Riesz-type criteria and theta transformation analogues. (English) Zbl 1396.11105
Summary: We give character analogues of a generalization of a result due to Ramanujan, Hardy and Littlewood, and provide Riesz-type criteria for Riemann Hypotheses for the Riemann zeta function and Dirichlet \(L\)-functions. We also provide analogues of the general theta transformation formula and of recent generalizations of the transformation formulas of W. L. Ferrar and G. H. Hardy for real primitive Dirichlet characters.
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
Keywords:
Möbius function; Riemann zeta function; Dirichlet \(L\)-function; Riemann hypothesis; hypergeometric function; Ramanujan; Hardy; FerrarReferences:
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