Recent Issues
Volume 18, 1 issue
Volume 17, 5 issues
Volume 17
Issue 5, 723–899
Issue 4, 543–722
Issue 3, 363–541
Issue 2, 183–362
Issue 1, 1–182
Volume 16, 5 issues
Volume 16
Issue 5, 727–903
Issue 4, 547–726
Issue 3, 365–546
Issue 2, 183–364
Issue 1, 1–182
Volume 15, 5 issues
Volume 15
Issue 5, 727–906
Issue 4, 547–726
Issue 3, 367–546
Issue 2, 185–365
Issue 1, 1–184
Volume 14, 5 issues
Volume 14
Issue 5, 723–905
Issue 4, 541–721
Issue 3, 361–540
Issue 2, 181–360
Issue 1, 1–179
Volume 13, 5 issues
Volume 13
Issue 5, 721–900
Issue 4, 541–719
Issue 3, 361–539
Issue 2, 181–360
Issue 1, 1–180
Volume 12, 8 issues
Volume 12
Issue 8, 1261–1439
Issue 7, 1081–1260
Issue 6, 901–1080
Issue 5, 721–899
Issue 4, 541–720
Issue 3, 361–539
Issue 2, 181–360
Issue 1, 1–180
Volume 11, 5 issues
Volume 11
Issue 5, 721–900
Issue 4, 541–720
Issue 3, 361–540
Issue 2, 181–359
Issue 1, 1–179
Volume 10, 5 issues
Volume 10
Issue 5, 721–900
Issue 4, 541–720
Issue 3, 361–539
Issue 2, 181–360
Issue 1, 1–180
Volume 9, 5 issues
Volume 9
Issue 5, 721–899
Issue 4, 541–720
Issue 3, 361–540
Issue 2, 181–359
Issue 1, 1–180
Volume 8, 5 issues
Volume 8
Issue 5, 721–900
Issue 4, 541–719
Issue 3, 361–540
Issue 2, 181–359
Issue 1, 1–179
Volume 7, 6 issues
Volume 7
Issue 6, 713–822
Issue 5, 585–712
Issue 4, 431–583
Issue 3, 245–430
Issue 2, 125–244
Issue 1, 1–124
Volume 6, 4 issues
Volume 6
Issue 4, 383–510
Issue 3, 261–381
Issue 2, 127–260
Issue 1, 1–126
Volume 5, 4 issues
Volume 5
Issue 4, 379–504
Issue 3, 237–378
Issue 2, 115–236
Issue 1, 1–113
Volume 4, 4 issues
Volume 4
Issue 4, 307–416
Issue 3, 203–305
Issue 2, 103–202
Issue 1, 1–102
Volume 3, 4 issues
Volume 3
Issue 4, 349–474
Issue 3, 241–347
Issue 2, 129–240
Issue 1, 1–127
Volume 2, 5 issues
Volume 2
Issue 5, 495–628
Issue 4, 371–494
Issue 3, 249–370
Issue 2, 121–247
Issue 1, 1–120
Volume 1, 2 issues
Volume 1
Issue 2, 123–233
Issue 1, 1–121
Abstract
Ramanujan showed that
τ ( p )
≡ p 1 1
+ 1 ( mod 6 9 1 ) ,
where
τ ( n )
is the
n -th
Fourier coefficient of the unique normalized cusp form of weight
1 2 and full level, and the
prime
6 9 1 appears in the
numerator of
ζ ( 1 2 ) ∕ π 1 2 for the
Riemann zeta function
ζ ( s ) .
Searching for such congruences, it is shown that the prime
6 7 appears in the
numerator of
L ( 6 , χ ) ∕ ( π 6 5 ) ,
where
χ
is the unique nontrivial quadratic Dirichlet character modulo
5 and
L ( s , χ ) its Dirichlet
L -function, giving rise to
a congruence
f χ
≡ E 6 , χ ∘ ( mod 6 7 ) between
a cusp form
f χ and
an Eisenstein series
E 6 , χ ∘
of weight
6 on
Γ 0 ( 5 ) with nebentypus
character
χ .
Keywords
congruences, Dirichlet characters, $L$-functions
Mathematical Subject Classification
Primary: 11F33
Secondary: 11F67
Milestones
Received: 31 May 2023
Revised: 10 August 2023
Accepted: 12 August 2023
Published: 24 January 2025
Communicated by Ken Ono
© 2025 MSP (Mathematical Sciences
Publishers).