirichlet series'', 2026.
  28. J. Friedlander, H. Iwaniec, Opera de Cribro , AMS Colloquium Publications 57, 2010.
  29. J. Friedlander, H. Iwaniec, The polynomial X2+Y4X^2+Y^4 captures its primes, Ann. of Math. 148 (1998), 945--1040.
  30. K. Ford, A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, J. reine angew. Math. 579 (2005), 145--158.
  31. D. A. Goldston, S. M. Gonek, H. L. Montgomery, Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals, J. reine angew. Math. 537 (2001), 105--126.
  32. S. M. Gonek, C. P. Hughes, J. P. Keating, A hybrid Euler--Hadamard product for the Riemann zeta function, Duke Math. J. 136 (2007), 507--549.
  33. L. Guth, J. Maynard, New large value estimates for Dirichlet polynomials, Ann. of Math. 203 (2026), no. 2.
  34. M. Griffin, K. Ono, L. Rolen, D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, Proc. Natl. Acad. Sci. USA 116 (2019), 11103--11110.
  35. K. Hashimoto, Zeta functions of finite graphs and representations of pp-adic groups, in Automorphic Forms and Geometry of Arithmetic Varieties , Adv. Stud. Pure Math. 15 (1989), 211--280.
  36. D. R. Heath-Brown, Primes represented by x3+2y3x^3+2y^3, Acta Math. 186 (2001), 1--84.
  37. E. Hlawka, Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafunktion zusammenhängen, Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 184 (1975), 459--471.
  38. M. N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164--170.
  39. Y. Ihara, On discrete subgroups of the two by two projective linear group over pp-adic fields, J. Math. Soc. Japan 18 (1966), 219--235.
  40. H. Iwaniec, E. Kowalski, Analytic Number Theory , AMS Colloquium Publications 53, 2004.
  41. A. E. Ingham, The Distribution of Prime Numbers , Cambridge University Press, 1932.
  42. B. Julia, Statistical theory of numbers, in Number Theory and Physics (Les Houches, 1989), Springer Proc. Phys. 47 (1990), 276--293.
  43. H. Kadiri, Une région explicite sans zéros pour la fonction $ $ de Riemann, Acta Arith. 117 (2005), 303--339.
  44. J. Kaczorowski, M. Kulas, On the non-trivial zeros off the critical line for LL-functions from the extended Selberg class, Monatsh. Math. 150 (2007), 217--232.
  45. J. C. Lagarias, On a positivity property of the Riemann $ $-function, Acta Arith. 89 (1999), 217--234.
  46. E. Landau, Über die Nullstellen der Zetafunktion, Math. Ann. 71 (1912), 548--564.
  47. X.-J. Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory 65 (1997), 325--333.
  48. V. Liu, Certified Weil positivity beyond the unit window, preprint, 2026.
  49. A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261--277.
  50. T. D. Lee, C. N. Yang, Statistical theory of equations of state and phase transitions II: lattice gas and Ising model, Phys. Rev. 87 (1952), 410--419.
  51. H. L. Montgomery, The pair correlation of zeros of the zeta function, Proc. Sympos. Pure Math. 24 (1973), 181--193.
  52. H. L. Montgomery, Zeros of approximations to the zeta function, in Studies in Pure Mathematics , Birkhäuser, Basel, 1983, 497--506.
  53. M. J. Mossinghoff, T. S. Trudgian, Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function, J. Number Theory 157 (2015), 329--349.
  54. H. L. Montgomery, R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119--134.
  55. K. Matomäki, M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. 183 (2016), 1015--1056.
  56. H. L. Montgomery, R. C. Vaughan, Mean values of multiplicative functions, Period. Math. Hungar. 43 (2001), 199--214.
  57. C. M. Newman, Fourier transforms with only real zeros, Proc. Amer. Math. Soc. 61 (1976), 245--251.
  58. N. Ng, The distribution of the summatory function of the Möbius function, Proc. London Math. Soc. (3) 89 (2004), 361--389.
  59. B. Nyman, On the one-dimensional translation group and semi-group in certain function spaces , Thesis, Uppsala, 1950.
  60. A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), 273--308.
  61. D. H. J. Polymath, Effective approximation of heat flow evolution of the Riemann $ $ function, and a new upper bound for the de Bruijn--Newman constant, Res. Math. Sci. 6 (2019), 31.
  62. G. Pólya, Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd. 7 (1927), 3--33.
  63. K. Pratt, N. Robles, A. Zaharescu, D. Zeindler, More than five-twelfths of the zeros of $ $ are on the critical line, Res. Math. Sci. 7 (2020), Paper No. 2.
  64. D. Platt, T. Trudgian, The Riemann hypothesis is true up to $3ḑot10^ 12 $, Bull. Lond. Math. Soc. 53 (2021), 792--797.
  65. M. Rosen, Number Theory in Function Fields , Graduate Texts in Mathematics 210, Springer, 2002.
  66. B. Rodgers, T. Tao, The de Bruijn--Newman constant is non-negative, Forum Math. Pi 8 (2020), e6.
  67. J. Sondow, C. Dumitrescu, A monotonicity property of Riemann's xi function and a reformulation of the Riemann hypothesis, Period. Math. Hungar. 60 (2010), 37--40.
  68. A. Selberg, Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid. 48 (1946), 89--155.
  69. A. Sheth, Euler products at the centre and applications to Chebyshev's bias, Math. Proc. Cambridge Philos. Soc. 179 (2025), 331--349.
  70. S. A. Stepanov, On the number of points of a hyperelliptic curve over a finite prime field, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 1171--1181.
  71. S. B. Stechkin, Zeros of the Riemann zeta-function, Math. Notes 8 (1970), 706--711.
  72. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , 2nd ed., Oxford University Press, 1986.
  73. P. Turán, On some approximative Dirichlet polynomials in the theory of the zeta-function of Riemann, Danske Vid. Selsk. Mat.-Fys. Medd. 24 (1948), no. 17.
  74. V. I. Vasyunin, On a biorthogonal system associated with the Riemann hypothesis, St. Petersburg Math. J. 7 (1996), 405--419.
  75. A. Weil, Sur les ``formules explicites'' de la théorie des nombres premiers, Comm. Sém. Math. Univ. Lund (1952), 252--265.
  76. H. Yoshida, On Hermitian forms attached to zeta functions, in Zeta Functions in Geometry , Adv. Stud. Pure Math. 21 (1992), 281--325.

Version history

  1. v1Submitted by Pedro CaceresInitial depositCurrentOct 08, 2026