Pima County Sheriff Says He Believes Nancy Guthrie Is Alive

As the search continues for Savannah Guthrie's kidnapped mother ... Pima County Sheriff Chris Nanos painted a cautiously hopeful picture of where the investigation is at right now.

In a new interview Wednesday, Nanos told 'TODAY' ... "We have nothing else to go on but the belief that she is here, she's present, she's alive, and we want to save her."

As you know ... detectives suspect Savannah's 84-year-old mom Nancy was taken from her Arizona home in the middle of the night against her will.

While some details have come out, none of them have led to cracking the case or even identifying any persons of interest in the Tucson-area case.

As we reported ... Nancy has a pacemaker that reportedly stopped synching with her Apple devices early Sunday, before she was reported missing. While there is video of blood on Nancy's front porch, detectives have not said if she's believed to have suffered any injuries in her abduction. DNA evidence has been collected and sent for testing.

Additionally, TMZ received an alleged ransom note demanding millions of dollars in Bitcoin in exchange for Nancy's release ... and PCSD says they are aware of the alleged note and are coordinating with the FBI.